Formula appendix (Docling inventory)

This page lists formula-related content extracted from Docling markdown in the markdown_library workspace. Rendered blocks use the same $ / $$ conventions as Math rendering rules (KaTeX on this site; equivalent LaTeX typically works in MathJax).

Pipeline status

  • Enriched Docling run: a sample PDF was converted with --enrich-formula (see markdown_library/pdfs/ and automata/docling_md_formula_runs/). Drop additional licensed PDFs into pdfs/ and run scripts/formula_extraction/run_docling_enriched.py.
  • Legacy exports: many entries are <!-- formula-not-decoded --> placeholders from earlier conversions without --enrich-formula. Those entries show context in plain text; the rendered cell uses \text{...} as a stand-in until OCR LaTeX is available.

Inventory summary

Metric Value
Generated (UTC) 2026-04-19T07:44:22.220335+00:00
Total records 983
LaTeX-like $...$ / $$...$$ in markdown 0
Formula placeholders 983
Shown below 0 LaTeX extracts + 983 placeholder contexts (capped)

Full machine-readable inventory: assets/formula_inventory.json (or the copy committed alongside this site).

Entries (collapsible)

Placeholder regions (formula-not-decoded)

1. ph-81f2c804f2fbd226c057 — agents/docling_md/Agents in Trustworthy.md ### Plain (markdown context) m .' 'Who did __?' ``` ``` (5.51) [' VP Imperative.' 'I don't want to __.'] ' Get going .' 'I don't want to __.' (5.52) ['I see .' 'No, you don't __.'] 'I see .' 'No, you don't __.' ``` Although these constructions are presented as text strings for clarity's sake, they are actually recorded and pro cessed in terms of meaning repre senta tions, which allows for both lexical variability (paraphrasing) and the potential for intervening material, as in (5.53). Another kind of VP ellipsis that is handled during Extended Semantics is what we call repetition structures ,… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{m .' 'Who did \_\_?' ``` ``` (5.51) [' VP Imperative.' 'I don't want to \_\_.'] ' Get going .' 'I don't want to \_\_.' (5.52) ['I see .' 'No, you don't \_\_.'] 'I see .' 'No, you don't \_\_.' ``` Although these constructions are presented as text strings for clarity's sake, they are actua…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ agents/docling_md/Agents in Trustworthy.md:offset=343793 \begin{verbatim} m .' 'Who did __?' ``` ``` (5.51) [' VP Imperative.' 'I don't want to __.'] ' Get going .' 'I don't want to __.' (5.52) ['I see .' 'No, you don't __.'] 'I see .' 'No, you don't __.' ``` Although these constructions are presented as text strings for clarity's sake, they are actually recorded and pro cessed in terms of meaning repre senta tions, which allows for both lexical variability (paraphrasing) and the potential for intervening material, as in (5.53). Another kind of VP ellipsis that is handled during Extended Semantics is what we call repetition structures ,… \end{verbatim} ``` </details>
2. ph-be94782a0ed3a41ed509 — agents/docling_md/Agents in Trustworthy_chapters/Agents in Trustworthy_chapter_2.5.md ### Plain (markdown context) m .' 'Who did __?' ``` ``` (5.51) [' VP Imperative.' 'I don't want to __.'] ' Get going .' 'I don't want to __.' (5.52) ['I see .' 'No, you don't __.'] 'I see .' 'No, you don't __.' ``` Although these constructions are presented as text strings for clarity's sake, they are actually recorded and pro cessed in terms of meaning repre senta tions, which allows for both lexical variability (paraphrasing) and the potential for intervening material, as in (5.53). Another kind of VP ellipsis that is handled during Extended Semantics is what we call repetition structures ,… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{m .' 'Who did \_\_?' ``` ``` (5.51) [' VP Imperative.' 'I don't want to \_\_.'] ' Get going .' 'I don't want to \_\_.' (5.52) ['I see .' 'No, you don't \_\_.'] 'I see .' 'No, you don't \_\_.' ``` Although these constructions are presented as text strings for clarity's sake, they are actua…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ agents/docling_md/Agents in Trustworthy_chapters/Agents in Trustworthy_chapter_2.5.md:offset=14695 \begin{verbatim} m .' 'Who did __?' ``` ``` (5.51) [' VP Imperative.' 'I don't want to __.'] ' Get going .' 'I don't want to __.' (5.52) ['I see .' 'No, you don't __.'] 'I see .' 'No, you don't __.' ``` Although these constructions are presented as text strings for clarity's sake, they are actually recorded and pro cessed in terms of meaning repre senta tions, which allows for both lexical variability (paraphrasing) and the potential for intervening material, as in (5.53). Another kind of VP ellipsis that is handled during Extended Semantics is what we call repetition structures ,… \end{verbatim} ``` </details>
3. ph-cc2293253ed9e4052084 — agents/docling_md/PromptEngineeringForDeveloper.md ### Plain (markdown context) stant", "content": "When Harry was 25 yea \ rs old, his sister was 29 years old. There's 4 year differences between them and the sister is older than Harry. When Harry is 30 years old, h is sister should be 34 years old."}, {"role": "user", "content": f" { message } "}, ], temperature=1, max_tokens=256, top_p=1, frequency_penalty=0, presence_penalty=0 ) try : print("Felix:", response.choices[0].message.content) except : print("Felix: Sorry, a problem occurred. Please try again l \ ``` ater.") ## When asked the following question, the answer was correct: Felix : Hi… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{stant", "content": "When Harry was 25 yea \textbackslash rs old, his sister was 29 years old. There's 4 year differences between them and the sister is older than Harry. When Harry is 30 years old, h is sister should be 34 years old."\}, \{"role": "user", "content": f" \{ message \} "\}, ], tempe…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ agents/docling_md/PromptEngineeringForDeveloper.md:offset=121499 \begin{verbatim} stant", "content": "When Harry was 25 yea \ rs old, his sister was 29 years old. There's 4 year differences between them and the sister is older than Harry. When Harry is 30 years old, h is sister should be 34 years old."}, {"role": "user", "content": f" { message } "}, ], temperature=1, max_tokens=256, top_p=1, frequency_penalty=0, presence_penalty=0 ) try : print("Felix:", response.choices[0].message.content) except : print("Felix: Sorry, a problem occurred. Please try again l \ ``` ater.") ## When asked the following question, the answer was correct: Felix : Hi… \end{verbatim} ``` </details>
4. ph-72962440b734fbdfbbc9 — agents/docling_md/PromptEngineeringForDeveloper.md ### Plain (markdown context) 's not very 'surprised' by the actual next word because it expected it. Conversely, if it struggles to predict the next word, its perplexity is high - it's more 'surprised' by the actual word that comes next. Think of this as giving someone a task in a language they are fluent in versus a language they've just started learning. Naturally, they'd perform better with the familiar language. ## 12.2 - How to Calculate Perplexity? In general, perplexity is calculated using the following formula: Where: - s is the sample - wx is a word from the sample - N is the number … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{'s not very 'surprised' by the actual next word because it expected it. Conversely, if it struggles to predict the next word, its perplexity is high - it's more 'surprised' by the actual word that comes next. Think of this as giving someone a task in a language they are fluent i…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ agents/docling_md/PromptEngineeringForDeveloper.md:offset=169510 \begin{verbatim} 's not very 'surprised' by the actual next word because it expected it. Conversely, if it struggles to predict the next word, its perplexity is high - it's more 'surprised' by the actual word that comes next. Think of this as giving someone a task in a language they are fluent in versus a language they've just started learning. Naturally, they'd perform better with the familiar language. ## 12.2 - How to Calculate Perplexity? In general, perplexity is calculated using the following formula: Where: - s is the sample - wx is a word from the sample - N is the number … \end{verbatim} ``` </details>
5. ph-258a141e459e78681212 — agents/docling_md/PromptEngineeringForDeveloper_chapters/PromptEngineeringForDeveloper_chapter_1.6.md ### Plain (markdown context) stant", "content": "When Harry was 25 yea \ rs old, his sister was 29 years old. There's 4 year differences between them and the sister is older than Harry. When Harry is 30 years old, h is sister should be 34 years old."}, {"role": "user", "content": f" { message } "}, ], temperature=1, max_tokens=256, top_p=1, frequency_penalty=0, presence_penalty=0 ) try : print("Felix:", response.choices[0].message.content) except : print("Felix: Sorry, a problem occurred. Please try again l \ ``` ater.") ## When asked the following question, the answer was correct: Felix : Hi… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{stant", "content": "When Harry was 25 yea \textbackslash rs old, his sister was 29 years old. There's 4 year differences between them and the sister is older than Harry. When Harry is 30 years old, h is sister should be 34 years old."\}, \{"role": "user", "content": f" \{ message \} "\}, ], tempe…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ agents/docling_md/PromptEngineeringForDeveloper_chapters/PromptEngineeringForDeveloper_chapter_1.6.md:offset=3150 \begin{verbatim} stant", "content": "When Harry was 25 yea \ rs old, his sister was 29 years old. There's 4 year differences between them and the sister is older than Harry. When Harry is 30 years old, h is sister should be 34 years old."}, {"role": "user", "content": f" { message } "}, ], temperature=1, max_tokens=256, top_p=1, frequency_penalty=0, presence_penalty=0 ) try : print("Felix:", response.choices[0].message.content) except : print("Felix: Sorry, a problem occurred. Please try again l \ ``` ater.") ## When asked the following question, the answer was correct: Felix : Hi… \end{verbatim} ``` </details>
6. ph-86eefa320d188eb79d8c — agents/docling_md/PromptEngineeringForDeveloper_chapters/PromptEngineeringForDeveloper_chapter_1.6.md ### Plain (markdown context) 's not very 'surprised' by the actual next word because it expected it. Conversely, if it struggles to predict the next word, its perplexity is high - it's more 'surprised' by the actual word that comes next. Think of this as giving someone a task in a language they are fluent in versus a language they've just started learning. Naturally, they'd perform better with the familiar language. ## 12.2 - How to Calculate Perplexity? In general, perplexity is calculated using the following formula: Where: - s is the sample - wx is a word from the sample - N is the number … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{'s not very 'surprised' by the actual next word because it expected it. Conversely, if it struggles to predict the next word, its perplexity is high - it's more 'surprised' by the actual word that comes next. Think of this as giving someone a task in a language they are fluent i…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ agents/docling_md/PromptEngineeringForDeveloper_chapters/PromptEngineeringForDeveloper_chapter_1.6.md:offset=51161 \begin{verbatim} 's not very 'surprised' by the actual next word because it expected it. Conversely, if it struggles to predict the next word, its perplexity is high - it's more 'surprised' by the actual word that comes next. Think of this as giving someone a task in a language they are fluent in versus a language they've just started learning. Naturally, they'd perform better with the familiar language. ## 12.2 - How to Calculate Perplexity? In general, perplexity is calculated using the following formula: Where: - s is the sample - wx is a word from the sample - N is the number … \end{verbatim} ``` </details>
7. ph-4f053d7443ccb1797caa — automata/docling_md/AWK-implementation.md ### Plain (markdown context) he nodes with 2 or more possible next nodes will be of type 2. This can lead to significant memory waste for a node. For example, if a node has only six possible next nodes, the type 2 node will have 250 NULL pointers in its array. In this work, we compare different trie configurations using (1), where P is the performance, C is the number of clock cycles per character and M is the overall memory used. This relation allows to find the best compromise between C and M . ## 3. Methods To compare various trie configurations, we used a virtual machine with Xubuntu 14.0… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{he nodes with 2 or more possible next nodes will be of type 2. This can lead to significant memory waste for a node. For example, if a node has only six possible next nodes, the type 2 node will have 250 NULL pointers in its array. In this work, we compare different trie configu…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AWK-implementation.md:offset=8109 \begin{verbatim} he nodes with 2 or more possible next nodes will be of type 2. This can lead to significant memory waste for a node. For example, if a node has only six possible next nodes, the type 2 node will have 250 NULL pointers in its array. In this work, we compare different trie configurations using (1), where P is the performance, C is the number of clock cycles per character and M is the overall memory used. This relation allows to find the best compromise between C and M . ## 3. Methods To compare various trie configurations, we used a virtual machine with Xubuntu 14.0… \end{verbatim} ``` </details>
8. ph-f8231af044f65f20f782 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) s is the set of all strings of symbols in a given alphabet. Definition 1.8 Let A be a set. A ′ = U -A is the set of all elements not in A. Example 1.3 Let A be the set of even integers and U be the set of integers. Then A ′ is the set of odd integers. Example 1.4 Let A = { x : x collects coins } , then A ′ = { x : x does not collect coins } . The proof of the following theorem is left to the reader. Theorem 1.1 Let A, B, and C be subsets of the universal set U (a) Distributive properties - (b) Idempotent properties - (d) De Morgan's la… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{s is the set of all strings of symbols in a given alphabet. Definition 1.8 Let A be a set. A ′ = U -A is the set of all elements not in A. Example 1.3 Let A be the set of even integers and U be the set of integers. Then A ′ is the set of odd integers. Example 1.4 Let A = \{ x : x…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=19051 \begin{verbatim} s is the set of all strings of symbols in a given alphabet. Definition 1.8 Let A be a set. A ′ = U -A is the set of all elements not in A. Example 1.3 Let A be the set of even integers and U be the set of integers. Then A ′ is the set of odd integers. Example 1.4 Let A = { x : x collects coins } , then A ′ = { x : x does not collect coins } . The proof of the following theorem is left to the reader. Theorem 1.1 Let A, B, and C be subsets of the universal set U (a) Distributive properties - (b) Idempotent properties - (d) De Morgan's la… \end{verbatim} ``` </details>
9. ph-dee9d9d636688b57be58 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) symbols in a given alphabet. Definition 1.8 Let A be a set. A ′ = U -A is the set of all elements not in A. Example 1.3 Let A be the set of even integers and U be the set of integers. Then A ′ is the set of odd integers. Example 1.4 Let A = { x : x collects coins } , then A ′ = { x : x does not collect coins } . The proof of the following theorem is left to the reader. Theorem 1.1 Let A, B, and C be subsets of the universal set U (a) Distributive properties - (b) Idempotent properties - (d) De Morgan's laws - (e) Commutative properties… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{symbols in a given alphabet. Definition 1.8 Let A be a set. A ′ = U -A is the set of all elements not in A. Example 1.3 Let A be the set of even integers and U be the set of integers. Then A ′ is the set of odd integers. Example 1.4 Let A = \{ x : x collects coins \} , then A ′ = …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=19081 \begin{verbatim} symbols in a given alphabet. Definition 1.8 Let A be a set. A ′ = U -A is the set of all elements not in A. Example 1.3 Let A be the set of even integers and U be the set of integers. Then A ′ is the set of odd integers. Example 1.4 Let A = { x : x collects coins } , then A ′ = { x : x does not collect coins } . The proof of the following theorem is left to the reader. Theorem 1.1 Let A, B, and C be subsets of the universal set U (a) Distributive properties - (b) Idempotent properties - (d) De Morgan's laws - (e) Commutative properties… \end{verbatim} ``` </details>
10. ph-d7f4682645e561714633 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) Example 1.3 Let A be the set of even integers and U be the set of integers. Then A ′ is the set of odd integers. Example 1.4 Let A = { x : x collects coins } , then A ′ = { x : x does not collect coins } . The proof of the following theorem is left to the reader. Theorem 1.1 Let A, B, and C be subsets of the universal set U (a) Distributive properties - (b) Idempotent properties - (d) De Morgan's laws - (e) Commutative properties - (f) Associative laws - (g) Identity properties … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{Example 1.3 Let A be the set of even integers and U be the set of integers. Then A ′ is the set of odd integers. Example 1.4 Let A = \{ x : x collects coins \} , then A ′ = \{ x : x does not collect coins \} . The proof of the following theorem is left to the reader. Theorem 1.1 Let…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=19192 \begin{verbatim} Example 1.3 Let A be the set of even integers and U be the set of integers. Then A ′ is the set of odd integers. Example 1.4 Let A = { x : x collects coins } , then A ′ = { x : x does not collect coins } . The proof of the following theorem is left to the reader. Theorem 1.1 Let A, B, and C be subsets of the universal set U (a) Distributive properties - (b) Idempotent properties - (d) De Morgan's laws - (e) Commutative properties - (f) Associative laws - (g) Identity properties … \end{verbatim} ``` </details>
11. ph-ef5bd99d8def10e5468a — automata/docling_md/AutomataTheory.md ### Plain (markdown context) the set of integers. Then A ′ is the set of odd integers. Example 1.4 Let A = { x : x collects coins } , then A ′ = { x : x does not collect coins } . The proof of the following theorem is left to the reader. Theorem 1.1 Let A, B, and C be subsets of the universal set U (a) Distributive properties - (b) Idempotent properties - (d) De Morgan's laws - (e) Commutative properties - (f) Associative laws - (g) Identity properties - (h) Complement propertie… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{the set of integers. Then A ′ is the set of odd integers. Example 1.4 Let A = \{ x : x collects coins \} , then A ′ = \{ x : x does not collect coins \} . The proof of the following theorem is left to the reader. Theorem 1.1 Let A, B, and C be subsets of the universal set U (a) Dist…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=19246 \begin{verbatim} the set of integers. Then A ′ is the set of odd integers. Example 1.4 Let A = { x : x collects coins } , then A ′ = { x : x does not collect coins } . The proof of the following theorem is left to the reader. Theorem 1.1 Let A, B, and C be subsets of the universal set U (a) Distributive properties - (b) Idempotent properties - (d) De Morgan's laws - (e) Commutative properties - (f) Associative laws - (g) Identity properties - (h) Complement propertie… \end{verbatim} ``` </details>
12. ph-63a34ca549693e0b0322 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) . Example 1.4 Let A = { x : x collects coins } , then A ′ = { x : x does not collect coins } . The proof of the following theorem is left to the reader. Theorem 1.1 Let A, B, and C be subsets of the universal set U (a) Distributive properties - (b) Idempotent properties - (d) De Morgan's laws - (e) Commutative properties - (f) Associative laws - (g) Identity properties - (h) Complement properties - (c) Double Complement p… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{. Example 1.4 Let A = \{ x : x collects coins \} , then A ′ = \{ x : x does not collect coins \} . The proof of the following theorem is left to the reader. Theorem 1.1 Let A, B, and C be subsets of the universal set U (a) Distributive properties - (b) Idempotent properties - (d) De Morgan's laws - (e) Commutative properties - (f) Associative laws - (g) Identity properties - (h) Complement properties - (c) Double Complement p… \end{verbatim} ```
13. ph-87de056ad9c386a6bf33 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) = { x : x does not collect coins } . The proof of the following theorem is left to the reader. Theorem 1.1 Let A, B, and C be subsets of the universal set U (a) Distributive properties - (b) Idempotent properties - (d) De Morgan's laws - (e) Commutative properties - (f) Associative laws - (g) Identity properties - (h) Complement properties - (c) Double Complement property - (b) Idempotent properties - (d) De…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=19362 \begin{verbatim} = { x : x does not collect coins } . The proof of the following theorem is left to the reader. Theorem 1.1 Let A, B, and C be subsets of the universal set U (a) Distributive properties - (b) Idempotent properties - (d) De Morgan's laws - (e) Commutative properties - (f) Associative laws - (g) Identity properties - (h) Complement properties - (c) Double Complement property - (b) Idempotent properties - (d) De Morgan's laws - (e) Commutative properties - (f) Associative laws - (g) Identity properties - (h) Complement properties - (c) Double Complement property Definition 1.9 The size or … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{theorem is left to the reader. Theorem 1.1 Let A, B, and C be subsets of the universal set U (a) Distributive properties - (b) Idempotent properties - (d) De Morgan's laws - (e) Commutative properties - (b) Idempotent properties - (d) De Morgan's laws - (e) Commutative properties - (f) Associative laws - (g) Identity properties - (h) Complement properties - (c) Double Complement property Definition 1.9 The size or … \end{verbatim} ``` </details>
15. ph-475b215467922c1908a6 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) . Theorem 1.1 Let A, B, and C be subsets of the universal set U (a) Distributive properties - (b) Idempotent properties - (d) De Morgan's laws - (e) Commutative properties - (f) Associative laws - (g) Identity properties - (h) Complement properties - (c) Double Complement property Definition 1.9 The size or cardinality of a finite set A… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{. Theorem 1.1 Let A, B, and C be subsets of the universal set U (a) Distributive properties - (b) Idempotent properties - (d) De Morgan's laws - (e) Commutative properties - (f) Associative la…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=19456 \begin{verbatim} . Theorem 1.1 Let A, B, and C be subsets of the universal set U (a) Distributive properties - (b) Idempotent properties - (d) De Morgan's laws - (e) Commutative properties - (f) Associative laws - (g) Identity properties - (h) Complement properties - (c) Double Complement property Definition 1.9 The size or cardinality of a finite set A… \end{verbatim} ``` </details>
16. ph-3aa788208ba01c01ea81 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) be subsets of the universal set U (a) Distributive properties - (b) Idempotent properties - (d) De Morgan's laws - (e) Commutative properties - (f) Associative laws - (g) Identity properties - (h) Complement properties - (c) Double Complement property Definition 1.9 The size or cardinality of a finite set A, denoted by | A | , is the nu… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{be subsets of the universal set U (a) Distributive properties - (b) Idempotent properties - (d) De Morgan's laws - (e) Commutative properties - (f) Associative laws - (b) Idempotent properties - (d) De Morgan's laws - (e) Commutative properties - (f) Associative laws - (g) Identity properties - (h) Complement properties - (c) Double Complement property Definition 1.9 The size or cardinality of a finite set A, denoted by | A | , is the nu… \end{verbatim} ```
17. ph-d74d98f681db563c7baf — automata/docling_md/AutomataTheory.md ### Plain (markdown context) countable set. We see that there are two infinite sets, the countable sets and the uncountable sets with different cardinality; however, we shall soon see that there are an infinite number of infinite sets of different cardinality. Further discussion of cardinality will be continued in the appendices. Definition 1.10 Let A and B be sets. The Cartesian product of A and B, denoted by A × B is the set { ( a , b ) : a ∈ A and b ∈ B } . For example, let A = { a , b } and B = { 1 , 2 , 3 } , then ThefamiliarCartesianplane R × R is the set of all ordered pairs of real nu… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{countable set. We see that there are two infinite sets, the countable sets and the uncountable sets with different cardinality; however, we shall soon see that there are an infinite number of infinite sets of different cardinality. Further discussion of cardinality will be conti…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=21143 \begin{verbatim} countable set. We see that there are two infinite sets, the countable sets and the uncountable sets with different cardinality; however, we shall soon see that there are an infinite number of infinite sets of different cardinality. Further discussion of cardinality will be continued in the appendices. Definition 1.10 Let A and B be sets. The Cartesian product of A and B, denoted by A × B is the set { ( a , b ) : a ∈ A and b ∈ B } . For example, let A = { a , b } and B = { 1 , 2 , 3 } , then ThefamiliarCartesianplane R × R is the set of all ordered pairs of real nu… \end{verbatim} ``` </details>
18. ph-f9dc26fbcc1aa4952716 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) nition 1.10 Let A and B be sets. The Cartesian product of A and B, denoted by A × B is the set { ( a , b ) : a ∈ A and b ∈ B } . For example, let A = { a , b } and B = { 1 , 2 , 3 } , then ThefamiliarCartesianplane R × R is the set of all ordered pairs of real numbers. Note that for finite sets | A × B | = | A | × | B | . Definition 1.11 The power set of a set A, denoted by P ( A ) , is the set of all subsets of A. For example the power set of { a , b , c } is In the finite case, it can be easily shown that | P ( A ) | = 2 | A | . ## E… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{nition 1.10 Let A and B be sets. The Cartesian product of A and B, denoted by A × B is the set \{ ( a , b ) : a ∈ A and b ∈ B \} . For example, let A = \{ a , b \} and B = \{ 1 , 2 , 3 \} , then ThefamiliarCartesianplane R × R is the set of all ordered pai…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=21452 \begin{verbatim} nition 1.10 Let A and B be sets. The Cartesian product of A and B, denoted by A × B is the set { ( a , b ) : a ∈ A and b ∈ B } . For example, let A = { a , b } and B = { 1 , 2 , 3 } , then ThefamiliarCartesianplane R × R is the set of all ordered pairs of real numbers. Note that for finite sets | A × B | = | A | × | B | . Definition 1.11 The power set of a set A, denoted by P ( A ) , is the set of all subsets of A. For example the power set of { a , b , c } is In the finite case, it can be easily shown that | P ( A ) | = 2 | A | . ## E… \end{verbatim} ``` </details>
19. ph-b7975e77e998f69a5110 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) , b , c } is In the finite case, it can be easily shown that | P ( A ) | = 2 | A | . ## Exercises - (1) State which of the following are true and which are false: 2. (a) {∅} ⊆ A for an arbitrary set A 3. (c) { a , b , c } ⊆ { a , b , { a , b , c }} . 4. (b) ∅ ⊆ A for an arbitrary set A . 5. (d) { a , b , c } ∈ { a , b , { a , b , c }} . - (2) Prove Theorem 1.1. Let A , B , and C be subsets of the universal set U . 7. (e) A ∈ P ( A ). 8. (a) Idempotent property - (b) Double Complement property In the finite case, it can be easily shown that | P ( A ) | = 2 | A | . \#\# Exercises - (1) State which of the following are true and which are false: 2. (a) \{∅\} ⊆ A for an arbitrary set A 3. (c) \{ a , b , c \} ⊆ \{ a , b , \{ a , b , c \}\} .…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=21937 \begin{verbatim} , b , c } is In the finite case, it can be easily shown that | P ( A ) | = 2 | A | . ## Exercises - (1) State which of the following are true and which are false: 2. (a) {∅} ⊆ A for an arbitrary set A 3. (c) { a , b , c } ⊆ { a , b , { a , b , c }} . 4. (b) ∅ ⊆ A for an arbitrary set A . 5. (d) { a , b , c } ∈ { a , b , { a , b , c }} . - (2) Prove Theorem 1.1. Let A , B , and C be subsets of the universal set U . 7. (e) A ∈ P ( A ). 8. (a) Idempotent property - (b) Double Complement property - (b) Double Complement property - . - (c) De Morgan's laws ## (d) Commutative p… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{it can be easily shown that | P ( A ) | = 2 | A | . \#\# Exercises - (1) State which of the following are true and which are false: 2. (a) \{∅\} ⊆ A for an arbitrary set A 3. (c) \{ a , b , c \} ⊆ \{ a , b , \{ a , b , c \}\} . 4. (b) ∅ ⊆ A for an arbitrary set A . 5. (d) \{ a , b , c \} ∈ …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=22001 \begin{verbatim} it can be easily shown that | P ( A ) | = 2 | A | . ## Exercises - (1) State which of the following are true and which are false: 2. (a) {∅} ⊆ A for an arbitrary set A 3. (c) { a , b , c } ⊆ { a , b , { a , b , c }} . 4. (b) ∅ ⊆ A for an arbitrary set A . 5. (d) { a , b , c } ∈ { a , b , { a , b , c }} . - (2) Prove Theorem 1.1. Let A , B , and C be subsets of the universal set U . 7. (e) A ∈ P ( A ). 8. (a) Idempotent property - (b) Double Complement property - . - (c) De Morgan's laws ## (d) Commutative p… \end{verbatim} ``` </details>
21. ph-d567ed1997f25b39d430 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) P ( A ) | = 2 | A | . ## Exercises - (1) State which of the following are true and which are false: 2. (a) {∅} ⊆ A for an arbitrary set A 3. (c) { a , b , c } ⊆ { a , b , { a , b , c }} . 4. (b) ∅ ⊆ A for an arbitrary set A . 5. (d) { a , b , c } ∈ { a , b , { a , b , c }} . - (2) Prove Theorem 1.1. Let A , B , and C be subsets of the universal set U . 7. (e) A ∈ P ( A ). 8. (a) Idempotent property - (b) Double Complement property - . - (c) De Morgan's laws ## (d) Commutative properties - (b) Double Complement property - . - (c) De Morgan's laws ## (d) Commutative properties - (b) Double Complement property - . - (c) De Morgan's laws ## (d) Commutative properties - (e) Associative properties - (f) Distributive … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ch are false: 2. (a) \{∅\} ⊆ A for an arbitrary set A 3. (c) \{ a , b , c \} ⊆ \{ a , b , \{ a , b , c \}\} . 4. (b) ∅ ⊆ A for an arbitrary set A . 5. (d) \{ a , b , c \} ∈ \{ a , b , \{ a , b , c \}\} . - (2) Prove Theorem 1.1. Let A , B , and C be subsets of the universal set U . 7. (e) A ∈…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=22120 \begin{verbatim} ch are false: 2. (a) {∅} ⊆ A for an arbitrary set A 3. (c) { a , b , c } ⊆ { a , b , { a , b , c }} . 4. (b) ∅ ⊆ A for an arbitrary set A . 5. (d) { a , b , c } ∈ { a , b , { a , b , c }} . - (2) Prove Theorem 1.1. Let A , B , and C be subsets of the universal set U . 7. (e) A ∈ P ( A ). 8. (a) Idempotent property - (b) Double Complement property - . - (c) De Morgan's laws ## (d) Commutative properties - (e) Associative properties - (f) Distributive … \end{verbatim} ``` </details>
23. ph-6e4246e008ba8f3e3c70 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) a , b , c } ⊆ { a , b , { a , b , c }} . 4. (b) ∅ ⊆ A for an arbitrary set A . 5. (d) { a , b , c } ∈ { a , b , { a , b , c }} . - (2) Prove Theorem 1.1. Let A , B , and C be subsets of the universal set U . 7. (e) A ∈ P ( A ). 8. (a) Idempotent property - (b) Double Complement property - . - (c) De Morgan's laws ## (d) Commutative properties - (e) Associative properties - (f) Distributive properties - (b) Double Complement property - . - (c) De Morgan's laws ## (d) Commutative properties - (e) Associative properties - (f) Distributive properties - (b) Double Complement property - . - (c) De Morgan's laws ## (d) Commutative properties - (e) Associative properties - (f) Distributive properties - (g) Identity properties ## (h) Complement propertie… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{arbitrary set A . 5. (d) \{ a , b , c \} ∈ \{ a , b , \{ a , b , c \}\} . - (2) Prove Theorem 1.1. Let A , B , and C be subsets of the universal set U . 7. (e) A ∈ P ( A ). 8. (a) Idempotent property - (b) Double Complement property - (b) Double Complement property - . - (c) De Morgan's laws ## (d) Commutative properties - (e) Associative properties - (f) Distributive properties - (g) Identity properties ## (h) Complement propertie… \end{verbatim} ```
25. ph-19abd2424b1478deb377 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) , b , c } ∈ { a , b , { a , b , c }} . - (2) Prove Theorem 1.1. Let A , B , and C be subsets of the universal set U . 7. (e) A ∈ P ( A ). 8. (a) Idempotent property - (b) Double Complement property - . - (c) De Morgan's laws ## (d) Commutative properties - (e) Associative properties - (f) Distributive properties - (g) Identity properties ## (h) Complement properties - (4) If A B , what is A /D… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{, b , c \} ∈ \{ a , b , \{ a , b , c \}\} . - (2) Prove Theorem 1.1. Let A , B , and C be subsets of the universal set U . 7. (e) A ∈ P ( A ). 8. (a) Idempotent property - (b) Double Complement property - (b) Double Complement property - . - (c) De Morgan's laws ## (d) Commutative properties - (e) Associative properties - (f) Distributive properties - (g) Identity properties ## (h) Complement properties - (4) If A B , what is A /D… \end{verbatim} ```
26. ph-d6955887337419bc6030 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) A ∈ P ( A ). 8. (a) Idempotent property - (b) Double Complement property - . - (c) De Morgan's laws ## (d) Commutative properties - (e) Associative properties - (f) Distributive properties - (g) Identity properties ## (h) Complement properties - (4) If A B , what is A /Delta1 B ? - (3) Given a set A ∈ P ( C ), find a set B such that A /… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{A ∈ P ( A ). 8. (a) Idempotent property - (b) Double Complement property - . - (c) De Morgan's laws \#\# (d) Commutative properties - (e) Associative properties - (b) Double Complement property - . - (c) De Morgan's laws ## (d) Commutative properties - (e) Associative properties - (f) Distributive properties - (g) Identity properties ## (h) Complement properties - (4) If A B , what is A /Delta1 B ? - (3) Given a set A ∈ P ( C ), find a set B such that A /… \end{verbatim} ```
27. ph-04619380f9ea62386d0f — automata/docling_md/AutomataTheory.md ### Plain (markdown context) property - (b) Double Complement property - . - (c) De Morgan's laws ## (d) Commutative properties - (e) Associative properties - (f) Distributive properties - (g) Identity properties ## (h) Complement properties - (4) If A B , what is A /Delta1 B ? - (3) Given a set A ∈ P ( C ), find a set B such that A /Delta1 B = ∅ . - (5) Using the … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{property - (b) Double Complement property - . - (c) De Morgan's laws \#\# (d) Commutative properties - (e) Associative properties - (f) …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=22426 \begin{verbatim} property - (b) Double Complement property - . - (c) De Morgan's laws ## (d) Commutative properties - (e) Associative properties - (f) Distributive properties - (g) Identity properties ## (h) Complement properties - (4) If A B , what is A /Delta1 B ? - (3) Given a set A ∈ P ( C ), find a set B such that A /Delta1 B = ∅ . - (5) Using the … \end{verbatim} ``` </details>
28. ph-c7b3d9fda01e970fb153 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) -decoded --> - (3) Given a set A ∈ P ( C ), find a set B such that A /Delta1 B = ∅ . - (5) Using the properties in Theorem 1.1 prove that A ∩ ( B /Delta1 C ) = ( A ∩ B ) /Delta1 ( A ∩ C ) . 3. ⊆ - (6) Use induction to prove that for any finite set A , | A | < | P ( A ) | . 5. (8) 6. Prove using the properties in Theorem 1.1 - (7) (Russell's Paradox) Let S be the set of all sets. Then S ∈ S . Obviously ∅ / ∈ ∅ . Let W = { A : A / ∈ A } . Discuss whether W ∈ W . - (9) Use the fact that A ∩ ( A ∪ B ) = A to prove that A ∪ ( A ∩ B ) = A… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{-decoded --> - (3) Given a set A ∈ P ( C ), find a set B such that A /Delta1 B = ∅ . - (5) Using the properties in Theorem 1.1 prove that A ∩ ( B /Delta1 C ) = ( A ∩ B ) /Delta1 ( A ∩ C ) . 3. ⊆ - (6) Use induction to prove that for any finite set A …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=22912 \begin{verbatim} -decoded --> - (3) Given a set A ∈ P ( C ), find a set B such that A /Delta1 B = ∅ . - (5) Using the properties in Theorem 1.1 prove that A ∩ ( B /Delta1 C ) = ( A ∩ B ) /Delta1 ( A ∩ C ) . 3. ⊆ - (6) Use induction to prove that for any finite set A , | A | < | P ( A ) | . 5. (8) 6. Prove using the properties in Theorem 1.1 - (7) (Russell's Paradox) Let S be the set of all sets. Then S ∈ S . Obviously ∅ / ∈ ∅ . Let W = { A : A / ∈ A } . Discuss whether W ∈ W . - (9) Use the fact that A ∩ ( A ∪ B ) = A to prove that A ∪ ( A ∩ B ) = A… \end{verbatim} ``` </details>
29. ph-9bc5aaf6a7e525d7b926 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) B = ∅ . - (5) Using the properties in Theorem 1.1 prove that A ∩ ( B /Delta1 C ) = ( A ∩ B ) /Delta1 ( A ∩ C ) . 3. ⊆ - (6) Use induction to prove that for any finite set A , | A | < | P ( A ) | . 5. (8) 6. Prove using the properties in Theorem 1.1 - (7) (Russell's Paradox) Let S be the set of all sets. Then S ∈ S . Obviously ∅ / ∈ ∅ . Let W = { A : A / ∈ A } . Discuss whether W ∈ W . - (9) Use the fact that A ∩ ( A ∪ B ) = A to prove that A ∪ ( A ∩ B ) = A . - (10) Prove that if two disjoint sets are countable, then their union is … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{B = ∅ . - (5) Using the properties in Theorem 1.1 prove that A ∩ ( B /Delta1 C ) = ( A ∩ B ) /Delta1 ( A ∩ C ) . 3. ⊆ - (6) Use induction to prove that for any finite set A , | A | \< | P ( A ) | . 5. (8) 6. Prove using the properties in Theorem 1.1 - (7) (Russell's Paradox) L…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=23019 \begin{verbatim} B = ∅ . - (5) Using the properties in Theorem 1.1 prove that A ∩ ( B /Delta1 C ) = ( A ∩ B ) /Delta1 ( A ∩ C ) . 3. ⊆ - (6) Use induction to prove that for any finite set A , | A | < | P ( A ) | . 5. (8) 6. Prove using the properties in Theorem 1.1 - (7) (Russell's Paradox) Let S be the set of all sets. Then S ∈ S . Obviously ∅ / ∈ ∅ . Let W = { A : A / ∈ A } . Discuss whether W ∈ W . - (9) Use the fact that A ∩ ( A ∪ B ) = A to prove that A ∪ ( A ∩ B ) = A . - (10) Prove that if two disjoint sets are countable, then their union is … \end{verbatim} ``` </details>
30. ph-4275a05a48c6d144dd9e — automata/docling_md/AutomataTheory.md ### Plain (markdown context) S . Obviously ∅ / ∈ ∅ . Let W = { A : A / ∈ A } . Discuss whether W ∈ W . - (9) Use the fact that A ∩ ( A ∪ B ) = A to prove that A ∪ ( A ∩ B ) = A . - (10) Prove that if two disjoint sets are countable, then their union is countable. ## 1.2 Relations Definition 1.12 Given sets A and B, any subset R of A × B is a relation between A and B. If ( a , b ) ∈ R , this is often denoted by a R b. If A = B, R is said to be a relation on A. Note that relations need not have any particular property nor even be describ… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{S . Obviously ∅ / ∈ ∅ . Let W = \{ A : A / ∈ A \} . Discuss whether W ∈ W . - (9) Use the fact that A ∩ ( A ∪ B ) = A to prove that A ∪ ( A ∩ B ) = A . - (10) Prove that if two disjoint sets are countable, then their union …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=23337 \begin{verbatim} S . Obviously ∅ / ∈ ∅ . Let W = { A : A / ∈ A } . Discuss whether W ∈ W . - (9) Use the fact that A ∩ ( A ∪ B ) = A to prove that A ∪ ( A ∩ B ) = A . - (10) Prove that if two disjoint sets are countable, then their union is countable. ## 1.2 Relations Definition 1.12 Given sets A and B, any subset R of A × B is a relation between A and B. If ( a , b ) ∈ R , this is often denoted by a R b. If A = B, R is said to be a relation on A. Note that relations need not have any particular property nor even be describ… \end{verbatim} ``` </details>
31. ph-74809c467ed3f44bc582 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) sets are countable, then their union is countable. ## 1.2 Relations Definition 1.12 Given sets A and B, any subset R of A × B is a relation between A and B. If ( a , b ) ∈ R , this is often denoted by a R b. If A = B, R is said to be a relation on A. Note that relations need not have any particular property nor even be describable. Obviously we will be interested in those relations which are describable and have particular properties which will be shown later. is a relation between A and B . Example 1.7 If … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{sets are countable, then their union is countable. \#\# 1.2 Relations Definition 1.12 Given sets A and B, any subset R of A × B is a relation between A and B. If ( a , b ) ∈ R , this is often denoted by a R b. If A = B, R is said to be a relation on A. …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=23582 \begin{verbatim} sets are countable, then their union is countable. ## 1.2 Relations Definition 1.12 Given sets A and B, any subset R of A × B is a relation between A and B. If ( a , b ) ∈ R , this is often denoted by a R b. If A = B, R is said to be a relation on A. Note that relations need not have any particular property nor even be describable. Obviously we will be interested in those relations which are describable and have particular properties which will be shown later. is a relation between A and B . Example 1.7 If … \end{verbatim} ``` </details>
32. ph-94aba14c4794010c12c6 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) ations Definition 1.12 Given sets A and B, any subset R of A × B is a relation between A and B. If ( a , b ) ∈ R , this is often denoted by a R b. If A = B, R is said to be a relation on A. Note that relations need not have any particular property nor even be describable. Obviously we will be interested in those relations which are describable and have particular properties which will be shown later. is a relation between A and B . Example 1.7 If A is the set of people, then a R b if a and b are cousins is … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ations Definition 1.12 Given sets A and B, any subset R of A × B is a relation between A and B. If ( a , b ) ∈ R , this is often denoted by a R b. If A = B, R is said to be a relation on A. Note that relations need not have any particular property no…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=23645 \begin{verbatim} ations Definition 1.12 Given sets A and B, any subset R of A × B is a relation between A and B. If ( a , b ) ∈ R , this is often denoted by a R b. If A = B, R is said to be a relation on A. Note that relations need not have any particular property nor even be describable. Obviously we will be interested in those relations which are describable and have particular properties which will be shown later. is a relation between A and B . Example 1.7 If A is the set of people, then a R b if a and b are cousins is … \end{verbatim} ``` </details>
33. ph-3460b63f4957f5791cc6 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) Example 1.8 The domain and range of the relation { ( x , y ) : x 2 + y 2 = 4 } are -2 ≤ x ≤ 2 and -2 ≤ y ≤ 2 respectively. Example 1.9 The relation R is on the set of people. The domain and range of R is the set of people who have cousins. Definition 1.14 Let R be a relation between A and B. The inverse of the relation R denoted by R -1 is a relation been B and A, defined by R -1 = { ( b , a ) : ( a , b ) ∈ R } . Example 1.10 If A = { a , b , c , d , e } and B = { 1 , 2 , 3 , 4 , 5 } , and is a relation between A and B then is a relati… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{Example 1.8 The domain and range of the relation \{ ( x , y ) : x 2 + y 2 = 4 \} are -2 ≤ x ≤ 2 and -2 ≤ y ≤ 2 respectively. Example 1.9 The relation R is on the set of people. The domain and range of R is the set of people who have cousins. Definition 1.14 Let R be a relation bet…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=24499 \begin{verbatim} Example 1.8 The domain and range of the relation { ( x , y ) : x 2 + y 2 = 4 } are -2 ≤ x ≤ 2 and -2 ≤ y ≤ 2 respectively. Example 1.9 The relation R is on the set of people. The domain and range of R is the set of people who have cousins. Definition 1.14 Let R be a relation between A and B. The inverse of the relation R denoted by R -1 is a relation been B and A, defined by R -1 = { ( b , a ) : ( a , b ) ∈ R } . Example 1.10 If A = { a , b , c , d , e } and B = { 1 , 2 , 3 , 4 , 5 } , and is a relation between A and B then is a relati… \end{verbatim} ``` </details>
34. ph-5f4571d20dee85c44fc4 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) 2 + y 2 = 4 } are -2 ≤ x ≤ 2 and -2 ≤ y ≤ 2 respectively. Example 1.9 The relation R is on the set of people. The domain and range of R is the set of people who have cousins. Definition 1.14 Let R be a relation between A and B. The inverse of the relation R denoted by R -1 is a relation been B and A, defined by R -1 = { ( b , a ) : ( a , b ) ∈ R } . Example 1.10 If A = { a , b , c , d , e } and B = { 1 , 2 , 3 , 4 , 5 } , and is a relation between A and B then is a relation between B and A . Definition 1.15… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{2 + y 2 = 4 \} are -2 ≤ x ≤ 2 and -2 ≤ y ≤ 2 respectively. Example 1.9 The relation R is on the set of people. The domain and range of R is the set of people who have cousins. Definition 1.14 Let R be a relation between A and B. The inverse of the relation R denoted by R -1 is a …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=24565 \begin{verbatim} 2 + y 2 = 4 } are -2 ≤ x ≤ 2 and -2 ≤ y ≤ 2 respectively. Example 1.9 The relation R is on the set of people. The domain and range of R is the set of people who have cousins. Definition 1.14 Let R be a relation between A and B. The inverse of the relation R denoted by R -1 is a relation been B and A, defined by R -1 = { ( b , a ) : ( a , b ) ∈ R } . Example 1.10 If A = { a , b , c , d , e } and B = { 1 , 2 , 3 , 4 , 5 } , and is a relation between A and B then is a relation between B and A . Definition 1.15… \end{verbatim} ``` </details>
35. ph-d36ebda1d19c1942c397 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) ple 1.9 The relation R is on the set of people. The domain and range of R is the set of people who have cousins. Definition 1.14 Let R be a relation between A and B. The inverse of the relation R denoted by R -1 is a relation been B and A, defined by R -1 = { ( b , a ) : ( a , b ) ∈ R } . Example 1.10 If A = { a , b , c , d , e } and B = { 1 , 2 , 3 , 4 , 5 } , and is a relation between A and B then is a relation between B and A . Definition 1.15 Let R be a relation between A and B, and let S be a relation … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ple 1.9 The relation R is on the set of people. The domain and range of R is the set of people who have cousins. Definition 1.14 Let R be a relation between A and B. The inverse of the relation R denoted by R -1 is a relation been B and A, defined by R -1 = \{ ( b , a ) : ( a , b…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=24628 \begin{verbatim} ple 1.9 The relation R is on the set of people. The domain and range of R is the set of people who have cousins. Definition 1.14 Let R be a relation between A and B. The inverse of the relation R denoted by R -1 is a relation been B and A, defined by R -1 = { ( b , a ) : ( a , b ) ∈ R } . Example 1.10 If A = { a , b , c , d , e } and B = { 1 , 2 , 3 , 4 , 5 } , and is a relation between A and B then is a relation between B and A . Definition 1.15 Let R be a relation between A and B, and let S be a relation … \end{verbatim} ``` </details>
36. ph-64bb8d9bfab2524d51d0 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) ple 1.10 If A = { a , b , c , d , e } and B = { 1 , 2 , 3 , 4 , 5 } , and is a relation between A and B then is a relation between B and A . Definition 1.15 Let R be a relation between A and B, and let S be a relation between B and C. The composition of R and S , denoted by S ◦ R is a relation between A and C defined by ( a , c ) ∈ S ◦ R if there exists b ∈ B such that ( a , b ) ∈ R and ( b , c ) ∈ S . be a relation between A and B . Then, as shown above is a relation between A and B then is a relation between B and A . Definition 1.15 Let R be a relation between A and B…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=24924 \begin{verbatim} ple 1.10 If A = { a , b , c , d , e } and B = { 1 , 2 , 3 , 4 , 5 } , and is a relation between A and B then is a relation between B and A . Definition 1.15 Let R be a relation between A and B, and let S be a relation between B and C. The composition of R and S , denoted by S ◦ R is a relation between A and C defined by ( a , c ) ∈ S ◦ R if there exists b ∈ B such that ( a , b ) ∈ R and ( b , c ) ∈ S . be a relation between A and B . Then, as shown above is a relation between A and B then is a relation between B and A . Definition 1.15 Let R be a relation between A and B, and let S be a relation between B and C. The composition of R and S , denoted by S ◦ R is a relation between A and C defined by ( a , c ) ∈ S ◦ R if there exists b ∈ B such that ( a , b ) ∈ R and ( b , c ) ∈ S . be a relation between A and B . Then, as shown above is a relation between B , and A , is a relation o… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ula-not-decoded --> is a relation between A and B then is a relation between B and A . Definition 1.15 Let R be a relation between A and B, and let S be a relation between B and C. The composition of R and S , denoted by …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=25008 \begin{verbatim} ula-not-decoded --> is a relation between A and B then is a relation between B and A . Definition 1.15 Let R be a relation between A and B, and let S be a relation between B and C. The composition of R and S , denoted by S ◦ R is a relation between A and C defined by ( a , c ) ∈ S ◦ R if there exists b ∈ B such that ( a , b ) ∈ R and ( b , c ) ∈ S . be a relation between A and B . Then, as shown above is a relation between B , and A , is a relation o… \end{verbatim} ``` </details>
38. ph-9d488c2d859596c3609d — automata/docling_md/AutomataTheory.md ### Plain (markdown context) mula-not-decoded --> is a relation between B and A . Definition 1.15 Let R be a relation between A and B, and let S be a relation between B and C. The composition of R and S , denoted by S ◦ R is a relation between A and C defined by ( a , c ) ∈ S ◦ R if there exists b ∈ B such that ( a , b ) ∈ R and ( b , c ) ∈ S . be a relation between A and B . Then, as shown above is a relation between B , and A , is a relation on B , and is a relation on A . Exa… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{mula-not-decoded --> is a relation between B and A . Definition 1.15 Let R be a relation between A and B, and let S be a relation between B and C. The composition of R and S , denoted by S ◦ R is a relation between A and C defined by ( a , c ) ∈ S ◦ …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=25073 \begin{verbatim} mula-not-decoded --> is a relation between B and A . Definition 1.15 Let R be a relation between A and B, and let S be a relation between B and C. The composition of R and S , denoted by S ◦ R is a relation between A and C defined by ( a , c ) ∈ S ◦ R if there exists b ∈ B such that ( a , b ) ∈ R and ( b , c ) ∈ S . be a relation between A and B . Then, as shown above is a relation between B , and A , is a relation on B , and is a relation on A . Exa… \end{verbatim} ``` </details>
39. ph-8c81b6c8834008ad4787 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) !-- formula-not-decoded --> Definition 1.15 Let R be a relation between A and B, and let S be a relation between B and C. The composition of R and S , denoted by S ◦ R is a relation between A and C defined by ( a , c ) ∈ S ◦ R if there exists b ∈ B such that ( a , b ) ∈ R and ( b , c ) ∈ S . be a relation between A and B . Then, as shown above is a relation between B , and A , is a relation on B , and is a relation on A . Example 1.13 If R = { ( x , y ) : y = x + 5 } and S = { (… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{!-- formula-not-decoded --> Definition 1.15 Let R be a relation between A and B, and let S be a relation between B and C. The composition of R and S , denoted by S ◦ R is a relation between A and C defined by ( a , c ) ∈ S ◦ R if there exists b ∈ B such that ( a , b ) ∈ R and ( …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=25129 \begin{verbatim} !-- formula-not-decoded --> Definition 1.15 Let R be a relation between A and B, and let S be a relation between B and C. The composition of R and S , denoted by S ◦ R is a relation between A and C defined by ( a , c ) ∈ S ◦ R if there exists b ∈ B such that ( a , b ) ∈ R and ( b , c ) ∈ S . be a relation between A and B . Then, as shown above is a relation between B , and A , is a relation on B , and is a relation on A . Example 1.13 If R = { ( x , y ) : y = x + 5 } and S = { (… \end{verbatim} ``` </details>
40. ph-6c86b22eaf4da3d170e7 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) conclude that a and a are siblings, which we know is not true. Example 1.15 Let A be the set of all people and a R b if a and b have the same parents. The relation R is reflexive since everyone has the same parents as themselves. It is symmetric since if a and b have the same parents, b and a have the same parents. It is also transitive since if a and b have the same parents and b and c have the same parents, then a and c have the same parents. Example 1.16 Let A = { a , b , c , d , e } and R is not reflexive since ( e , e ) / ∈ R . It is not symmetric because ( a… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{conclude that a and a are siblings, which we know is not true. Example 1.15 Let A be the set of all people and a R b if a and b have the same parents. The relation R is reflexive since everyone has the same parents as themselves. It is symmetric since if a and b have the same pa…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=27270 \begin{verbatim} conclude that a and a are siblings, which we know is not true. Example 1.15 Let A be the set of all people and a R b if a and b have the same parents. The relation R is reflexive since everyone has the same parents as themselves. It is symmetric since if a and b have the same parents, b and a have the same parents. It is also transitive since if a and b have the same parents and b and c have the same parents, then a and c have the same parents. Example 1.16 Let A = { a , b , c , d , e } and R is not reflexive since ( e , e ) / ∈ R . It is not symmetric because ( a… \end{verbatim} ``` </details>
41. ph-7beb04fc04e000d14bb5 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) , a ) ∈ R , but d /negationslash= a . It is not transitive since ( a , c ) , ( c , e ) , but ( a , e ) / . ∈ R ∈ R Example 1.17 Let R betherelation on Z defined by a R b if a -b is a multiple of 5. Certainly a -a = 0 is a multiple of 5, so R is reflexive. If a -b is a multiple of 5, then a -b = 5 k for some integer k . Hence b -a = 5( -k ) is a multiple of 5, so R is symmetric. If a -b is a multiple of 5 and b -c is a multiple of 5, then a -b = 5 k and b -c = 5 m for some integers k and m . so that a -c is a multiple of 5. Hence R is transitive. Definition 1.17 Ar… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{, a ) ∈ R , but d /negationslash= a . It is not transitive since ( a , c ) , ( c , e ) , but ( a , e ) / . ∈ R ∈ R Example 1.17 Let R betherelation on Z defined by a R b if a -b is a multiple of 5. Certainly a -a = 0 is a multiple of 5, so R is reflexive. If a -b is a multiple o…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=27954 \begin{verbatim} , a ) ∈ R , but d /negationslash= a . It is not transitive since ( a , c ) , ( c , e ) , but ( a , e ) / . ∈ R ∈ R Example 1.17 Let R betherelation on Z defined by a R b if a -b is a multiple of 5. Certainly a -a = 0 is a multiple of 5, so R is reflexive. If a -b is a multiple of 5, then a -b = 5 k for some integer k . Hence b -a = 5( -k ) is a multiple of 5, so R is symmetric. If a -b is a multiple of 5 and b -c is a multiple of 5, then a -b = 5 k and b -c = 5 m for some integers k and m . so that a -c is a multiple of 5. Hence R is transitive. Definition 1.17 Ar… \end{verbatim} ``` </details>
42. ph-c17d7f08137a8fe5ea6e — automata/docling_md/AutomataTheory.md ### Plain (markdown context) R 1 be the relation on Z defined by R 1 = { ( m , n ) : m -n } is divisible by 5. R 1 is shown above to be an equivalence relation on the integers. Example 1.19 Let A be the set of all people. Define R 2 by a R 2 b if a and b are the same age. This is easily shown to be an equivalence relation. An equivalence relation on a set A divides A into nonempty subsets that are mutually exclusive or disjoint , meaning that no two of them have an element in common. In the first example above, the sets contain elements that are related to each other and no element in one set… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{R 1 be the relation on Z defined by R 1 = \{ ( m , n ) : m -n \} is divisible by 5. R 1 is shown above to be an equivalence relation on the integers. Example 1.19 Let A be the set of all people. Define R 2 by a R 2 b if a and b are the same age. This is easily shown to be an equiv…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=28695 \begin{verbatim} R 1 be the relation on Z defined by R 1 = { ( m , n ) : m -n } is divisible by 5. R 1 is shown above to be an equivalence relation on the integers. Example 1.19 Let A be the set of all people. Define R 2 by a R 2 b if a and b are the same age. This is easily shown to be an equivalence relation. An equivalence relation on a set A divides A into nonempty subsets that are mutually exclusive or disjoint , meaning that no two of them have an element in common. In the first example above, the sets contain elements that are related to each other and no element in one set… \end{verbatim} ``` </details>
43. ph-eb0e13a183677a7f1d84 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) two sets. (See the definition of partition below.) Notation 1.1 Let R be an equivalence relation on a set A and a ∈ A . Then [ a ] R = { x : x R a } . If the relation is understood, then [ a ] R is simply denoted by [ a ]. Let [ A ] R = { [ a ] R : a ∈ A } . Definition 1.18 Let A and I be nonempty sets and 〈 A 〉 = { Ai : i ∈ I } be a set of nonempty subsets of A. The set 〈 A 〉 is called a partition of A if both of the following are satisfied: - (a) Ai ∩ Aj = ∅ for all i /negationslash= j . Theorem 1.3 A nonempty set of subsets 〈 A 〉 of a set A is a partition of A … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{two sets. (See the definition of partition below.) Notation 1.1 Let R be an equivalence relation on a set A and a ∈ A . Then [ a ] R = \{ x : x R a \} . If the relation is understood, then [ a ] R is simply denoted by [ a ]. Let [ A ] R = \{ [ a ] R : a ∈ A \} . Definition 1.18 Let …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=29528 \begin{verbatim} two sets. (See the definition of partition below.) Notation 1.1 Let R be an equivalence relation on a set A and a ∈ A . Then [ a ] R = { x : x R a } . If the relation is understood, then [ a ] R is simply denoted by [ a ]. Let [ A ] R = { [ a ] R : a ∈ A } . Definition 1.18 Let A and I be nonempty sets and 〈 A 〉 = { Ai : i ∈ I } be a set of nonempty subsets of A. The set 〈 A 〉 is called a partition of A if both of the following are satisfied: - (a) Ai ∩ Aj = ∅ for all i /negationslash= j . Theorem 1.3 A nonempty set of subsets 〈 A 〉 of a set A is a partition of A … \end{verbatim} ``` </details>
44. ph-60553470583019d89c25 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) t A ( if it exists ) is called the greatest element of A. For a subset B of a poset A, an element a of A is a lower bound of B if a ≤ b (or b ≥ a ) for all b in B. The element a is called a greatest lower bound (glb) of B if ( i ) a is a lower bound of B and ( ii ) if any other element a ′ of A is a lower bound of B, then a ≥ a ′ . The greatest lower bound for the entire poset A ( if it exists ) is called the least element of A. Example 1.22 Let C = { a , b , c } and X be the power set of C . Define the relation ≤ on X by T ≤ V if T ⊆ V . By definition, { a , b } … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{t A ( if it exists ) is called the greatest element of A. For a subset B of a poset A, an element a of A is a lower bound of B if a ≤ b (or b ≥ a ) for all b in B. The element a is called a greatest lower bound (glb) of B if ( i ) a is a lower bound of B and ( ii ) if any other …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=32591 \begin{verbatim} t A ( if it exists ) is called the greatest element of A. For a subset B of a poset A, an element a of A is a lower bound of B if a ≤ b (or b ≥ a ) for all b in B. The element a is called a greatest lower bound (glb) of B if ( i ) a is a lower bound of B and ( ii ) if any other element a ′ of A is a lower bound of B, then a ≥ a ′ . The greatest lower bound for the entire poset A ( if it exists ) is called the least element of A. Example 1.22 Let C = { a , b , c } and X be the power set of C . Define the relation ≤ on X by T ≤ V if T ⊆ V . By definition, { a , b } … \end{verbatim} ``` </details>
45. ph-0dc370ffc35420cebd9c — automata/docling_md/AutomataTheory.md ### Plain (markdown context) ement b ∈ B, there is an element a ∈ A so that b = f ( a ) . Definition 1.27 If f : A → B, and f ( a ) = f ( a ′ ) ⇒ a = a ′ for all a , a ′ ∈ A then f is one-to-one . It is also called a monomorphism or injection . Definition 1.28 If f : A → B is one-to-one and onto, then f is called a one-to-one correspondence or bijection . If A is finite, then f is also called a permutation . Notation 1.2 If f is a permutation on the set { 1 , 2 , 3 , . . . , n } , then it can be represented in the form Thus if f ( a ) = b , f ( b ) = d , f ( c ) = a , and f ( d ) = c , we may… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ement b ∈ B, there is an element a ∈ A so that b = f ( a ) . Definition 1.27 If f : A → B, and f ( a ) = f ( a ′ ) ⇒ a = a ′ for all a , a ′ ∈ A then f is one-to-one . It is also called a monomorphism or injection . Definition 1.28 If f : A → B is one-to-one and onto, then f is …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=36724 \begin{verbatim} ement b ∈ B, there is an element a ∈ A so that b = f ( a ) . Definition 1.27 If f : A → B, and f ( a ) = f ( a ′ ) ⇒ a = a ′ for all a , a ′ ∈ A then f is one-to-one . It is also called a monomorphism or injection . Definition 1.28 If f : A → B is one-to-one and onto, then f is called a one-to-one correspondence or bijection . If A is finite, then f is also called a permutation . Notation 1.2 If f is a permutation on the set { 1 , 2 , 3 , . . . , n } , then it can be represented in the form Thus if f ( a ) = b , f ( b ) = d , f ( c ) = a , and f ( d ) = c , we may… \end{verbatim} ``` </details>
46. ph-7a47e9633a91369b07fd — automata/docling_md/AutomataTheory.md ### Plain (markdown context) f ( f -1 ( b )) = b , f is onto. By symmetry, f -1 Assume f : A → B is a bijection. Define the relation f -1 on B × A by f -1 ( b ) = a if f ( a ) = b . Let b ∈ B and choose a so that f ( a ) = b . This is possible since f is onto. Therefore f -1 ( b ) = a and f -1 has domain B . If f -1 ( b ) = a and f -1 ( b ) = a ′ , then f ( a ) = b and f ( a ′ ) = b . But since f is one-to-one, a = a ′ . Therefore f -1 is well defined and hence f -1 is a function. By definition f ◦ f -1 = f -1 ◦ f = I . The proof of the following theorem is left to the reader: Theorem 1.7 Let… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{f ( f -1 ( b )) = b , f is onto. By symmetry, f -1 Assume f : A → B is a bijection. Define the relation f -1 on B × A by f -1 ( b ) = a if f ( a ) = b . Let b ∈ B and choose a so that f ( a ) = b . This is possible since f is onto. Therefore f -1 ( b ) = a and f -1 has domain B …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=39093 \begin{verbatim} f ( f -1 ( b )) = b , f is onto. By symmetry, f -1 Assume f : A → B is a bijection. Define the relation f -1 on B × A by f -1 ( b ) = a if f ( a ) = b . Let b ∈ B and choose a so that f ( a ) = b . This is possible since f is onto. Therefore f -1 ( b ) = a and f -1 has domain B . If f -1 ( b ) = a and f -1 ( b ) = a ′ , then f ( a ) = b and f ( a ′ ) = b . But since f is one-to-one, a = a ′ . Therefore f -1 is well defined and hence f -1 is a function. By definition f ◦ f -1 = f -1 ◦ f = I . The proof of the following theorem is left to the reader: Theorem 1.7 Let… \end{verbatim} ``` </details>
47. ph-3213bbfd61a14ba2f46c — automata/docling_md/AutomataTheory.md ### Plain (markdown context) f ◦ g ) -1 = g -1 ◦ f -1 . - (3) Give an example of a function f and sets A 1 , A 2 ⊆ A such that f ( A 1 ∩ A 2 ) /negationslash= f ( A 1 ) ∩ f ( A 2 ). - (5) Prove that if f ◦ g is onto, then f is onto. - (4) Prove that if f ◦ g is one-to-one then g is one-to-one. ## 1.4 Semigroups In the following function /star : S × S → S we shall use the notation a /star a ′ for /star (( a , a ′ )). Definition 1.32 A semigroup is a nonempty set S together with a function /star from S × S → S such that The function or operation /star with this property is called associative . … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{f ◦ g ) -1 = g -1 ◦ f -1 . - (3) Give an example of a function f and sets A 1 , A 2 ⊆ A such that f ( A 1 ∩ A 2 ) /negationslash= f ( A 1 ) ∩ f ( A 2 ). - (5) Prove that if f ◦ g is onto, then f is onto. - (4) Prove that if f ◦ g is one-to-one then g is one-to-one. \#\# 1.4 Semigr…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=42452 \begin{verbatim} f ◦ g ) -1 = g -1 ◦ f -1 . - (3) Give an example of a function f and sets A 1 , A 2 ⊆ A such that f ( A 1 ∩ A 2 ) /negationslash= f ( A 1 ) ∩ f ( A 2 ). - (5) Prove that if f ◦ g is onto, then f is onto. - (4) Prove that if f ◦ g is one-to-one then g is one-to-one. ## 1.4 Semigroups In the following function /star : S × S → S we shall use the notation a /star a ′ for /star (( a , a ′ )). Definition 1.32 A semigroup is a nonempty set S together with a function /star from S × S → S such that The function or operation /star with this property is called associative . … \end{verbatim} ``` </details>
48. ph-8dbedd042f1d747dced4 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) th the operation defined in the previous definition and φ R : S → S / R defined by φ R ( s ) = s R is a homomorphism. Theorem 1.16 Let f : A → B be a homomorphism and R be the congruence a R a ′ iff f ( a ) = f ( a ′ ) , then there exists a homomorphism g : A / R → B defined by g ( a R ) = f ( a ) . Hence g ◦ φ R = f . ![Image](./AutomataTheory_artifacts/image_000005_bf80f879566ee4059c3c198a8d7b04785ae4ecab8e0f64eb8a143017edeba4ab.png) R Proof We showed in Theorem 1.9 that g is a function. /square Theorem 1.17 Let f : A / R → B be a function and S ⊆ R , so if a S … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{th the operation defined in the previous definition and φ R : S → S / R defined by φ R ( s ) = s R is a homomorphism. Theorem 1.16 Let f : A → B be a homomorphism and R be the congruence a R a ′ iff f ( a ) = f ( a ′ ) , then there exists a homomorphism g : A / R → B defined by …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=52550 \begin{verbatim} th the operation defined in the previous definition and φ R : S → S / R defined by φ R ( s ) = s R is a homomorphism. Theorem 1.16 Let f : A → B be a homomorphism and R be the congruence a R a ′ iff f ( a ) = f ( a ′ ) , then there exists a homomorphism g : A / R → B defined by g ( a R ) = f ( a ) . Hence g ◦ φ R = f . ![Image](./AutomataTheory_artifacts/image_000005_bf80f879566ee4059c3c198a8d7b04785ae4ecab8e0f64eb8a143017edeba4ab.png) R Proof We showed in Theorem 1.9 that g is a function. /square Theorem 1.17 Let f : A / R → B be a function and S ⊆ R , so if a S … \end{verbatim} ``` </details>
49. ph-45b475758c590705285b — automata/docling_md/AutomataTheory.md ### Plain (markdown context) , so if a S a ′ implies a R a ′ , then there exist functions g : A / S → B and i : A / S → A / R such that f ◦ i = g. ![Image](./AutomataTheory_artifacts/image_000006_2053b4815736ff133061fdd5f00a5bfe86cf8dfe428aba6bb478524767b6009d.png) S Proof Let i : A / S → A / R be defined by i ( a S ) = a R and g : A / S → B by g ( a S ) = f ( a R ) . The function i is trivially well defined and a homomorphism. The proof that g is a function is similar to the proof of Theorem 1.9. (See Theorem 1.10). /square We already know that the set of all functions from a set A to itself… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{, so if a S a ′ implies a R a ′ , then there exist functions g : A / S → B and i : A / S → A / R such that f ◦ i = g. ![Image](./AutomataTheory\_artifacts/image\_000006\_2053b4815736ff133061fdd5f00a5bfe86cf8dfe428aba6bb478524767b6009d.png) S Proof Let i : A / S → A / R be defined b…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=53143 \begin{verbatim} , so if a S a ′ implies a R a ′ , then there exist functions g : A / S → B and i : A / S → A / R such that f ◦ i = g. ![Image](./AutomataTheory_artifacts/image_000006_2053b4815736ff133061fdd5f00a5bfe86cf8dfe428aba6bb478524767b6009d.png) S Proof Let i : A / S → A / R be defined by i ( a S ) = a R and g : A / S → B by g ( a S ) = f ( a R ) . The function i is trivially well defined and a homomorphism. The proof that g is a function is similar to the proof of Theorem 1.9. (See Theorem 1.10). /square We already know that the set of all functions from a set A to itself… \end{verbatim} ``` </details>
50. ph-f8e7dcc6b185ba8dd0ea — automata/docling_md/AutomataTheory.md ### Plain (markdown context) already know that the set of all functions from a set A to itself form a semigroup since for a ∈ A , and functions f , g , and h from A to itself, (( f ◦ ( g ◦ h ))( a ) = (( f ◦ g ) ◦ h )( a ) = f ( g ( h ( a ). Also since f , g , and h are relations we have already proven that ( f ( g h ) ( f g ) h . Conversely, given a semigroup S , and s ∈ S we can define a function φ s : S → S by φ s ( t ) = st for all t ∈ S . Let T S = { φ s : S → S for s ∈ S } . For all s , t , and u in S , ◦ ◦ = ◦ ◦ and φ st = ( φ s ◦ φ t ). Let τ : S → T S be defined by τ ( s ) = φ s . Th… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{already know that the set of all functions from a set A to itself form a semigroup since for a ∈ A , and functions f , g , and h from A to itself, (( f ◦ ( g ◦ h ))( a ) = (( f ◦ g ) ◦ h )( a ) = f ( g ( h ( a ). Also since f , g , and h are relations we have already proven that…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=53684 \begin{verbatim} already know that the set of all functions from a set A to itself form a semigroup since for a ∈ A , and functions f , g , and h from A to itself, (( f ◦ ( g ◦ h ))( a ) = (( f ◦ g ) ◦ h )( a ) = f ( g ( h ( a ). Also since f , g , and h are relations we have already proven that ( f ( g h ) ( f g ) h . Conversely, given a semigroup S , and s ∈ S we can define a function φ s : S → S by φ s ( t ) = st for all t ∈ S . Let T S = { φ s : S → S for s ∈ S } . For all s , t , and u in S , ◦ ◦ = ◦ ◦ and φ st = ( φ s ◦ φ t ). Let τ : S → T S be defined by τ ( s ) = φ s . Th… \end{verbatim} ``` </details>
51. ph-4051a69818f20233e019 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) self, (( f ◦ ( g ◦ h ))( a ) = (( f ◦ g ) ◦ h )( a ) = f ( g ( h ( a ). Also since f , g , and h are relations we have already proven that ( f ( g h ) ( f g ) h . Conversely, given a semigroup S , and s ∈ S we can define a function φ s : S → S by φ s ( t ) = st for all t ∈ S . Let T S = { φ s : S → S for s ∈ S } . For all s , t , and u in S , ◦ ◦ = ◦ ◦ and φ st = ( φ s ◦ φ t ). Let τ : S → T S be defined by τ ( s ) = φ s . The function τ is a homomorphism since Theorem 1.18 Every semigroup is isomorphic to a semigroup of functions from… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{self, (( f ◦ ( g ◦ h ))( a ) = (( f ◦ g ) ◦ h )( a ) = f ( g ( h ( a ). Also since f , g , and h are relations we have already proven that ( f ( g h ) ( f g ) h . Conversely, given a semigroup S , and s ∈ S we can define a function φ s : S → S by φ s ( t ) = st for all t ∈ S . L…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=53826 \begin{verbatim} self, (( f ◦ ( g ◦ h ))( a ) = (( f ◦ g ) ◦ h )( a ) = f ( g ( h ( a ). Also since f , g , and h are relations we have already proven that ( f ( g h ) ( f g ) h . Conversely, given a semigroup S , and s ∈ S we can define a function φ s : S → S by φ s ( t ) = st for all t ∈ S . Let T S = { φ s : S → S for s ∈ S } . For all s , t , and u in S , ◦ ◦ = ◦ ◦ and φ st = ( φ s ◦ φ t ). Let τ : S → T S be defined by τ ( s ) = φ s . The function τ is a homomorphism since Theorem 1.18 Every semigroup is isomorphic to a semigroup of functions from… \end{verbatim} ``` </details>
52. ph-7bbd5f73f8caaa75d3b6 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) quence a 1 a 2 a 3 a 4 . . . an where ai ∈ /Sigma1 . Thus if /Sigma1 = { a , b } , then aab , a , baba , bbbbb , and baaaaa would all be strings of symbols of /Sigma1 . In addition we include an empty string denoted by λ which has no symbols in it. Definition 2.2 Let /Sigma1 ∗ denote the set of all strings of /Sigma1 including the empty string. Define the binary operation ◦ called concatenation on /Sigma1 ∗ as follows: If a 1 a 2 a 3 a 4 . . . an and b 1 b 2 b 3 b 4 . . . bm ∈ /Sigma1 ∗ then If S and T are subsets of /Sigma1 ∗ then S ◦ T = { s ◦ t : s ∈ S , t ∈ T … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{quence a 1 a 2 a 3 a 4 . . . an where ai ∈ /Sigma1 . Thus if /Sigma1 = \{ a , b \} , then aab , a , baba , bbbbb , and baaaaa would all be strings of symbols of /Sigma1 . In addition we include an empty string denoted by λ which has no symbols in it. Definition 2.2 Let /Sigma1 ∗ d…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=57501 \begin{verbatim} quence a 1 a 2 a 3 a 4 . . . an where ai ∈ /Sigma1 . Thus if /Sigma1 = { a , b } , then aab , a , baba , bbbbb , and baaaaa would all be strings of symbols of /Sigma1 . In addition we include an empty string denoted by λ which has no symbols in it. Definition 2.2 Let /Sigma1 ∗ denote the set of all strings of /Sigma1 including the empty string. Define the binary operation ◦ called concatenation on /Sigma1 ∗ as follows: If a 1 a 2 a 3 a 4 . . . an and b 1 b 2 b 3 b 4 . . . bm ∈ /Sigma1 ∗ then If S and T are subsets of /Sigma1 ∗ then S ◦ T = { s ◦ t : s ∈ S , t ∈ T … \end{verbatim} ``` </details>
53. ph-ed9dcf24163e545dc5f0 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) the submonoid generated by a subset of a monoid described in Chapter 1. Definition 2.3 Let B be a subset of /Sigma1 ∗ then B ∗ is the set of all strings or words formed by concatenating words from B together with the empty string, i.e. B ∗ = { w 1 w 2 . . . w n : w i ∈ B } ∪ { λ } . If ∅ denotes the empty set then ∅ ∗ = { λ } . The symbol ∗ is called the Kleene star and is named after the mathematician and logician Stephen Cole Kleene. Note that /Sigma1 ∗ is consistent with this definition. Let A + be the set consisting of all finite p… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{the submonoid generated by a subset of a monoid described in Chapter 1. Definition 2.3 Let B be a subset of /Sigma1 ∗ then B ∗ is the set of all strings or words formed by concatenating words from B together with the empty string, i.e. B ∗ = \{ w 1 w 2 . . . w n : w i ∈ B \} ∪ \{ λ…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=58470 \begin{verbatim} the submonoid generated by a subset of a monoid described in Chapter 1. Definition 2.3 Let B be a subset of /Sigma1 ∗ then B ∗ is the set of all strings or words formed by concatenating words from B together with the empty string, i.e. B ∗ = { w 1 w 2 . . . w n : w i ∈ B } ∪ { λ } . If ∅ denotes the empty set then ∅ ∗ = { λ } . The symbol ∗ is called the Kleene star and is named after the mathematician and logician Stephen Cole Kleene. Note that /Sigma1 ∗ is consistent with this definition. Let A + be the set consisting of all finite p… \end{verbatim} ``` </details>
54. ph-69ad2bc19fdff4de8e90 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) ubset of a monoid described in Chapter 1. Definition 2.3 Let B be a subset of /Sigma1 ∗ then B ∗ is the set of all strings or words formed by concatenating words from B together with the empty string, i.e. B ∗ = { w 1 w 2 . . . w n : w i ∈ B } ∪ { λ } . If ∅ denotes the empty set then ∅ ∗ = { λ } . The symbol ∗ is called the Kleene star and is named after the mathematician and logician Stephen Cole Kleene. Note that /Sigma1 ∗ is consistent with this definition. Let A + be the set consisting of all finite products of elements of a nonem… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ubset of a monoid described in Chapter 1. Definition 2.3 Let B be a subset of /Sigma1 ∗ then B ∗ is the set of all strings or words formed by concatenating words from B together with the empty string, i.e. B ∗ = \{ w 1 w 2 . . . w n : w i ∈ B \} ∪ \{ λ \} . If ∅ denotes the empty se…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=58500 \begin{verbatim} ubset of a monoid described in Chapter 1. Definition 2.3 Let B be a subset of /Sigma1 ∗ then B ∗ is the set of all strings or words formed by concatenating words from B together with the empty string, i.e. B ∗ = { w 1 w 2 . . . w n : w i ∈ B } ∪ { λ } . If ∅ denotes the empty set then ∅ ∗ = { λ } . The symbol ∗ is called the Kleene star and is named after the mathematician and logician Stephen Cole Kleene. Note that /Sigma1 ∗ is consistent with this definition. Let A + be the set consisting of all finite products of elements of a nonem… \end{verbatim} ``` </details>
55. ph-4de411ab92dea32624bd — automata/docling_md/AutomataTheory.md ### Plain (markdown context) Sigma1 and the symbols ∅ , λ, ∗ , ∨ , ( , and ) . The symbol λ is used to denote the symbol ∅ ∗ . - (i) The symbol ∅ is a regular expression and for every a ∈ /Sigma1 , the symbol a is a regular expression. - (ii) If w 1 and w 2 are regular expressions, then w 1 w 2 , w 1 ∨ w 2 , w ∗ 1 , and ( w 1 ) are regular expressions. - (iii) There are no regular expressions which are not generated by ( i ) and ( ii ) . Each expression corresponds to a set with the following correspondence £ defined by The image of a regular expression is a regul… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{Sigma1 and the symbols ∅ , λ, ∗ , ∨ , ( , and ) . The symbol λ is used to denote the symbol ∅ ∗ . - (i) The symbol ∅ is a regular expression and for every a ∈ /Sigma1 , the symbol a is a regular expression. - (ii) If w 1 and w 2 are regular expressions, then w 1 w 2 , w 1 ∨ w 2 …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=60991 \begin{verbatim} Sigma1 and the symbols ∅ , λ, ∗ , ∨ , ( , and ) . The symbol λ is used to denote the symbol ∅ ∗ . - (i) The symbol ∅ is a regular expression and for every a ∈ /Sigma1 , the symbol a is a regular expression. - (ii) If w 1 and w 2 are regular expressions, then w 1 w 2 , w 1 ∨ w 2 , w ∗ 1 , and ( w 1 ) are regular expressions. - (iii) There are no regular expressions which are not generated by ( i ) and ( ii ) . Each expression corresponds to a set with the following correspondence £ defined by The image of a regular expression is a regul… \end{verbatim} ``` </details>
56. ph-fbe6259853f2f78b7b20 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) ∗ , ∨ , ( , and ) . The symbol λ is used to denote the symbol ∅ ∗ . - (i) The symbol ∅ is a regular expression and for every a ∈ /Sigma1 , the symbol a is a regular expression. - (ii) If w 1 and w 2 are regular expressions, then w 1 w 2 , w 1 ∨ w 2 , w ∗ 1 , and ( w 1 ) are regular expressions. - (iii) There are no regular expressions which are not generated by ( i ) and ( ii ) . Each expression corresponds to a set with the following correspondence £ defined by The image of a regular expression is a regular language. Regular languages… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{∗ , ∨ , ( , and ) . The symbol λ is used to denote the symbol ∅ ∗ . - (i) The symbol ∅ is a regular expression and for every a ∈ /Sigma1 , the symbol a is a regular expression. - (ii) If w 1 and w 2 are regular expressions, then w 1 w 2 , w 1 ∨ w 2 , w ∗ 1 , and ( w 1 ) are regu…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=61021 \begin{verbatim} ∗ , ∨ , ( , and ) . The symbol λ is used to denote the symbol ∅ ∗ . - (i) The symbol ∅ is a regular expression and for every a ∈ /Sigma1 , the symbol a is a regular expression. - (ii) If w 1 and w 2 are regular expressions, then w 1 w 2 , w 1 ∨ w 2 , w ∗ 1 , and ( w 1 ) are regular expressions. - (iii) There are no regular expressions which are not generated by ( i ) and ( ii ) . Each expression corresponds to a set with the following correspondence £ defined by The image of a regular expression is a regular language. Regular languages… \end{verbatim} ``` </details>
57. ph-ee0acc1ee30386d7ed76 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) A : h ( a ) = ua v where only mortal symbols occur in u and v } . For each a in X, let Na be the least nonnegative integer for which h Na ( u v ) = λ . Let H = { h Na ( a ) : a ∈ X } . The fixed language L of h is the submonoid of A ∗ generated by H. The correspondence a ↔ h Na ( a ) is a one-to-one correspon- dence between X and H. Proof (1) H ∗ ⊆ L : Since h is a homomorphism it is sufficient to verify that each element h N ( a ) ( a ) of H is in L , which is confirmed by the calculation: (2) L ⊆ H ∗ : Let w be in L . Let a 1 , a 2 , a 3 , . . . , an be the subs… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{A : h ( a ) = ua v where only mortal symbols occur in u and v \} . For each a in X, let Na be the least nonnegative integer for which h Na ( u v ) = λ . Let H = \{ h Na ( a ) : a ∈ X \} . The fixed language L of h is the submonoid of A ∗ generated by H. The correspondence a ↔ h Na …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=76342 \begin{verbatim} A : h ( a ) = ua v where only mortal symbols occur in u and v } . For each a in X, let Na be the least nonnegative integer for which h Na ( u v ) = λ . Let H = { h Na ( a ) : a ∈ X } . The fixed language L of h is the submonoid of A ∗ generated by H. The correspondence a ↔ h Na ( a ) is a one-to-one correspon- dence between X and H. Proof (1) H ∗ ⊆ L : Since h is a homomorphism it is sufficient to verify that each element h N ( a ) ( a ) of H is in L , which is confirmed by the calculation: (2) L ⊆ H ∗ : Let w be in L . Let a 1 , a 2 , a 3 , . . . , an be the subs… \end{verbatim} ``` </details>
58. ph-88706cb48ec207ed6c8d — automata/docling_md/AutomataTheory.md ### Plain (markdown context) rds of length > 2 is not free. Corollary 2.2 If A is an alphabet having exactly n symbols, then no inclusion chain of distinct retracts of A ∗ has more than n + 1 retracts even when the retract { λ } is included. Corollary 2.3 If X is a key code and x n lies in X ∗ , then so does x. Corollary 2.4 If X is a key code and both u v and v u lie in X ∗ , then so do u and v . Let A = { a 1 , a 2 , a 3 , . . . , an } . A simple example of a longest possible inclusion chain of retracts in A ∗ is Each of these retracts, except the first, is maximal among the retracts con… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{rds of length \> 2 is not free. Corollary 2.2 If A is an alphabet having exactly n symbols, then no inclusion chain of distinct retracts of A ∗ has more than n + 1 retracts even when the retract \{ λ \} is included. Corollary 2.3 If X is a key code and x n lies in X ∗ , then so …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=79091 \begin{verbatim} rds of length > 2 is not free. Corollary 2.2 If A is an alphabet having exactly n symbols, then no inclusion chain of distinct retracts of A ∗ has more than n + 1 retracts even when the retract { λ } is included. Corollary 2.3 If X is a key code and x n lies in X ∗ , then so does x. Corollary 2.4 If X is a key code and both u v and v u lie in X ∗ , then so do u and v . Let A = { a 1 , a 2 , a 3 , . . . , an } . A simple example of a longest possible inclusion chain of retracts in A ∗ is Each of these retracts, except the first, is maximal among the retracts con… \end{verbatim} ``` </details>
59. ph-92591839f1e9a8e4de4f — automata/docling_md/AutomataTheory.md ### Plain (markdown context) ng the retracts contained in its predecessor. In each case the number of generators of the subretract is one less than the number of generators of its predecessor. However, maximal proper subretracts of a retract can have many fewer generators: Proposition 2.2 Let n be a positive integer and A = { a 1 , a 2 , a 3 , . . . , an } an alphabet of n symbols. Let m be any positive integer less than n. Then A ∗ contains a maximal proper retract generated by exactly m words. Proof The set of m words is a key code for which K ∗ is a maximal proper retract of A ∗ . The veri… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ng the retracts contained in its predecessor. In each case the number of generators of the subretract is one less than the number of generators of its predecessor. However, maximal proper subretracts of a retract can have many fewer generators: Proposition 2.2 Let n be a positiv…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=79677 \begin{verbatim} ng the retracts contained in its predecessor. In each case the number of generators of the subretract is one less than the number of generators of its predecessor. However, maximal proper subretracts of a retract can have many fewer generators: Proposition 2.2 Let n be a positive integer and A = { a 1 , a 2 , a 3 , . . . , an } an alphabet of n symbols. Let m be any positive integer less than n. Then A ∗ contains a maximal proper retract generated by exactly m words. Proof The set of m words is a key code for which K ∗ is a maximal proper retract of A ∗ . The veri… \end{verbatim} ``` </details>
60. ph-a56fdfcba5b3132b450c — automata/docling_md/AutomataTheory.md ### Plain (markdown context) and both u v and v u lie in X ∗ , then so do u and v . ## 2.3 Semiretracts and lattices (Optional) The intersection of two retracts of the free monoid on a finite set A need not be a retract if A contains four or more symbols. Possibly the simplest example is the following one adapted from [ 7 ]: Let A = { a , b , c , d } . The sets { ab , ac , d } and { ba , c , da } are key codes and consequently the submonoids R and R ′ that they generated are retracts of A ∗ . However, their intersection is not only not a retract; it is not even finitely generated. The set d (… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{and both u v and v u lie in X ∗ , then so do u and v . \#\# 2.3 Semiretracts and lattices (Optional) The intersection of two retracts of the free monoid on a finite set A need not be a retract if A contains four or more symbols. Possibly the simplest example is the following one a…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=81415 \begin{verbatim} and both u v and v u lie in X ∗ , then so do u and v . ## 2.3 Semiretracts and lattices (Optional) The intersection of two retracts of the free monoid on a finite set A need not be a retract if A contains four or more symbols. Possibly the simplest example is the following one adapted from [ 7 ]: Let A = { a , b , c , d } . The sets { ab , ac , d } and { ba , c , da } are key codes and consequently the submonoids R and R ′ that they generated are retracts of A ∗ . However, their intersection is not only not a retract; it is not even finitely generated. The set d (… \end{verbatim} ``` </details>
61. ph-b31ea07bb712add67363 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) always a complete lattice. By broadening our attention slightly we obtain a similarly attractive stability result for what we call semiretracts of free monoids: Definition 2.14 By a semiretract of A ∗ , we mean an intersection of a finite number of retracts of A ∗ . Each retract of A ∗ is also a semiretract. The clearest example of a semiretract that is not a retract is the example given previously: R ∩ R ′ = ( d ( ab ) ∗ ac ) ∗ . Some pairs of retracts have as their intersection a retract: As stated above, but not to be demonstrated here, if fewer than four alpha… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{always a complete lattice. By broadening our attention slightly we obtain a similarly attractive stability result for what we call semiretracts of free monoids: Definition 2.14 By a semiretract of A ∗ , we mean an intersection of a finite number of retracts of A ∗ . Each retract…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=82496 \begin{verbatim} always a complete lattice. By broadening our attention slightly we obtain a similarly attractive stability result for what we call semiretracts of free monoids: Definition 2.14 By a semiretract of A ∗ , we mean an intersection of a finite number of retracts of A ∗ . Each retract of A ∗ is also a semiretract. The clearest example of a semiretract that is not a retract is the example given previously: R ∩ R ′ = ( d ( ab ) ∗ ac ) ∗ . Some pairs of retracts have as their intersection a retract: As stated above, but not to be demonstrated here, if fewer than four alpha… \end{verbatim} ``` </details>
62. ph-e8166ee4c007a92fc63b — automata/docling_md/AutomataTheory.md ### Plain (markdown context) e diagram ![Image](./AutomataTheory_artifacts/image_000014_c7ddbf19f5bcbf685107f350d041cd0044eb2ce56089a62dbd44bbeec04cf4da.png) In this automaton, if three consecutive b s are read, then the automaton is in state s 3, which is a sink state and is not an acceptance state. This is the only way to get to s 3 and every other state is an acceptance state. Thus the language accepted by this automaton consists of all words which do not have three consecutive b s. An expression for this language is As previously mentioned, the automata that we have been discussing are ca… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{e diagram ![Image](./AutomataTheory\_artifacts/image\_000014\_c7ddbf19f5bcbf685107f350d041cd0044eb2ce56089a62dbd44bbeec04cf4da.png) In this automaton, if three consecutive b s are read, then the automaton is in state s 3, which is a sink state and is not an acceptance state. This i…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=93045 \begin{verbatim} e diagram ![Image](./AutomataTheory_artifacts/image_000014_c7ddbf19f5bcbf685107f350d041cd0044eb2ce56089a62dbd44bbeec04cf4da.png) In this automaton, if three consecutive b s are read, then the automaton is in state s 3, which is a sink state and is not an acceptance state. This is the only way to get to s 3 and every other state is an acceptance state. Thus the language accepted by this automaton consists of all words which do not have three consecutive b s. An expression for this language is As previously mentioned, the automata that we have been discussing are ca… \end{verbatim} ``` </details>
63. ph-39b8eedbccfc645f0e48 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) ϒ as a set of rules, given a ∈ /Sigma1 and s ∈ Q , the rules may allow advancement to each of several states or there may not be a rule which does not allow it to go to any state after reading a in state s . In the latter case, the automaton is 'hung up' and can proceed no further. This cannot occur with a deterministic automaton. Although the definition of a nondeterministic automaton varies, we shall use the following definition: ## Definition 3.2 A nondeterministic automaton , denoted by consists of a finite alphabet /Sigma1 , a finite set Q of states, and a fu… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ϒ as a set of rules, given a ∈ /Sigma1 and s ∈ Q , the rules may allow advancement to each of several states or there may not be a rule which does not allow it to go to any state after reading a in state s . In the latter case, the automaton is 'hung up' and can proceed no furth…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=93973 \begin{verbatim} ϒ as a set of rules, given a ∈ /Sigma1 and s ∈ Q , the rules may allow advancement to each of several states or there may not be a rule which does not allow it to go to any state after reading a in state s . In the latter case, the automaton is 'hung up' and can proceed no further. This cannot occur with a deterministic automaton. Although the definition of a nondeterministic automaton varies, we shall use the following definition: ## Definition 3.2 A nondeterministic automaton , denoted by consists of a finite alphabet /Sigma1 , a finite set Q of states, and a fu… \end{verbatim} ``` </details>
64. ph-d1c5eb81e044ee1a45b6 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) there may not be a rule which does not allow it to go to any state after reading a in state s . In the latter case, the automaton is 'hung up' and can proceed no further. This cannot occur with a deterministic automaton. Although the definition of a nondeterministic automaton varies, we shall use the following definition: ## Definition 3.2 A nondeterministic automaton , denoted by consists of a finite alphabet /Sigma1 , a finite set Q of states, and a function called the transition function . The set Q contains an element s 0 and a sub… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{there may not be a rule which does not allow it to go to any state after reading a in state s . In the latter case, the automaton is 'hung up' and can proceed no further. This cannot occur with a deterministic automaton. Although the definition of a nondeterministic automaton va…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=94085 \begin{verbatim} there may not be a rule which does not allow it to go to any state after reading a in state s . In the latter case, the automaton is 'hung up' and can proceed no further. This cannot occur with a deterministic automaton. Although the definition of a nondeterministic automaton varies, we shall use the following definition: ## Definition 3.2 A nondeterministic automaton , denoted by consists of a finite alphabet /Sigma1 , a finite set Q of states, and a function called the transition function . The set Q contains an element s 0 and a sub… \end{verbatim} ``` </details>
65. ph-525bba209b8cb7717ecf — automata/docling_md/AutomataTheory.md ### Plain (markdown context) of P ( Q ), i.e. the set of subsets of Q , as states for the deterministic automaton which we are constructing. Some of these states may not be used since they do not occur on any path which leads to acceptance state. Hence they could be removed and greatly simplify the deterministic automaton created. However, for our purpose, we are only interested in showing that a deterministic automaton can be created. In general we have the following procedure for constructing a deterministic automaton from a nondeterministic automaton. - (1) Beg… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{of P ( Q ), i.e. the set of subsets of Q , as states for the deterministic automaton which we are constructing. Some of these states may not be used since they do not occur on any path which leads to acceptance state. Hence they could be removed and greatly simplify the determin…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=97604 \begin{verbatim} of P ( Q ), i.e. the set of subsets of Q , as states for the deterministic automaton which we are constructing. Some of these states may not be used since they do not occur on any path which leads to acceptance state. Hence they could be removed and greatly simplify the deterministic automaton created. However, for our purpose, we are only interested in showing that a deterministic automaton can be created. In general we have the following procedure for constructing a deterministic automaton from a nondeterministic automaton. - (1) Beg… \end{verbatim} ``` </details>
66. ph-dacc66717f95d38063f1 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) rministic automaton which we are constructing. Some of these states may not be used since they do not occur on any path which leads to acceptance state. Hence they could be removed and greatly simplify the deterministic automaton created. However, for our purpose, we are only interested in showing that a deterministic automaton can be created. In general we have the following procedure for constructing a deterministic automaton from a nondeterministic automaton. - (1) Begin with the state { s 0 } where s 0 is the start state of the non… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{rministic automaton which we are constructing. Some of these states may not be used since they do not occur on any path which leads to acceptance state. Hence they could be removed and greatly simplify the deterministic automaton created. However, for our purpose, we are only in…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=97670 \begin{verbatim} rministic automaton which we are constructing. Some of these states may not be used since they do not occur on any path which leads to acceptance state. Hence they could be removed and greatly simplify the deterministic automaton created. However, for our purpose, we are only interested in showing that a deterministic automaton can be created. In general we have the following procedure for constructing a deterministic automaton from a nondeterministic automaton. - (1) Begin with the state { s 0 } where s 0 is the start state of the non… \end{verbatim} ``` </details>
67. ph-604d706857011b1e1697 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) till read abbb . Assume that we have ( si , a w ) w ∈ /Sigma1 + . Thus the automaton is in state si and must still read a followed by w . The notation ( si , a w ) /turnstileleft ( s j , w ) means that the automaton has read a and moved from state si to state s j . Therefore ϒ ( si , a ) = s j . In the automaton ![Image](./AutomataTheory_artifacts/image_000025_12d71b8e061a1adc45ccfe10f6a2453c723b9c3fd128d7412c290c5ffe402835.png) we have ( s 2 , bab ) /turnstileleft ( s 3 , ab ). We also have If we have ( si , w i ) /turnstileleft ( s j , w j ) /turnstileleft · · ·… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{till read abbb . Assume that we have ( si , a w ) w ∈ /Sigma1 + . Thus the automaton is in state si and must still read a followed by w . The notation ( si , a w ) /turnstileleft ( s j , w ) means that the automaton has read a and moved from state si to state s j . Therefore ϒ (…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=101415 \begin{verbatim} till read abbb . Assume that we have ( si , a w ) w ∈ /Sigma1 + . Thus the automaton is in state si and must still read a followed by w . The notation ( si , a w ) /turnstileleft ( s j , w ) means that the automaton has read a and moved from state si to state s j . Therefore ϒ ( si , a ) = s j . In the automaton ![Image](./AutomataTheory_artifacts/image_000025_12d71b8e061a1adc45ccfe10f6a2453c723b9c3fd128d7412c290c5ffe402835.png) we have ( s 2 , bab ) /turnstileleft ( s 3 , ab ). We also have If we have ( si , w i ) /turnstileleft ( s j , w j ) /turnstileleft · · ·… \end{verbatim} ``` </details>
68. ph-024438a054482f985d16 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) onstruction, the set of states of M ′ is a subset of P ( Q ). The state s ′ 0 = E ( s 0), and F ′ is a set containing an element of F . For each element a of /Sigma1 , define ϒ ′ by ϒ ′ ( P , a ) = ⋃ p ∈ P E ( ϒ ( p , a )). We first show that M ′ is deterministic. It is certainly single valued. Further ϒ ′ ( P , a ) will always have a value even if it is the empty set. We must now show that M ( L ) = M ′ ( L ). To do this we show that for any states p and q in Q , and any word w in /Sigma1 ∗ for some P containing q . From this it will follow that for some P containing q . From this it will follow that for some P containing q . From this it will follow that for some P containing f , where f ∈ F . and it must be shown that We prove … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{), and F ′ is a set containing an element of F . For each element a of /Sigma1 , define ϒ ′ by ϒ ′ ( P , a ) = ⋃ p ∈ P E ( ϒ ( p , a )). We first show that M ′ is deterministic. It is certainly single valued. Further ϒ ′ ( P , a ) will always have a value even if it is the empty…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=107783 \begin{verbatim} ), and F ′ is a set containing an element of F . For each element a of /Sigma1 , define ϒ ′ by ϒ ′ ( P , a ) = ⋃ p ∈ P E ( ϒ ( p , a )). We first show that M ′ is deterministic. It is certainly single valued. Further ϒ ′ ( P , a ) will always have a value even if it is the empty set. We must now show that M ( L ) = M ′ ( L ). To do this we show that for any states p and q in Q , and any word w in /Sigma1 ∗ for some P containing q . From this it will follow that for some P containing f , where f ∈ F . and it must be shown that We prove … \end{verbatim} ``` </details>
70. ph-855aeef646ca513f135b — automata/docling_md/AutomataTheory.md ### Plain (markdown context) . It is certainly single valued. Further ϒ ′ ( P , a ) will always have a value even if it is the empty set. We must now show that M ( L ) = M ′ ( L ). To do this we show that for any states p and q in Q , and any word w in /Sigma1 ∗ for some P containing q . From this it will follow that for some P containing f , where f ∈ F . and it must be shown that We prove this using induction of the length of w . If | w | = 0, then w = λ , for some P containing q . Now ( p , λ ) /turnstileleft ∗ ( q , λ ) if and only… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{. It is certainly single valued. Further ϒ ′ ( P , a ) will always have a value even if it is the empty set. We must now show that M ( L ) = M ′ ( L ). To do this we show that for any states p and q in Q , and any word w in /Sigma1 ∗ for some P conta…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=107960 \begin{verbatim} . It is certainly single valued. Further ϒ ′ ( P , a ) will always have a value even if it is the empty set. We must now show that M ( L ) = M ′ ( L ). To do this we show that for any states p and q in Q , and any word w in /Sigma1 ∗ for some P containing q . From this it will follow that for some P containing f , where f ∈ F . and it must be shown that We prove this using induction of the length of w . If | w | = 0, then w = λ , for some P containing q . Now ( p , λ ) /turnstileleft ∗ ( q , λ ) if and only… \end{verbatim} ``` </details>
71. ph-dfd14e06ec821aaf4feb — automata/docling_md/AutomataTheory.md ### Plain (markdown context) ula-not-decoded --> for some P containing q . Now ( p , λ ) /turnstileleft ∗ ( q , λ ) if and only if q ∈ E ( p ); but since M ′ is deterministic and no letter is read, then P = E ( p ) and p ∈ E ( p ) . Therefore the statement is true if | w | = 0. ⇒ : Assume w = v a for some letter a and w and ( p , w ) /turnstileleft ∗ ( q , λ ) so that Assume the statement is true for all strings having nonnegative length k . We now have to prove the statement is true for any string w with length k + 1. where at the end, possibly no letters of the alphabet are read. Since ( p … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ula-not-decoded --> for some P containing q . Now ( p , λ ) /turnstileleft ∗ ( q , λ ) if and only if q ∈ E ( p ); but since M ′ is deterministic and no letter is read, then P = E ( p ) and p ∈ E ( p ) . Therefore the statement is true if | w | = 0. ⇒ : Assume w = v a for some l…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=108469 \begin{verbatim} ula-not-decoded --> for some P containing q . Now ( p , λ ) /turnstileleft ∗ ( q , λ ) if and only if q ∈ E ( p ); but since M ′ is deterministic and no letter is read, then P = E ( p ) and p ∈ E ( p ) . Therefore the statement is true if | w | = 0. ⇒ : Assume w = v a for some letter a and w and ( p , w ) /turnstileleft ∗ ( q , λ ) so that Assume the statement is true for all strings having nonnegative length k . We now have to prove the statement is true for any string w with length k + 1. where at the end, possibly no letters of the alphabet are read. Since ( p … \end{verbatim} ``` </details>
72. ph-182761a057d36799e165 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) states. If they do, we can always relabel them. Since we want to begin in M 1, we let s 0 be the starting state of M so that s ′′ 0 = s 0. Since we want to finish in M 2, we let the set of acceptance states be F ′ so that F ′′ = F ′ . We define the rules for ϒ ′′ as follows. If the rule If in state si and a is read, go to state s j is in ϒ and s j is not an acceptance state then include this rule in ϒ ′′ . If s j is an acceptance state then include this rule in ϒ ′′ and also include the rule Hence there is the option of going to the state s j or skipping over to s… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{states. If they do, we can always relabel them. Since we want to begin in M 1, we let s 0 be the starting state of M so that s ′′ 0 = s 0. Since we want to finish in M 2, we let the set of acceptance states be F ′ so that F ′′ = F ′ . We define the rules for ϒ ′′ as follows. If …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=116795 \begin{verbatim} states. If they do, we can always relabel them. Since we want to begin in M 1, we let s 0 be the starting state of M so that s ′′ 0 = s 0. Since we want to finish in M 2, we let the set of acceptance states be F ′ so that F ′′ = F ′ . We define the rules for ϒ ′′ as follows. If the rule If in state si and a is read, go to state s j is in ϒ and s j is not an acceptance state then include this rule in ϒ ′′ . If s j is an acceptance state then include this rule in ϒ ′′ and also include the rule Hence there is the option of going to the state s j or skipping over to s… \end{verbatim} ``` </details>
73. ph-0c75b6e70ee8b3236db8 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) ge](./AutomataTheory_artifacts/image_000061_475e3e83df6fe2ff7c71da51a51ef85161d8f8c7b8f17a885e0294af64d160d3.png) occurs, where e 1 , e 2 , e 3 , · · · , e k are regular expressions, then replace it with the diagram ![Image](./AutomataTheory_artifacts/image_000062_ad14692ce11cff48238ca8a33d702d3aa5aa2219c1008349b9b866e1e0773544.png) ![Image](./AutomataTheory_artifacts/image_000063_09a265b2e78d8222da71fc563beaf311e119039a9d8c7470becc6958af4dbabe.png) occurs, then replace it with the diagram More generally if the diagram ![Image](./AutomataTheory_artifacts/image_000… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ge](./AutomataTheory\_artifacts/image\_000061\_475e3e83df6fe2ff7c71da51a51ef85161d8f8c7b8f17a885e0294af64d160d3.png) occurs, where e 1 , e 2 , e 3 , · · · , e k are regular expressions, then replace it with the diagram ![Image](./AutomataTheory\_artifacts/image\_000062\_ad14692ce11cff…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=130620 \begin{verbatim} ge](./AutomataTheory_artifacts/image_000061_475e3e83df6fe2ff7c71da51a51ef85161d8f8c7b8f17a885e0294af64d160d3.png) occurs, where e 1 , e 2 , e 3 , · · · , e k are regular expressions, then replace it with the diagram ![Image](./AutomataTheory_artifacts/image_000062_ad14692ce11cff48238ca8a33d702d3aa5aa2219c1008349b9b866e1e0773544.png) ![Image](./AutomataTheory_artifacts/image_000063_09a265b2e78d8222da71fc563beaf311e119039a9d8c7470becc6958af4dbabe.png) occurs, then replace it with the diagram More generally if the diagram ![Image](./AutomataTheory_artifacts/image_000… \end{verbatim} ``` </details>
74. ph-3eb47af54e956a6d1cac — automata/docling_md/AutomataTheory.md ### Plain (markdown context) 4692ce11cff48238ca8a33d702d3aa5aa2219c1008349b9b866e1e0773544.png) ![Image](./AutomataTheory_artifacts/image_000063_09a265b2e78d8222da71fc563beaf311e119039a9d8c7470becc6958af4dbabe.png) occurs, then replace it with the diagram More generally if the diagram ![Image](./AutomataTheory_artifacts/image_000064_644de6e8f1660091c5d4019410542b2e049c5e8bd56e5fc133c0600717a04eb8.png) occurs, where e 1 , e 2 , e 3 are regular expressions, then replace it with the diagram In particular, when e 2 = λ , then e 1 e ∗ 2 e 3 becomes e 1 e 3 so that the … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{4692ce11cff48238ca8a33d702d3aa5aa2219c1008349b9b866e1e0773544.png) ![Image](./AutomataTheory\_artifacts/image\_000063\_09a265b2e78d8222da71fc563beaf311e119039a9d8c7470becc6958af4dbabe.png) occurs, then replace it with the diagram More generally if the d…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=130890 \begin{verbatim} 4692ce11cff48238ca8a33d702d3aa5aa2219c1008349b9b866e1e0773544.png) ![Image](./AutomataTheory_artifacts/image_000063_09a265b2e78d8222da71fc563beaf311e119039a9d8c7470becc6958af4dbabe.png) occurs, then replace it with the diagram More generally if the diagram ![Image](./AutomataTheory_artifacts/image_000064_644de6e8f1660091c5d4019410542b2e049c5e8bd56e5fc133c0600717a04eb8.png) occurs, where e 1 , e 2 , e 3 are regular expressions, then replace it with the diagram In particular, when e 2 = λ , then e 1 e ∗ 2 e 3 becomes e 1 e 3 so that the … \end{verbatim} ``` </details>
75. ph-5168fd32d43991ff1108 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) _09a265b2e78d8222da71fc563beaf311e119039a9d8c7470becc6958af4dbabe.png) occurs, then replace it with the diagram More generally if the diagram ![Image](./AutomataTheory_artifacts/image_000064_644de6e8f1660091c5d4019410542b2e049c5e8bd56e5fc133c0600717a04eb8.png) occurs, where e 1 , e 2 , e 3 are regular expressions, then replace it with the diagram In particular, when e 2 = λ , then e 1 e ∗ 2 e 3 becomes e 1 e 3 so that the diagram is replaced by the diagram ![Image](./AutomataTheory_artifacts/image_000065_50… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{\_09a265b2e78d8222da71fc563beaf311e119039a9d8c7470becc6958af4dbabe.png) occurs, then replace it with the diagram More generally if the diagram ![Image](./AutomataTheory\_artifacts/image\_000064\_644de6e8f1660091c5d4019410542b2e049c5e8bd56e5fc133c0600717a…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=131006 \begin{verbatim} _09a265b2e78d8222da71fc563beaf311e119039a9d8c7470becc6958af4dbabe.png) occurs, then replace it with the diagram More generally if the diagram ![Image](./AutomataTheory_artifacts/image_000064_644de6e8f1660091c5d4019410542b2e049c5e8bd56e5fc133c0600717a04eb8.png) occurs, where e 1 , e 2 , e 3 are regular expressions, then replace it with the diagram In particular, when e 2 = λ , then e 1 e ∗ 2 e 3 becomes e 1 e 3 so that the diagram is replaced by the diagram ![Image](./AutomataTheory_artifacts/image_000065_50… \end{verbatim} ``` </details>
76. ph-775e62be74475c114f1b — automata/docling_md/AutomataTheory.md ### Plain (markdown context) plete the proof, we need to show that R ( i , p , j ) is regular for 1 ≤ p ≤ n + 1 . We do this using induction. If p = 1, then there are no interior states in the path so R ( i , p , j ) = { a ∈ /Sigma1 : δ ( qi , a ) = qj } if i /negationslash= j and { λ } ∪ { a ∈ /Sigma1 : δ ( qi , a ) = qj } if i = j . Hence we have a finite set of elements of /Sigma1 and possibly λ in the set so it is a regular set. Assume R ( i , k , j ) is regular. The set of words R ( i , k + 1 , j ) can be defined as where the path from qi to qj may not pass through a state qm where m ≥ k… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{plete the proof, we need to show that R ( i , p , j ) is regular for 1 ≤ p ≤ n + 1 . We do this using induction. If p = 1, then there are no interior states in the path so R ( i , p , j ) = \{ a ∈ /Sigma1 : δ ( qi , a ) = qj \} if i /negationslash= j and \{ λ \} ∪ \{ a ∈ /Sigma1 : δ …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=136477 \begin{verbatim} plete the proof, we need to show that R ( i , p , j ) is regular for 1 ≤ p ≤ n + 1 . We do this using induction. If p = 1, then there are no interior states in the path so R ( i , p , j ) = { a ∈ /Sigma1 : δ ( qi , a ) = qj } if i /negationslash= j and { λ } ∪ { a ∈ /Sigma1 : δ ( qi , a ) = qj } if i = j . Hence we have a finite set of elements of /Sigma1 and possibly λ in the set so it is a regular set. Assume R ( i , k , j ) is regular. The set of words R ( i , k + 1 , j ) can be defined as where the path from qi to qj may not pass through a state qm where m ≥ k… \end{verbatim} ``` </details>
77. ph-b89a830f1e42472b5d35 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) g an interior state qm where m ≥ k . Since R ( i , k + 1 , j ) is formed using union, concatenation, and Kleene star of regular states, it is regular and hence L is regular. Since we have now shown that every regular expression is accepted by an automaton and that the language accepted by an automaton is regular, we have proven Kleene's Theorem. As a result of Kleene's Theorem, we discover two new properties about the regular languages: Theorem 3.4 If L 1 and L 2 are regular languages, then and L 1 ∩ L 2 are regular languages. Proof To show /Sigma1 ∗ -L 1 is regul… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{g an interior state qm where m ≥ k . Since R ( i , k + 1 , j ) is formed using union, concatenation, and Kleene star of regular states, it is regular and hence L is regular. Since we have now shown that every regular expression is accepted by an automaton and that the language a…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=137256 \begin{verbatim} g an interior state qm where m ≥ k . Since R ( i , k + 1 , j ) is formed using union, concatenation, and Kleene star of regular states, it is regular and hence L is regular. Since we have now shown that every regular expression is accepted by an automaton and that the language accepted by an automaton is regular, we have proven Kleene's Theorem. As a result of Kleene's Theorem, we discover two new properties about the regular languages: Theorem 3.4 If L 1 and L 2 are regular languages, then and L 1 ∩ L 2 are regular languages. Proof To show /Sigma1 ∗ -L 1 is regul… \end{verbatim} ``` </details>
78. ph-24b56b491b76bd202dec — automata/docling_md/AutomataTheory.md ### Plain (markdown context) t the automaton for /Sigma1 ∗ -L 1, simply change all of the terminal states in M 1 to nonterminal states and all of the nonterminal states to terminal states. As a result, all words that were accepted because the automaton stopped in a terminal state, are no longer accepted and all words which were not accepted are now accepted since the automaton will now stop in a terminal state after reading this word. To show that L 1 ∩ L 2 is a regular language we simply use the set theory property that This is most easily seen by thinking of /Sigma1 ∗ as the universe so tha… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{t the automaton for /Sigma1 ∗ -L 1, simply change all of the terminal states in M 1 to nonterminal states and all of the nonterminal states to terminal states. As a result, all words that were accepted because the automaton stopped in a terminal state, are no longer accepted and…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=137922 \begin{verbatim} t the automaton for /Sigma1 ∗ -L 1, simply change all of the terminal states in M 1 to nonterminal states and all of the nonterminal states to terminal states. As a result, all words that were accepted because the automaton stopped in a terminal state, are no longer accepted and all words which were not accepted are now accepted since the automaton will now stop in a terminal state after reading this word. To show that L 1 ∩ L 2 is a regular language we simply use the set theory property that This is most easily seen by thinking of /Sigma1 ∗ as the universe so tha… \end{verbatim} ``` </details>
79. ph-51a04293e9b636c85b2a — automata/docling_md/AutomataTheory.md ### Plain (markdown context) , and we are finished. In the graph shown below, only one element is picked from each equivalence class. ## Therefore a minimal deterministic automaton is the automaton ![Image](./AutomataTheory_artifacts/image_000098_402bf5d85ac8e549b35ed4667cc383f7893f54705f75577e9b943cfeef6c2343.png) ## Example 3.24 Let M be the deterministic automaton ![Image](./AutomataTheory_artifacts/image_000099_b9837eae8db81f54654a098c80428de31704d024be0dbeef1dcffbeca985b0c2.png) The unmarked pairs in step 1 are The unmarked pairs in the first use of step 2 are The unmarked pairs in the first use of step 2 are The unmarked pairs in the first use of step 2 are The second use of step 2 produces no new results so the equivalence classes are… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{each equivalence class. \#\# Therefore a minimal deterministic automaton is the automaton ![Image](./AutomataTheory\_artifacts/image\_000098\_402bf5d85ac8e549b35ed4667cc383f7893f54705f75577e9b943cfeef6c2343.png) \#\# Example 3.24 Let M be the deterministic automaton ![Image](./Automata…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=152398 \begin{verbatim} each equivalence class. ## Therefore a minimal deterministic automaton is the automaton ![Image](./AutomataTheory_artifacts/image_000098_402bf5d85ac8e549b35ed4667cc383f7893f54705f75577e9b943cfeef6c2343.png) ## Example 3.24 Let M be the deterministic automaton ![Image](./AutomataTheory_artifacts/image_000099_b9837eae8db81f54654a098c80428de31704d024be0dbeef1dcffbeca985b0c2.png) The unmarked pairs in step 1 are The unmarked pairs in the first use of step 2 are The second use of step 2 produces no new results so the equivalence classes are… \end{verbatim} ``` </details>
81. ph-90c4ef168412ba253b5f — automata/docling_md/AutomataTheory.md ### Plain (markdown context) is minimized version of the arbitrary automaton recognizing the language L is virtually identical with the intrinsic automaton of the language. Theorem 3.5 For a given regular language L, the two minimal reduced automaton developed above accepting language L are isomorphic. Proof M = ( /Sigma1 , Q , s 0 , ϒ ′ , F ), the minimal reduced automaton developed by the collapsing method is isomorphic to the intrinsic minimal automaton. So Mi = ( /Sigma1 , Qi , [1] , ϒ i , Fi ). Define f : Q → Qi by Assume [ x ] = … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{is minimized version of the arbitrary automaton recognizing the language L is virtually identical with the intrinsic automaton of the language. Theorem 3.5 For a given regular language L, the two minimal reduced automaton developed above accepting language L are isomorphic. Proo…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=153339 \begin{verbatim} is minimized version of the arbitrary automaton recognizing the language L is virtually identical with the intrinsic automaton of the language. Theorem 3.5 For a given regular language L, the two minimal reduced automaton developed above accepting language L are isomorphic. Proof M = ( /Sigma1 , Q , s 0 , ϒ ′ , F ), the minimal reduced automaton developed by the collapsing method is isomorphic to the intrinsic minimal automaton. So Mi = ( /Sigma1 , Qi , [1] , ϒ i , Fi ). Define f : Q → Qi by Assume [ x ] = … \end{verbatim} ``` </details>
82. ph-96aef37dc15089b8db3d — automata/docling_md/AutomataTheory.md ### Plain (markdown context) bitrary automaton recognizing the language L is virtually identical with the intrinsic automaton of the language. Theorem 3.5 For a given regular language L, the two minimal reduced automaton developed above accepting language L are isomorphic. Proof M = ( /Sigma1 , Q , s 0 , ϒ ′ , F ), the minimal reduced automaton developed by the collapsing method is isomorphic to the intrinsic minimal automaton. So Mi = ( /Sigma1 , Qi , [1] , ϒ i , Fi ). Define f : Q → Qi by Assume [ x ] = [ y ], then ϒ ( x , u ) ∈ F if… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{bitrary automaton recognizing the language L is virtually identical with the intrinsic automaton of the language. Theorem 3.5 For a given regular language L, the two minimal reduced automaton developed above accepting language L are isomorphic. Proof M = ( /Sigma1 , Q , s 0 , ϒ …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=153369 \begin{verbatim} bitrary automaton recognizing the language L is virtually identical with the intrinsic automaton of the language. Theorem 3.5 For a given regular language L, the two minimal reduced automaton developed above accepting language L are isomorphic. Proof M = ( /Sigma1 , Q , s 0 , ϒ ′ , F ), the minimal reduced automaton developed by the collapsing method is isomorphic to the intrinsic minimal automaton. So Mi = ( /Sigma1 , Qi , [1] , ϒ i , Fi ). Define f : Q → Qi by Assume [ x ] = [ y ], then ϒ ( x , u ) ∈ F if… \end{verbatim} ``` </details>
83. ph-9178717f3c2ff7e467d0 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) the language L is virtually identical with the intrinsic automaton of the language. Theorem 3.5 For a given regular language L, the two minimal reduced automaton developed above accepting language L are isomorphic. Proof M = ( /Sigma1 , Q , s 0 , ϒ ′ , F ), the minimal reduced automaton developed by the collapsing method is isomorphic to the intrinsic minimal automaton. So Mi = ( /Sigma1 , Qi , [1] , ϒ i , Fi ). Define f : Q → Qi by Assume [ x ] = [ y ], then ϒ ( x , u ) ∈ F if and only if ϒ ( y , u ) ∈ F f… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{the language L is virtually identical with the intrinsic automaton of the language. Theorem 3.5 For a given regular language L, the two minimal reduced automaton developed above accepting language L are isomorphic. Proof M = ( /Sigma1 , Q , s 0 , ϒ ′ , F ), the minimal reduced a…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=153399 \begin{verbatim} the language L is virtually identical with the intrinsic automaton of the language. Theorem 3.5 For a given regular language L, the two minimal reduced automaton developed above accepting language L are isomorphic. Proof M = ( /Sigma1 , Q , s 0 , ϒ ′ , F ), the minimal reduced automaton developed by the collapsing method is isomorphic to the intrinsic minimal automaton. So Mi = ( /Sigma1 , Qi , [1] , ϒ i , Fi ). Define f : Q → Qi by Assume [ x ] = [ y ], then ϒ ( x , u ) ∈ F if and only if ϒ ( y , u ) ∈ F f… \end{verbatim} ``` </details>
84. ph-a150cd5430af176eae7c — automata/docling_md/AutomataTheory.md ### Plain (markdown context) u ) ∈ F for u , v ∈ /Sigma1 ∗ . Let f ([ x ]) = [ w ] and f ([ y ]) = ([ w ′ ]). Then w u ∈ L if and only if w ′ u ∈ L ( = Fi ). Hence [ w ] = [ w ′ ] and f is well defined. Conversely, assume f ([ x ]) = f ([ y ]) then w u ∈ L if and only if w ′ u ∈ L ( = Fi ) where ϒ ( s 0 , w ) = x and ϒ ( s 0 , w ′ ) = y . Hence ϒ ( x , u ) ∈ F if and only if ϒ ( y , u ) ∈ F and [ x ] = [ y ]. Hence f is well defined and one-to-one. Finally we must show that f ( ϒ ′ ([ x ] , a )) = ϒ i ( f ([ x ]) , a ), Let w ∈ f ([ x ] , then ϒ ( s 0 , w ) = x fo… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{u ) ∈ F for u , v ∈ /Sigma1 ∗ . Let f ([ x ]) = [ w ] and f ([ y ]) = ([ w ′ ]). Then w u ∈ L if and only if w ′ u ∈ L ( = Fi ). Hence [ w ] = [ w ′ ] and f is well defined. Conversely, assume f ([ x ]) = f ([ y ]) then w u ∈ L if and only if w ′ u ∈ L ( = Fi ) where ϒ ( s 0 , w…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=153994 \begin{verbatim} u ) ∈ F for u , v ∈ /Sigma1 ∗ . Let f ([ x ]) = [ w ] and f ([ y ]) = ([ w ′ ]). Then w u ∈ L if and only if w ′ u ∈ L ( = Fi ). Hence [ w ] = [ w ′ ] and f is well defined. Conversely, assume f ([ x ]) = f ([ y ]) then w u ∈ L if and only if w ′ u ∈ L ( = Fi ) where ϒ ( s 0 , w ) = x and ϒ ( s 0 , w ′ ) = y . Hence ϒ ( x , u ) ∈ F if and only if ϒ ( y , u ) ∈ F and [ x ] = [ y ]. Hence f is well defined and one-to-one. Finally we must show that f ( ϒ ′ ([ x ] , a )) = ϒ i ( f ([ x ]) , a ), Let w ∈ f ([ x ] , then ϒ ( s 0 , w ) = x fo… \end{verbatim} ``` </details>
85. ph-92cf1fb211c4069f5c44 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) . Let f ([ x ]) = [ w ] and f ([ y ]) = ([ w ′ ]). Then w u ∈ L if and only if w ′ u ∈ L ( = Fi ). Hence [ w ] = [ w ′ ] and f is well defined. Conversely, assume f ([ x ]) = f ([ y ]) then w u ∈ L if and only if w ′ u ∈ L ( = Fi ) where ϒ ( s 0 , w ) = x and ϒ ( s 0 , w ′ ) = y . Hence ϒ ( x , u ) ∈ F if and only if ϒ ( y , u ) ∈ F and [ x ] = [ y ]. Hence f is well defined and one-to-one. Finally we must show that f ( ϒ ′ ([ x ] , a )) = ϒ i ( f ([ x ]) , a ), Let w ∈ f ([ x ] , then ϒ ( s 0 , w ) = x for x ∈ [ x ]. Let Let w ∈ f ([ x ] , then ϒ ( s 0 , w ) = x for x ∈ [ x ]. Let Let w ∈ f ([ x ] , then ϒ ( s 0 , w ) = x for x ∈ [ x ]. Let and [ y ] = ϒ ′ ([ x ]] , a ) . Now ϒ ( s 0 , w a ) = y , so Let w ∈ f ([ x ] , then ϒ ( s 0 , w ) = x for x ∈ [ x ]. Let and [ y ] = ϒ ′ ([ x ]] , a ) . Now ϒ ( s 0 , w a ) = y , so Let w ∈ f ([ x ] , then ϒ ( s 0 , w ) = x for x ∈ [ x ]. Let and [ y ] = ϒ ′ ([ x ]] , a ) . Now ϒ ( s 0 , w a ) = y , so and so f ( ϒ ′ ([ x ] , a )) = ϒ i ( f ([ x ]) , a ). /square Corollary 3.1 … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{]) then w u ∈ L if and only if w ′ u ∈ L ( = Fi ) where ϒ ( s 0 , w ) = x and ϒ ( s 0 , w ′ ) = y . Hence ϒ ( x , u ) ∈ F if and only if ϒ ( y , u ) ∈ F and [ x ] = [ y ]. Hence f is well defined and one-to-one. Finally we must show that f ( ϒ ′ ([ x ] , a )) = ϒ i ( f ([ x ]) ,…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=154208 \begin{verbatim} ]) then w u ∈ L if and only if w ′ u ∈ L ( = Fi ) where ϒ ( s 0 , w ) = x and ϒ ( s 0 , w ′ ) = y . Hence ϒ ( x , u ) ∈ F if and only if ϒ ( y , u ) ∈ F and [ x ] = [ y ]. Hence f is well defined and one-to-one. Finally we must show that f ( ϒ ′ ([ x ] , a )) = ϒ i ( f ([ x ]) , a ), Let w ∈ f ([ x ] , then ϒ ( s 0 , w ) = x for x ∈ [ x ]. Let and [ y ] = ϒ ′ ([ x ]] , a ) . Now ϒ ( s 0 , w a ) = y , so and so f ( ϒ ′ ([ x ] , a )) = ϒ i ( f ([ x ]) , a ). /square Corollary 3.1 … \end{verbatim} ``` </details>
88. ph-2dd3340f90ea9a5fdbba — automata/docling_md/AutomataTheory.md ### Plain (markdown context) ] = ϒ ′ ([ x ]] , a ) . Now ϒ ( s 0 , w a ) = y , so and so f ( ϒ ′ ([ x ] , a )) = ϒ i ( f ([ x ]) , a ). /square Corollary 3.1 For a given regular language, all reduced automata which accept that language are unique up to isomorphism. Instead of looking at the syntactic monoid from the intrinsic point of view, as defined above we examine it using an automaton. In particular we look at minimal automata. The transformation monoid of a deterministic automaton is the image of a homomorphism ϕ from /Sigma1 ∗ to a submonoid TM of the monoi… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{] = ϒ ′ ([ x ]] , a ) . Now ϒ ( s 0 , w a ) = y , so and so f ( ϒ ′ ([ x ] , a )) = ϒ i ( f ([ x ]) , a ). /square Corollary 3.1 For a given regular language, all reduced automata which accept that language are unique up to isomorphism. Instead of lo…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=154653 \begin{verbatim} ] = ϒ ′ ([ x ]] , a ) . Now ϒ ( s 0 , w a ) = y , so and so f ( ϒ ′ ([ x ] , a )) = ϒ i ( f ([ x ]) , a ). /square Corollary 3.1 For a given regular language, all reduced automata which accept that language are unique up to isomorphism. Instead of looking at the syntactic monoid from the intrinsic point of view, as defined above we examine it using an automaton. In particular we look at minimal automata. The transformation monoid of a deterministic automaton is the image of a homomorphism ϕ from /Sigma1 ∗ to a submonoid TM of the monoi… \end{verbatim} ``` </details>
89. ph-d022934853d0582f7a1c — automata/docling_md/AutomataTheory.md ### Plain (markdown context) , ¯ a ( si ) = s j if there is an a -arrow from si to s j , i.e. ϒ ( si , a ) = s j . If a , b ∈ /Sigma1 , then ¯ ab = ¯ ab where ¯ ab ( s ) = ¯ a ( b ( s )). More specifically, for u ∈ /Sigma1 ∗ , u ( si ) = s j Thus and if ( si , u ) /turnstileleft ∗ ( s j , λ ). In other words, if the machine is in state si and reads u , then it is in state s j . Let M be the automaton ![Image](./AutomataTheory_artifacts/image_000101_7506cbfa55a9433240114a642749aa274accce799c3ef7e315e0f0d61494460b.png) For convenience, permutation notation is used here although the functions ar… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{, ¯ a ( si ) = s j if there is an a -arrow from si to s j , i.e. ϒ ( si , a ) = s j . If a , b ∈ /Sigma1 , then ¯ ab = ¯ ab where ¯ ab ( s ) = ¯ a ( b ( s )). More specifically, for u ∈ /Sigma1 ∗ , u ( si ) = s j Thus and if ( si , u ) /turnstileleft ∗ ( s j , λ ). In other word…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=155351 \begin{verbatim} , ¯ a ( si ) = s j if there is an a -arrow from si to s j , i.e. ϒ ( si , a ) = s j . If a , b ∈ /Sigma1 , then ¯ ab = ¯ ab where ¯ ab ( s ) = ¯ a ( b ( s )). More specifically, for u ∈ /Sigma1 ∗ , u ( si ) = s j Thus and if ( si , u ) /turnstileleft ∗ ( s j , λ ). In other words, if the machine is in state si and reads u , then it is in state s j . Let M be the automaton ![Image](./AutomataTheory_artifacts/image_000101_7506cbfa55a9433240114a642749aa274accce799c3ef7e315e0f0d61494460b.png) For convenience, permutation notation is used here although the functions ar… \end{verbatim} ``` </details>
90. ph-0fbba7942c206b2d1909 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) for u ∈ /Sigma1 ∗ , u ( si ) = s j Thus and if ( si , u ) /turnstileleft ∗ ( s j , λ ). In other words, if the machine is in state si and reads u , then it is in state s j . Let M be the automaton ![Image](./AutomataTheory_artifacts/image_000101_7506cbfa55a9433240114a642749aa274accce799c3ef7e315e0f0d61494460b.png) For convenience, permutation notation is used here although the functions are not usually permutations, since they are not one-to-one. Thus we have which we shall shorten to Bydefinition let ¯ λ =… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{for u ∈ /Sigma1 ∗ , u ( si ) = s j Thus and if ( si , u ) /turnstileleft ∗ ( s j , λ ). In other words, if the machine is in state si and reads u , then it is in state s j . Let M be the automaton ![Image](./AutomataTheory\_artifacts/image\_000101\_7506cbfa55a9433240114a642749aa274…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=155530 \begin{verbatim} for u ∈ /Sigma1 ∗ , u ( si ) = s j Thus and if ( si , u ) /turnstileleft ∗ ( s j , λ ). In other words, if the machine is in state si and reads u , then it is in state s j . Let M be the automaton ![Image](./AutomataTheory_artifacts/image_000101_7506cbfa55a9433240114a642749aa274accce799c3ef7e315e0f0d61494460b.png) For convenience, permutation notation is used here although the functions are not usually permutations, since they are not one-to-one. Thus we have which we shall shorten to Bydefinition let ¯ λ =… \end{verbatim} ``` </details>
91. ph-77f81205d116f419e495 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) ) /turnstileleft ∗ ( s j , λ ). In other words, if the machine is in state si and reads u , then it is in state s j . Let M be the automaton ![Image](./AutomataTheory_artifacts/image_000101_7506cbfa55a9433240114a642749aa274accce799c3ef7e315e0f0d61494460b.png) For convenience, permutation notation is used here although the functions are not usually permutations, since they are not one-to-one. Thus we have which we shall shorten to Bydefinition let ¯ λ = ( 0 1 2 3 0 1 2 3 ) . We now perform the following prod… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{) /turnstileleft ∗ ( s j , λ ). In other words, if the machine is in state si and reads u , then it is in state s j . Let M be the automaton ![Image](./AutomataTheory\_artifacts/image\_000101\_7506cbfa55a9433240114a642749aa274accce799c3ef7e315e0f0d61494460b.png) For convenience, permutation notation is used here although the functions are not usually permutations, since they are not one-to-one. Thus we have which we shall shorten to Bydefinition let ¯ λ = ( 0 1 2 3 0 1 2 3 ) . We now perform the following prod… \end{verbatim} ```
92. ph-8b399549d1197584e5a8 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) j . Let M be the automaton ![Image](./AutomataTheory_artifacts/image_000101_7506cbfa55a9433240114a642749aa274accce799c3ef7e315e0f0d61494460b.png) For convenience, permutation notation is used here although the functions are not usually permutations, since they are not one-to-one. Thus we have which we shall shorten to Bydefinition let ¯ λ = ( 0 1 2 3 0 1 2 3 ) . We now perform the following products: For convenience, permutation notation is used here although the functions are not usually permutations, …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=155702 \begin{verbatim} j . Let M be the automaton ![Image](./AutomataTheory_artifacts/image_000101_7506cbfa55a9433240114a642749aa274accce799c3ef7e315e0f0d61494460b.png) For convenience, permutation notation is used here although the functions are not usually permutations, since they are not one-to-one. Thus we have which we shall shorten to Bydefinition let ¯ λ = ( 0 1 2 3 0 1 2 3 ) . We now perform the following products: For convenience, permutation notation is used here although the functions are not usually permutations, since they are not one-to-one. Thus we have which we shall shorten to Bydefinition let ¯ λ = ( 0 1 2 3 0 1 2 3 ) . We now perform the following products: For convenience, permutation notation is used here although the functions are not usually permutations, since they are not one-to-on…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=155732 \begin{verbatim} [Image](./AutomataTheory_artifacts/image_000101_7506cbfa55a9433240114a642749aa274accce799c3ef7e315e0f0d61494460b.png) For convenience, permutation notation is used here although the functions are not usually permutations, since they are not one-to-one. Thus we have which we shall shorten to Bydefinition let ¯ λ = ( 0 1 2 3 0 1 2 3 ) . We now perform the following products: For convenience, permutation notation is used here although the functions are not usually permutations, since they are not one-to-one. Thus we have which we shall shorten to Bydefinition let ¯ λ = ( 0 1 2 3 0 1 2 3 ) . We now perform the following products: For convenience, permutation notation is used here although the functions are not usually permutations, since they are not one-to-one. Thus we have For convenience, permutation notation is used here although the functions are not usually permutations, since they are not one-to-one. Thus we have which we shall shorten to Bydefinition let ¯ λ = ( 0 1 2 3 0 1 2 3 ) . We now perform the following products: For convenience, permutation notation is used here although the functions are not usually permutations, since they are not one-to-one. Thus we have which we shall shorten to Bydefinition let ¯ λ = ( 0 1 2 3 0 1 2 3 ) . We now perform the following products: For convenience, permutation notation is used here although the functions are not usually permutations, since they are not one-to-one. Thus we have which we shall …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=155792 \begin{verbatim} 433240114a642749aa274accce799c3ef7e315e0f0d61494460b.png) For convenience, permutation notation is used here although the functions are not usually permutations, since they are not one-to-one. Thus we have which we shall shorten to Bydefinition let ¯ λ = ( 0 1 2 3 0 1 2 3 ) . We now perform the following products: For convenience, permutation notation is used here although the functions are not usually permutations, since they are not one-to-one. Thus we have which we shall shorten to Bydefinition let ¯ λ = ( 0 1 2 3 0 1 2 3 ) . We now perform the following products: For convenience, permutation notation is used here although the functions are not usually permutations, since they are not one-to-one. Thus we have which we shall shorten to For convenience, permutation notation is used here although the functions are not usually permutations, since they are not one-to-one. Thus we have which we shall shorten to Bydefinition let ¯ λ = ( 0 1 2 3 0 1 2 3 ) . We now perform the following products: For convenience, permutation notation is used here although the functions are not usually permutations, since they are not one-to-one. Thus we have which we shall shorten to Bydefinition let ¯ λ = ( 0 1 2 3 0 1 2 3 ) . We now perform the following products: then Continuing this pr… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{!-- formula-not-decoded --> For convenience, permutation notation is used here although the functions are not usually permutations, since they are not one-to-one. Thus we have which we shall shorten to Bydefinition let ¯ …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=155852 \begin{verbatim} !-- formula-not-decoded --> For convenience, permutation notation is used here although the functions are not usually permutations, since they are not one-to-one. Thus we have which we shall shorten to Bydefinition let ¯ λ = ( 0 1 2 3 0 1 2 3 ) . We now perform the following products: then Continuing this pr… \end{verbatim} ``` </details>
98. ph-35a826ebd134437a594a — automata/docling_md/AutomataTheory.md ### Plain (markdown context) or convenience, permutation notation is used here although the functions are not usually permutations, since they are not one-to-one. Thus we have which we shall shorten to Bydefinition let ¯ λ = ( 0 1 2 3 0 1 2 3 ) . We now perform the following products: then Continuing this process and letting γ = ¯ ab , … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{or convenience, permutation notation is used here although the functions are not usually permutations, since they are not one-to-one. Thus we have which we shall shorten to Bydefinition let ¯ λ = ( 0 1 2 3 0 1 2 3 ) . We …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=155882 \begin{verbatim} or convenience, permutation notation is used here although the functions are not usually permutations, since they are not one-to-one. Thus we have which we shall shorten to Bydefinition let ¯ λ = ( 0 1 2 3 0 1 2 3 ) . We now perform the following products: then Continuing this process and letting γ = ¯ ab , … \end{verbatim} ``` </details>
99. ph-846780b39454786a309e — automata/docling_md/AutomataTheory.md ### Plain (markdown context) tation is used here although the functions are not usually permutations, since they are not one-to-one. Thus we have which we shall shorten to Bydefinition let ¯ λ = ( 0 1 2 3 0 1 2 3 ) . We now perform the following products: then Continuing this process and letting γ = ¯ ab , δ = ¯ a ¯ a , ε = ¯ b ¯ b , an… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{tation is used here although the functions are not usually permutations, since they are not one-to-one. Thus we have which we shall shorten to Bydefinition let ¯ λ = ( 0 1 2 3 0 1 2 3 ) . We now perform the following prod…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=155912 \begin{verbatim} tation is used here although the functions are not usually permutations, since they are not one-to-one. Thus we have which we shall shorten to Bydefinition let ¯ λ = ( 0 1 2 3 0 1 2 3 ) . We now perform the following products: then Continuing this process and letting γ = ¯ ab , δ = ¯ a ¯ a , ε = ¯ b ¯ b , an… \end{verbatim} ``` </details>
100. ph-73683672808840fbdcf8 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) b | γ | δ | ε | ζ | | ¯ a | ¯ a | δ | γ | γ | γ | γ | γ | | b | b | ζ | ε | ε | ε | ε | ε | | γ | γ | γ | γ | ε | γ | γ | γ | | δ | δ | γ | γ | γ | γ | γ | γ | | ε | ε | ε | ε | ε | ε | ε | ε | | ζ | ζ | ε | ε | ε | ε | γ | γ | Example 3.25 Let M be the automaton ![Image](./AutomataTheory_artifacts/image_000102_20f8b3cae2fe46086a361c7213d433fe5e6833c128955bf285fc1c3071bbb81d.png) The table for the transformation monoid TM is seen to be | | λ | ¯ a | b | γ | δ | ε | ζ | η | θ | ϑ | ι | κ | µ | |-----|-----|-------|-----|-----|-----|----… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{b | γ | δ | ε | ζ | | ¯ a | ¯ a | δ | γ | γ | γ | γ | γ | | b | b | ζ | ε | ε | ε | ε | ε | | γ | γ | γ | γ | ε | γ | γ | γ | | δ | δ | γ | γ | γ | γ | γ | γ | | ε | ε | ε | ε | ε | ε | ε | ε | | ζ | ζ | ε | ε | ε | ε | γ | γ | Example 3.25 Let M be the automaton ![Image](./Auto…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=156723 \begin{verbatim} b | γ | δ | ε | ζ | | ¯ a | ¯ a | δ | γ | γ | γ | γ | γ | | b | b | ζ | ε | ε | ε | ε | ε | | γ | γ | γ | γ | ε | γ | γ | γ | | δ | δ | γ | γ | γ | γ | γ | γ | | ε | ε | ε | ε | ε | ε | ε | ε | | ζ | ζ | ε | ε | ε | ε | γ | γ | Example 3.25 Let M be the automaton ![Image](./AutomataTheory_artifacts/image_000102_20f8b3cae2fe46086a361c7213d433fe5e6833c128955bf285fc1c3071bbb81d.png) The table for the transformation monoid TM is seen to be | | λ | ¯ a | b | γ | δ | ε | ζ | η | θ | ϑ | ι | κ | µ | |-----|-----|-------|-----|-----|-----|----… \end{verbatim} ``` </details>
101. ph-578c00a782ef90c946b0 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) | ¯ a | ¯ a | δ | γ | γ | γ | γ | γ | | b | b | ζ | ε | ε | ε | ε | ε | | γ | γ | γ | γ | ε | γ | γ | γ | | δ | δ | γ | γ | γ | γ | γ | γ | | ε | ε | ε | ε | ε | ε | ε | ε | | ζ | ζ | ε | ε | ε | ε | γ | γ | Example 3.25 Let M be the automaton ![Image](./AutomataTheory_artifacts/image_000102_20f8b3cae2fe46086a361c7213d433fe5e6833c128955bf285fc1c3071bbb81d.png) The table for the transformation monoid TM is seen to be | | λ | ¯ a | b | γ | δ | ε | ζ | η | θ | ϑ | ι | κ | µ | |-----|-----|-------|-----|-----|-----|-----|-----|-----|-----|… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{| ¯ a | ¯ a | δ | γ | γ | γ | γ | γ | | b | b | ζ | ε | ε | ε | ε | ε | | γ | γ | γ | γ | ε | γ | γ | γ | | δ | δ | γ | γ | γ | γ | γ | γ | | ε | ε | ε | ε | ε | ε | ε | ε | | ζ | ζ | ε | ε | ε | ε | γ | γ | Example 3.25 Let M be the automaton ![Image](./AutomataTheory\_artifacts…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=156753 \begin{verbatim} | ¯ a | ¯ a | δ | γ | γ | γ | γ | γ | | b | b | ζ | ε | ε | ε | ε | ε | | γ | γ | γ | γ | ε | γ | γ | γ | | δ | δ | γ | γ | γ | γ | γ | γ | | ε | ε | ε | ε | ε | ε | ε | ε | | ζ | ζ | ε | ε | ε | ε | γ | γ | Example 3.25 Let M be the automaton ![Image](./AutomataTheory_artifacts/image_000102_20f8b3cae2fe46086a361c7213d433fe5e6833c128955bf285fc1c3071bbb81d.png) The table for the transformation monoid TM is seen to be | | λ | ¯ a | b | γ | δ | ε | ζ | η | θ | ϑ | ι | κ | µ | |-----|-----|-------|-----|-----|-----|-----|-----|-----|-----|… \end{verbatim} ``` </details>
102. ph-7d649b81d16d9d9693ff — automata/docling_md/AutomataTheory.md ### Plain (markdown context) here exists u , v, w ∈ /Sigma1 ∗ , v /negationslash= λ such that z = u vw and u v k w ∈ L for all k ≥ 0 . The length of the string u w is less than or equal to n. Further if M is an automaton accepting the language L and M has q states, then n < q. It is possible to have the stronger statement that z = u vw where the length of u v is less than or equal to q. Proof Let L be accepted by the automaton M = ( /Sigma1 , Q , s 0 , ϒ, F ). Let ϒ ( si , ai ) = si + 1 for i = r to t ; denote this by Since L contains a word of length m , where m > q , say w = a 1 a 2 a… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{here exists u , v, w ∈ /Sigma1 ∗ , v /negationslash= λ such that z = u vw and u v k w ∈ L for all k ≥ 0 . The length of the string u w is less than or equal to n. Further if M is an automaton accepting the language L and M has q states, then n \< q. It is possible to have the …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=163583 \begin{verbatim} here exists u , v, w ∈ /Sigma1 ∗ , v /negationslash= λ such that z = u vw and u v k w ∈ L for all k ≥ 0 . The length of the string u w is less than or equal to n. Further if M is an automaton accepting the language L and M has q states, then n < q. It is possible to have the stronger statement that z = u vw where the length of u v is less than or equal to q. Proof Let L be accepted by the automaton M = ( /Sigma1 , Q , s 0 , ϒ, F ). Let ϒ ( si , ai ) = si + 1 for i = r to t ; denote this by Since L contains a word of length m , where m > q , say w = a 1 a 2 a… \end{verbatim} ``` </details>
103. ph-f5357ac6d31806095236 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) + 1 for i = r to t ; denote this by Since L contains a word of length m , where m > q , say w = a 1 a 2 a 3 . . . am . Note that if ( s 1 , a 1 a 2 a 3 . . . am ) /turnstileleft ∗ ( sm , λ ), then sm is an acceptance state. Since m > q , in reading w , M must pass through the same state twice. Therefore ( s 1 , a 1 a 2 a 3 . . . aj -1 ) /turnstileleft ∗ ( sk , λ ) and ( s 1 , a 1 a 2 a 3 . . . ak -1 ) /turnstileleft ∗ ( sk , λ ) = for some j < k and both Thus Also ( s j , aj a j + 2 . . . ak -1 ) /… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{+ 1 for i = r to t ; denote this by Since L contains a word of length m , where m \> q , say w = a 1 a 2 a 3 . . . am . Note that if ( s 1 , a 1 a 2 a 3 . . . am ) /turnstileleft ∗ ( sm , λ ), then sm is an acceptance state. Since m \> q , in rea…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=164046 \begin{verbatim} + 1 for i = r to t ; denote this by Since L contains a word of length m , where m > q , say w = a 1 a 2 a 3 . . . am . Note that if ( s 1 , a 1 a 2 a 3 . . . am ) /turnstileleft ∗ ( sm , λ ), then sm is an acceptance state. Since m > q , in reading w , M must pass through the same state twice. Therefore ( s 1 , a 1 a 2 a 3 . . . aj -1 ) /turnstileleft ∗ ( sk , λ ) and ( s 1 , a 1 a 2 a 3 . . . ak -1 ) /turnstileleft ∗ ( sk , λ ) = for some j < k and both Thus Also ( s j , aj a j + 2 . . . ak -1 ) /… \end{verbatim} ``` </details>
104. ph-54df7a133911093aa0b5 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) Since L contains a word of length m , where m > q , say w = a 1 a 2 a 3 . . . am . Note that if ( s 1 , a 1 a 2 a 3 . . . am ) /turnstileleft ∗ ( sm , λ ), then sm is an acceptance state. Since m > q , in reading w , M must pass through the same state twice. Therefore ( s 1 , a 1 a 2 a 3 . . . aj -1 ) /turnstileleft ∗ ( sk , λ ) and ( s 1 , a 1 a 2 a 3 . . . ak -1 ) /turnstileleft ∗ ( sk , λ ) = for some j < k and both Thus Also ( s j , aj a j + 2 . . . ak -1 ) /turnstileleft ∗ s j , so in reading … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ Since L contains a word of length m , where m \> q , say w = a 1 a 2 a 3 . . . am . Note that if ( s 1 , a 1 a 2 a 3 . . . am ) /turnstileleft ∗ ( sm , λ ), then sm is an acceptance state. Since m \> q , in reading w , M must pass through the sam…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=164082 \begin{verbatim} Since L contains a word of length m , where m > q , say w = a 1 a 2 a 3 . . . am . Note that if ( s 1 , a 1 a 2 a 3 . . . am ) /turnstileleft ∗ ( sm , λ ), then sm is an acceptance state. Since m > q , in reading w , M must pass through the same state twice. Therefore ( s 1 , a 1 a 2 a 3 . . . aj -1 ) /turnstileleft ∗ ( sk , λ ) and ( s 1 , a 1 a 2 a 3 . . . ak -1 ) /turnstileleft ∗ ( sk , λ ) = for some j < k and both Thus Also ( s j , aj a j + 2 . . . ak -1 ) /turnstileleft ∗ s j , so in reading … \end{verbatim} ``` </details>
105. ph-519dbc51c7590a1e7c49 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) urnstileleft ∗ ( sm , λ ), then sm is an acceptance state. Since m > q , in reading w , M must pass through the same state twice. Therefore ( s 1 , a 1 a 2 a 3 . . . aj -1 ) /turnstileleft ∗ ( sk , λ ) and ( s 1 , a 1 a 2 a 3 . . . ak -1 ) /turnstileleft ∗ ( sk , λ ) = for some j < k and both Thus Also ( s j , aj a j + 2 . . . ak -1 ) /turnstileleft ∗ s j , so in reading aj a j + 2 . . . ak -1 , M returns to the same state and Letting u = a 1 a 2 . . . aj -1 , v = aj a j + 2 . . . ak -1, and w = akak … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{urnstileleft ∗ ( sm , λ ), then sm is an acceptance state. Since m \> q , in reading w , M must pass through the same state twice. Therefore ( s 1 , a 1 a 2 a 3 . . . aj -1 ) /turnstileleft ∗ ( sk , λ ) and ( s 1 , a 1 a 2 a 3 . . . ak -1 ) /turnstileleft ∗ ( sk , λ ) = for so…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=164245 \begin{verbatim} urnstileleft ∗ ( sm , λ ), then sm is an acceptance state. Since m > q , in reading w , M must pass through the same state twice. Therefore ( s 1 , a 1 a 2 a 3 . . . aj -1 ) /turnstileleft ∗ ( sk , λ ) and ( s 1 , a 1 a 2 a 3 . . . ak -1 ) /turnstileleft ∗ ( sk , λ ) = for some j < k and both Thus Also ( s j , aj a j + 2 . . . ak -1 ) /turnstileleft ∗ s j , so in reading aj a j + 2 . . . ak -1 , M returns to the same state and Letting u = a 1 a 2 . . . aj -1 , v = aj a j + 2 . . . ak -1, and w = akak … \end{verbatim} ``` </details>
106. ph-8e7d4b70772c6b833b0e — automata/docling_md/AutomataTheory.md ### Plain (markdown context) k b m = a m + k b m ∈ L , which is a contradiction. Second, u = a m , v = b k , and w = b m -k . Byasimilarargument,wereachacontradiction.Third u = a m -k , v = a k b r , and w = b m -r . But then a m -k a k b r a k b r b m -r ∈ L , which is a contradiction. Hence L is not regular. /square ## Exercises For each of the following sets, determine if the set is regular. If it is, describe the set with a regular expression. If it is not a regular set, use the Pumping lemma to show that it is not. - (2) { a n b 2 n a n : n ≥ 1 } . (3) { ( ab ) n : n ≥ 1 } . (4) { a n b … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{k b m = a m + k b m ∈ L , which is a contradiction. Second, u = a m , v = b k , and w = b m -k . Byasimilarargument,wereachacontradiction.Third u = a m -k , v = a k b r , and w = b m -r . But then a m -k a k b r a k b r b m -r ∈ L , which is a contradiction. Hence L is not regul…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=165697 \begin{verbatim} k b m = a m + k b m ∈ L , which is a contradiction. Second, u = a m , v = b k , and w = b m -k . Byasimilarargument,wereachacontradiction.Third u = a m -k , v = a k b r , and w = b m -r . But then a m -k a k b r a k b r b m -r ∈ L , which is a contradiction. Hence L is not regular. /square ## Exercises For each of the following sets, determine if the set is regular. If it is, describe the set with a regular expression. If it is not a regular set, use the Pumping lemma to show that it is not. - (2) { a n b 2 n a n : n ≥ 1 } . (3) { ( ab ) n : n ≥ 1 } . (4) { a n b … \end{verbatim} ``` </details>
107. ph-9e2bca12c765fbc26ef9 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) e length is greater than n and less than 2 n. Proof First assume L is infinite. By the Pumping Lemma there exists u v m w ∈ L for all m ≥ 0. Further if M is an automaton accepting the language L and M has n states, then | u w | , the length of the string u w , is less than or equal to n . Assume that after u is read, the machine is in state s . If while reading v , the machine returns to s , let v ′ be the string that is read when the machine first returns to s and v ′ x = v . Thus if we have replace it with Thus M reads the string s 0… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{e length is greater than n and less than 2 n. Proof First assume L is infinite. By the Pumping Lemma there exists u v m w ∈ L for all m ≥ 0. Further if M is an automaton accepting the language L and M has n states, then | u w | , the length of the string u w , is less than or eq…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=169747 \begin{verbatim} e length is greater than n and less than 2 n. Proof First assume L is infinite. By the Pumping Lemma there exists u v m w ∈ L for all m ≥ 0. Further if M is an automaton accepting the language L and M has n states, then | u w | , the length of the string u w , is less than or equal to n . Assume that after u is read, the machine is in state s . If while reading v , the machine returns to s , let v ′ be the string that is read when the machine first returns to s and v ′ x = v . Thus if we have replace it with Thus M reads the string s 0… \end{verbatim} ``` </details>
108. ph-50d1c9a5fc728dbf1590 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) less than 2 n. Proof First assume L is infinite. By the Pumping Lemma there exists u v m w ∈ L for all m ≥ 0. Further if M is an automaton accepting the language L and M has n states, then | u w | , the length of the string u w , is less than or equal to n . Assume that after u is read, the machine is in state s . If while reading v , the machine returns to s , let v ′ be the string that is read when the machine first returns to s and v ′ x = v . Thus if we have replace it with Thus M reads the string s 0 , u ( v ′ ) n w for any nonneg… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{less than 2 n. Proof First assume L is infinite. By the Pumping Lemma there exists u v m w ∈ L for all m ≥ 0. Further if M is an automaton accepting the language L and M has n states, then | u w | , the length of the string u w , is less than or equal to n . Assume that after u …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=169777 \begin{verbatim} less than 2 n. Proof First assume L is infinite. By the Pumping Lemma there exists u v m w ∈ L for all m ≥ 0. Further if M is an automaton accepting the language L and M has n states, then | u w | , the length of the string u w , is less than or equal to n . Assume that after u is read, the machine is in state s . If while reading v , the machine returns to s , let v ′ be the string that is read when the machine first returns to s and v ′ x = v . Thus if we have replace it with Thus M reads the string s 0 , u ( v ′ ) n w for any nonneg… \end{verbatim} ``` </details>
109. ph-9921a391d7283eda51a8 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) onstructed. The PDA, beginning at the left, reads a letter at a time in the same manner as a standard automaton. The PDA may read a letter from the tape or pop (remove from the top) and read a symbol from the stack or both. Depending on its current state and the symbol(s) read, the PDA may change state, push a symbol in the stack, or both. ![Image](./AutomataTheory_artifacts/image_000112_b968bb9607a88ba814f66590a1b105c8356ddd63139d322652c16e02b93f6004.png) We now define a PDA more formally. ## Definition 3.8 A pushdown automaton is a sextuple ## Definition 3.8 A pushdown automaton is a sextuple ## Definition 3.8 A pushdown automaton is a sextuple where /Sigma1 is a finite alphabet, Q is a finite set of states, s is the i… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{nner as a standard automaton. The PDA may read a letter from the tape or pop (remove from the top) and read a symbol from the stack or both. Depending on its current state and the symbol(s) read, the PDA may change state, push a symbol in the stack, or both. ![Image](./AutomataT…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=175385 \begin{verbatim} nner as a standard automaton. The PDA may read a letter from the tape or pop (remove from the top) and read a symbol from the stack or both. Depending on its current state and the symbol(s) read, the PDA may change state, push a symbol in the stack, or both. ![Image](./AutomataTheory_artifacts/image_000112_b968bb9607a88ba814f66590a1b105c8356ddd63139d322652c16e02b93f6004.png) We now define a PDA more formally. ## Definition 3.8 A pushdown automaton is a sextuple where /Sigma1 is a finite alphabet, Q is a finite set of states, s is the i… \end{verbatim} ``` </details>
111. ph-6d07c4d5a7ba529cc8dc — automata/docling_md/AutomataTheory.md ### Plain (markdown context) Image](./AutomataTheory_artifacts/image_000112_b968bb9607a88ba814f66590a1b105c8356ddd63139d322652c16e02b93f6004.png) We now define a PDA more formally. ## Definition 3.8 A pushdown automaton is a sextuple where /Sigma1 is a finite alphabet, Q is a finite set of states, s is the initial or starting state, I is a finite of stack symbols, ϒ is the transition relation and F is the set of acceptance states. The relation ϒ is a subset of Thus the relation reads a letter from /Sigma1 λ , determines the state, and … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{Image](./AutomataTheory\_artifacts/image\_000112\_b968bb9607a88ba814f66590a1b105c8356ddd63139d322652c16e02b93f6004.png) We now define a PDA more formally. \#\# Definition 3.8 A pushdown automaton is a sextuple where /Sigma1 is…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=175647 \begin{verbatim} Image](./AutomataTheory_artifacts/image_000112_b968bb9607a88ba814f66590a1b105c8356ddd63139d322652c16e02b93f6004.png) We now define a PDA more formally. ## Definition 3.8 A pushdown automaton is a sextuple where /Sigma1 is a finite alphabet, Q is a finite set of states, s is the initial or starting state, I is a finite of stack symbols, ϒ is the transition relation and F is the set of acceptance states. The relation ϒ is a subset of Thus the relation reads a letter from /Sigma1 λ , determines the state, and … \end{verbatim} ``` </details>
112. ph-07948fe2ba33949366c7 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) t reads the language L = { w c w r : w ∈ { a , b } ∗ } . - (20) Let /Sigma1 = { a , b } . Construct a pushdown automaton that reads the language L = { w : The number of a s in w is equal to twice the number of b s or the number of b s in w is equal to three times the number of a s } . - (19) Let /Sigma1 = { a , b , c } . Construct a pushdown automaton that reads the language L = { w : The number of a s in w is equal to the sum of the number of b s and c s } . - (21) Given two pushdown automata over the same alphabet /Sigma1 and accepti… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{t reads the language L = \{ w c w r : w ∈ \{ a , b \} ∗ \} . - (20) Let /Sigma1 = \{ a , b \} . Construct a pushdown automaton that reads the language L = \{ w : The number of a s in w is equal to twice the number of b s or the number of b s in w is equal to three times the number of a…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=187796 \begin{verbatim} t reads the language L = { w c w r : w ∈ { a , b } ∗ } . - (20) Let /Sigma1 = { a , b } . Construct a pushdown automaton that reads the language L = { w : The number of a s in w is equal to twice the number of b s or the number of b s in w is equal to three times the number of a s } . - (19) Let /Sigma1 = { a , b , c } . Construct a pushdown automaton that reads the language L = { w : The number of a s in w is equal to the sum of the number of b s and c s } . - (21) Given two pushdown automata over the same alphabet /Sigma1 and accepti… \end{verbatim} ``` </details>
113. ph-ff60eb4a33f2aa44e9d1 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) w r : w ∈ { a , b } ∗ } . - (20) Let /Sigma1 = { a , b } . Construct a pushdown automaton that reads the language L = { w : The number of a s in w is equal to twice the number of b s or the number of b s in w is equal to three times the number of a s } . - (19) Let /Sigma1 = { a , b , c } . Construct a pushdown automaton that reads the language L = { w : The number of a s in w is equal to the sum of the number of b s and c s } . - (21) Given two pushdown automata over the same alphabet /Sigma1 and accepting languages L and L ′ respecti… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{w r : w ∈ \{ a , b \} ∗ \} . - (20) Let /Sigma1 = \{ a , b \} . Construct a pushdown automaton that reads the language L = \{ w : The number of a s in w is equal to twice the number of b s or the number of b s in w is equal to three times the number of a s \} . - (19) Let /Sigma1 = \{ a…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=187826 \begin{verbatim} w r : w ∈ { a , b } ∗ } . - (20) Let /Sigma1 = { a , b } . Construct a pushdown automaton that reads the language L = { w : The number of a s in w is equal to twice the number of b s or the number of b s in w is equal to three times the number of a s } . - (19) Let /Sigma1 = { a , b , c } . Construct a pushdown automaton that reads the language L = { w : The number of a s in w is equal to the sum of the number of b s and c s } . - (21) Given two pushdown automata over the same alphabet /Sigma1 and accepting languages L and L ′ respecti… \end{verbatim} ``` </details>
114. ph-224e5ae41948519d48a3 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) automata over the same alphabet /Sigma1 and accepting languages L and L ′ respectively, - (a) Describe how to construct a pushdown automaton /Gamma1 1 that accepts the language L ∪ L ′ . - (b) Construct a pushdown automaton /Gamma1 1 that accepts the language L ∪ L ′ where L is the language accepted by the automaton in Example 3.26 and L ′ is the language accepted by the automaton in Example 3.27. - (22) Given two pushdown automata over the same alphabet /Sigma1 and accepting la… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{automata over the same alphabet /Sigma1 and accepting languages L and L ′ respectively, - (a) Describe how to construct a pushdown automaton /Gamma1 1 that accepts the language L ∪ L ′ . - (b) Construct a pushdown automat…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=188285 \begin{verbatim} automata over the same alphabet /Sigma1 and accepting languages L and L ′ respectively, - (a) Describe how to construct a pushdown automaton /Gamma1 1 that accepts the language L ∪ L ′ . - (b) Construct a pushdown automaton /Gamma1 1 that accepts the language L ∪ L ′ where L is the language accepted by the automaton in Example 3.26 and L ′ is the language accepted by the automaton in Example 3.27. - (22) Given two pushdown automata over the same alphabet /Sigma1 and accepting la… \end{verbatim} ``` </details>
115. ph-a0fda5f83f6270e04c92 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) coded --> over the same alphabet /Sigma1 and accepting languages L and L ′ respectively, - (a) Describe how to construct a pushdown automaton /Gamma1 1 that accepts the language L ∪ L ′ . - (b) Construct a pushdown automaton /Gamma1 1 that accepts the language L ∪ L ′ where L is the language accepted by the automaton in Example 3.26 and L ′ is the language accepted by the automaton in Example 3.27. - (22) Given two pushdown automata over the same alphabet /Sigma1 and accepting languages L and L ′ respective… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{coded --> over the same alphabet /Sigma1 and accepting languages L and L ′ respectively, - (a) Describe how to construct a pushdown automaton /Gamma1 1 that accepts the language L ∪ L ′ . - (b) Construct a pushdown automaton /Gamma1 1 that accepts th…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=188315 \begin{verbatim} coded --> over the same alphabet /Sigma1 and accepting languages L and L ′ respectively, - (a) Describe how to construct a pushdown automaton /Gamma1 1 that accepts the language L ∪ L ′ . - (b) Construct a pushdown automaton /Gamma1 1 that accepts the language L ∪ L ′ where L is the language accepted by the automaton in Example 3.26 and L ′ is the language accepted by the automaton in Example 3.27. - (22) Given two pushdown automata over the same alphabet /Sigma1 and accepting languages L and L ′ respective… \end{verbatim} ``` </details>
116. ph-483db656693ea0f64a32 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) utomaton in Example 3.27 . and and - (23) Given a pushdown automaton /Gamma1 = ( N , ϒ, S , P ) over the alphabet /Sigma1 and accepting language L , 2. (a) Describe how to construct a pushdown automaton /Gamma1 3 which accepts the language L ∗ . 3. (b) Construct a pushdown automaton /Gamma1 3 that accepts the language L ∪ L ′ where L is the language accepted by the automaton in Example 3.26 and L ′ is the language accepted by the automaton in Example 3.27 . - (24) Given two pushdown automata over the same alphabet /Sigma1 and accepting… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{utomaton in Example 3.27 . and and - (23) Given a pushdown automaton /Gamma1 = ( N , ϒ, S , P ) over the alphabet /Sigma1 and accepting language L , 2. (a) Describe how to construct a pushdown automaton /Gamma1 3 which accepts the language L ∗ . 3. (b) Construct a pushdown autom…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=189204 \begin{verbatim} utomaton in Example 3.27 . and and - (23) Given a pushdown automaton /Gamma1 = ( N , ϒ, S , P ) over the alphabet /Sigma1 and accepting language L , 2. (a) Describe how to construct a pushdown automaton /Gamma1 3 which accepts the language L ∗ . 3. (b) Construct a pushdown automaton /Gamma1 3 that accepts the language L ∪ L ′ where L is the language accepted by the automaton in Example 3.26 and L ′ is the language accepted by the automaton in Example 3.27 . - (24) Given two pushdown automata over the same alphabet /Sigma1 and accepting… \end{verbatim} ``` </details>
117. ph-ad2aa4800f2b66d749ce — automata/docling_md/AutomataTheory.md ### Plain (markdown context) d and - (23) Given a pushdown automaton /Gamma1 = ( N , ϒ, S , P ) over the alphabet /Sigma1 and accepting language L , 2. (a) Describe how to construct a pushdown automaton /Gamma1 3 which accepts the language L ∗ . 3. (b) Construct a pushdown automaton /Gamma1 3 that accepts the language L ∪ L ′ where L is the language accepted by the automaton in Example 3.26 and L ′ is the language accepted by the automaton in Example 3.27 . - (24) Given two pushdown automata over the same alphabet /Sigma1 and accepting languages L and L ′ respecti… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{d and - (23) Given a pushdown automaton /Gamma1 = ( N , ϒ, S , P ) over the alphabet /Sigma1 and accepting language L , 2. (a) Describe how to construct a pushdown automaton /Gamma1 3 which accepts the language L ∗ . 3. (b) Construct a pushdown automaton /Gamma1 3 that accepts t…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=189234 \begin{verbatim} d and - (23) Given a pushdown automaton /Gamma1 = ( N , ϒ, S , P ) over the alphabet /Sigma1 and accepting language L , 2. (a) Describe how to construct a pushdown automaton /Gamma1 3 which accepts the language L ∗ . 3. (b) Construct a pushdown automaton /Gamma1 3 that accepts the language L ∪ L ′ where L is the language accepted by the automaton in Example 3.26 and L ′ is the language accepted by the automaton in Example 3.27 . - (24) Given two pushdown automata over the same alphabet /Sigma1 and accepting languages L and L ′ respecti… \end{verbatim} ``` </details>
118. ph-ffe7184ba866e7687caa — automata/docling_md/AutomataTheory.md ### Plain (markdown context) B can be replaced by other symbols while + and the integers cannot be replaced. The symbols that can be replaced by other symbols are called nonterminal symbols and the symbols that can not be replaced by other symbols are called terminal symbols . We generate an element of the language when the string consists only of terminal symbols. The rules which tell us how to replace symbols are called productions . We denote the production (or rule) which tells us that add can be replaced with A + B Thus the productions for our first example above are Thus the productions for our first example above are Thus the productions for our first example above are Below, we shall expand our rules to do arbitrary addition, subtraction, mul… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{symbols that can be replaced by other symbols are called nonterminal symbols and the symbols that can not be replaced by other symbols are called terminal symbols . We generate an element of the language when the string consists only of terminal symbols. The rules which tell us …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=212965 \begin{verbatim} symbols that can be replaced by other symbols are called nonterminal symbols and the symbols that can not be replaced by other symbols are called terminal symbols . We generate an element of the language when the string consists only of terminal symbols. The rules which tell us how to replace symbols are called productions . We denote the production (or rule) which tells us that add can be replaced with A + B Thus the productions for our first example above are Below, we shall expand our rules to do arbitrary addition, subtraction, mul… \end{verbatim} ``` </details>
120. ph-c877bc37e8214f20410a — automata/docling_md/AutomataTheory.md ### Plain (markdown context) set of nonterminal symbols N, a finite set of terminal symbols /Sigma1 , an element S ∈ N, called the start symbol and a finite set of productions P , which is a relation in ( N ∪ /Sigma1 ) ∗ such that each first element in an ordered pair of P contains a symbol from N and at least one production has S as the left string in some ordered pair. Definition 4.2 If W and W ′ are elements of ( N ∪ /Sigma1 ) ∗ , W = u vw , W ′ = u v ′ w , and v → v ′ is a production, this is denoted by W ⇒ W ′ . If for n ≥ 1 , then Wn is derived from W 1 . This is denoted by W 1 ⇒ ∗ n Wn… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{set of nonterminal symbols N, a finite set of terminal symbols /Sigma1 , an element S ∈ N, called the start symbol and a finite set of productions P , which is a relation in ( N ∪ /Sigma1 ) ∗ such that each first element in an ordered pair of P contains a symbol from N and at le…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=213798 \begin{verbatim} set of nonterminal symbols N, a finite set of terminal symbols /Sigma1 , an element S ∈ N, called the start symbol and a finite set of productions P , which is a relation in ( N ∪ /Sigma1 ) ∗ such that each first element in an ordered pair of P contains a symbol from N and at least one production has S as the left string in some ordered pair. Definition 4.2 If W and W ′ are elements of ( N ∪ /Sigma1 ) ∗ , W = u vw , W ′ = u v ′ w , and v → v ′ is a production, this is denoted by W ⇒ W ′ . If for n ≥ 1 , then Wn is derived from W 1 . This is denoted by W 1 ⇒ ∗ n Wn… \end{verbatim} ``` </details>
121. ph-d0b9d575f788a477ccf0 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) is denoted by W 1 ⇒ ∗ n Wn and is called a derivation . If the number of productions in not important we simply use W 1 ⇒ ∗ Wn. The set of all strings of elements of /Sigma1 which may be generated by the set of productions P is called the language generated by the grammar /Gamma1 and is denoted by /Gamma1 ( L ) . To generate a word from the grammar /Gamma1 , we keep using productions to derive new strings until we have a string consisting only of terminal elements. Thus in our example above, and where we will denote ( add , A + B ) by … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{is denoted by W 1 ⇒ ∗ n Wn and is called a derivation . If the number of productions in not important we simply use W 1 ⇒ ∗ Wn. The set of all strings of elements of /Sigma1 which may be generated by the set of productions P is called the language generated by the grammar /Gamma…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=214375 \begin{verbatim} is denoted by W 1 ⇒ ∗ n Wn and is called a derivation . If the number of productions in not important we simply use W 1 ⇒ ∗ Wn. The set of all strings of elements of /Sigma1 which may be generated by the set of productions P is called the language generated by the grammar /Gamma1 and is denoted by /Gamma1 ( L ) . To generate a word from the grammar /Gamma1 , we keep using productions to derive new strings until we have a string consisting only of terminal elements. Thus in our example above, and where we will denote ( add , A + B ) by … \end{verbatim} ``` </details>
122. ph-0ad595ad7b7bab59eb8e — automata/docling_md/AutomataTheory.md ### Plain (markdown context) alled a derivation . If the number of productions in not important we simply use W 1 ⇒ ∗ Wn. The set of all strings of elements of /Sigma1 which may be generated by the set of productions P is called the language generated by the grammar /Gamma1 and is denoted by /Gamma1 ( L ) . To generate a word from the grammar /Gamma1 , we keep using productions to derive new strings until we have a string consisting only of terminal elements. Thus in our example above, and where we will denote ( add , A + B ) by add → A + B , ( A , A + B ) by A → … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{alled a derivation . If the number of productions in not important we simply use W 1 ⇒ ∗ Wn. The set of all strings of elements of /Sigma1 which may be generated by the set of productions P is called the language generated by the grammar /Gamma1 and is denoted by /Gamma1 ( L ) .…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=214410 \begin{verbatim} alled a derivation . If the number of productions in not important we simply use W 1 ⇒ ∗ Wn. The set of all strings of elements of /Sigma1 which may be generated by the set of productions P is called the language generated by the grammar /Gamma1 and is denoted by /Gamma1 ( L ) . To generate a word from the grammar /Gamma1 , we keep using productions to derive new strings until we have a string consisting only of terminal elements. Thus in our example above, and where we will denote ( add , A + B ) by add → A + B , ( A , A + B ) by A → … \end{verbatim} ``` </details>
123. ph-533066d00d9762695439 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) derive new strings until we have a string consisting only of terminal elements. Thus in our example above, and where we will denote ( add , A + B ) by add → A + B , ( A , A + B ) by A → A + B , etc. If we eliminate the production ( B , A × B ), the language generated by /Gamma1 is the set of all formal expressions of finite sums of nonnegative integers less than 10. Example 4.1 In the grammar described above, derive the expression and where we will denote ( add , A + B ) by add → A + B , ( A , A + B ) by A → A + B , etc. If we eliminate the pr…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=214766 \begin{verbatim} derive new strings until we have a string consisting only of terminal elements. Thus in our example above, and where we will denote ( add , A + B ) by add → A + B , ( A , A + B ) by A → A + B , etc. If we eliminate the production ( B , A × B ), the language generated by /Gamma1 is the set of all formal expressions of finite sums of nonnegative integers less than 10. Example 4.1 In the grammar described above, derive the expression and where we will denote ( add , A + B ) by add → A + B , ( A , A + B ) by A → A + B , etc. If we eliminate the production ( B , A × B ), the language generated by /Gamma1 is the set of all formal expressions of finite sums of nonnegative integers less than 10. Example 4.1 In the grammar described above, derive the expression and where we will denote ( add , A + B ) by add → A + B , ( A , A + B ) by A → A + B , etc. If we eliminate the production ( B , A × B ), the la…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=214796 \begin{verbatim} ve a string consisting only of terminal elements. Thus in our example above, and where we will denote ( add , A + B ) by add → A + B , ( A , A + B ) by A → A + B , etc. If we eliminate the production ( B , A × B ), the language generated by /Gamma1 is the set of all formal expressions of finite sums of nonnegative integers less than 10. Example 4.1 In the grammar described above, derive the expression and where we will denote ( add , A + B ) by add → A + B , ( A , A + B ) by A → A + B , etc. If we eliminate the production ( B , A × B ), the language generated by /Gamma1 is the set of all formal expressions of finite sums of nonnegative integers less than 10. Example 4.1 In the grammar described above, derive the expression and where we will denote ( add , A + B ) by add → A + B , ( A , A + B ) by A → A + B , etc. If we eliminate the production ( B , A × B ), the language generated by /Gamma1 is …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=214826 \begin{verbatim} terminal elements. Thus in our example above, and where we will denote ( add , A + B ) by add → A + B , ( A , A + B ) by A → A + B , etc. If we eliminate the production ( B , A × B ), the language generated by /Gamma1 is the set of all formal expressions of finite sums of nonnegative integers less than 10. Example 4.1 In the grammar described above, derive the expression and where we will denote ( add , A + B ) by add → A + B , ( A , A + B ) by A → A + B , etc. If we eliminate the production ( B , A × B ), the language generated by /Gamma1 is the set of all formal expressions of finite sums of nonnegative integers less than 10. Example 4.1 In the grammar described above, derive the expression and where we will denote ( add , A + B ) by add → A + B , ( A , A + B ) by A → A + B , etc. If we eliminate the production ( B , A × B ), the language generated by /Gamma1 is the set of all formal expres…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=214856 \begin{verbatim} ur example above, and where we will denote ( add , A + B ) by add → A + B , ( A , A + B ) by A → A + B , etc. If we eliminate the production ( B , A × B ), the language generated by /Gamma1 is the set of all formal expressions of finite sums of nonnegative integers less than 10. Example 4.1 In the grammar described above, derive the expression and where we will denote ( add , A + B ) by add → A + B , ( A , A + B ) by A → A + B , etc. If we eliminate the production ( B , A × B ), the language generated by /Gamma1 is the set of all formal expressions of finite sums of nonnegative integers less than 10. Example 4.1 In the grammar described above, derive the expression Begin with the produc… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{a-not-decoded --> and where we will denote ( add , A + B ) by add → A + B , ( A , A + B ) by A → A + B , etc. If we eliminate the production ( B , A × B ), the language generated by /Gamma1 is the set of all formal expressions of finite sums of nonne…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=214886 \begin{verbatim} a-not-decoded --> and where we will denote ( add , A + B ) by add → A + B , ( A , A + B ) by A → A + B , etc. If we eliminate the production ( B , A × B ), the language generated by /Gamma1 is the set of all formal expressions of finite sums of nonnegative integers less than 10. Example 4.1 In the grammar described above, derive the expression Begin with the produc… \end{verbatim} ``` </details>
128. ph-689ca072626842b767dd — automata/docling_md/AutomataTheory.md ### Plain (markdown context) ormula-not-decoded --> where we will denote ( add , A + B ) by add → A + B , ( A , A + B ) by A → A + B , etc. If we eliminate the production ( B , A × B ), the language generated by /Gamma1 is the set of all formal expressions of finite sums of nonnegative integers less than 10. Example 4.1 In the grammar described above, derive the expression Begin with the production to derive Then use the … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ormula-not-decoded --> where we will denote ( add , A + B ) by add → A + B , ( A , A + B ) by A → A + B , etc. If we eliminate the production ( B , A × B ), the language generated by /Gamma1 is the set of all formal expressions of finite sums of nonnegative integers less than 10…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=214916 \begin{verbatim} ormula-not-decoded --> where we will denote ( add , A + B ) by add → A + B , ( A , A + B ) by A → A + B , etc. If we eliminate the production ( B , A × B ), the language generated by /Gamma1 is the set of all formal expressions of finite sums of nonnegative integers less than 10. Example 4.1 In the grammar described above, derive the expression Begin with the production to derive Then use the … \end{verbatim} ``` </details>
129. ph-cab931aff6db37cb08b4 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) we will denote ( add , A + B ) by add → A + B , ( A , A + B ) by A → A + B , etc. If we eliminate the production ( B , A × B ), the language generated by /Gamma1 is the set of all formal expressions of finite sums of nonnegative integers less than 10. Example 4.1 In the grammar described above, derive the expression Begin with the production to derive Then use the production to derive Then use… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{we will denote ( add , A + B ) by add → A + B , ( A , A + B ) by A → A + B , etc. If we eliminate the production ( B , A × B ), the language generated by /Gamma1 is the set of all formal expressions of finite sums of nonnegative integers less than 10. Example 4.1 In the grammar …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=214946 \begin{verbatim} we will denote ( add , A + B ) by add → A + B , ( A , A + B ) by A → A + B , etc. If we eliminate the production ( B , A × B ), the language generated by /Gamma1 is the set of all formal expressions of finite sums of nonnegative integers less than 10. Example 4.1 In the grammar described above, derive the expression Begin with the production to derive Then use the production to derive Then use… \end{verbatim} ``` </details>
130. ph-fa9f1315c531fee6e513 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) s the set of all formal expressions of finite sums of nonnegative integers less than 10. Example 4.1 In the grammar described above, derive the expression Begin with the production to derive Then use the production to derive Then use the production to derive Then use the productions to derive 2 + 4 + 7 × 6. Note that we cannot derive Ex… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{s the set of all formal expressions of finite sums of nonnegative integers less than 10. Example 4.1 In the grammar described above, derive the expression Begin with the production to derive Then use the production to derive Then use the production to derive Then use the productions to derive 2 + 4 + 7 × 6. Note that we cannot derive Ex… \end{verbatim} ```
131. ph-e963af01dadab966971c — automata/docling_md/AutomataTheory.md ### Plain (markdown context) n 10. Example 4.1 In the grammar described above, derive the expression Begin with the production to derive Then use the production to derive Then use the production to derive Then use the productions to derive 2 + 4 + 7 × 6. Note that we cannot derive Example 4.2 Suppose we want a grammar which derives arithmetic expressions for the se… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{n 10. Example 4.1 In the grammar described above, derive the expression Begi…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=215192 \begin{verbatim} n 10. Example 4.1 In the grammar described above, derive the expression Begin with the production to derive Then use the production to derive Then use the production to derive Then use the productions to derive 2 + 4 + 7 × 6. Note that we cannot derive Example 4.2 Suppose we want a grammar which derives arithmetic expressions for the se… \end{verbatim} ``` </details>
132. ph-906faee4813d170504c0 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) 7 , 8 , 9 } . Thus the language generated by the grammar is the set of all finite arithmetic expressions for the set of integers { 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 } . Examples would be 3 × (5 + 4) and (4 + 5) ÷ (3ˆ2), where ˆ denotes exponent. As mentioned above, we obviously want to exclude expressions such as 3 +× 6 and 3 +÷ 6 × 4 -5 . Let the set N = { S , A , B } and /Sigma1 = {+ , -, × , ÷ , ˆ , 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , ( , ) } . Wewill need the following productions: We will use the grammar to derive the arithmetic expression We will use the grammar to derive the arithmetic expression We will use the grammar to derive the arithmetic expression We begin with the production We will use the grammar to derive the arithmetic expression We begin with the production We will use the grammar to derive the arithmetic expression We begin with the production We will use the grammar to derive the arithmetic expression We begin with the production We will use the grammar to derive the arithmetic expression We begin with the production We then use the pro… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{would be 3 × (5 + 4) and (4 + 5) ÷ (3ˆ2), where ˆ denotes exponent. As mentioned above, we obviously want to exclude expressions such as 3 +× 6 and 3 +÷ 6 × 4 -5 . Let the set N = \{ S , A , B \} and /Sigma1 = \{+ , -, × , ÷ , ˆ , 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , ( , ) \} . W…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=216032 \begin{verbatim} would be 3 × (5 + 4) and (4 + 5) ÷ (3ˆ2), where ˆ denotes exponent. As mentioned above, we obviously want to exclude expressions such as 3 +× 6 and 3 +÷ 6 × 4 -5 . Let the set N = { S , A , B } and /Sigma1 = {+ , -, × , ÷ , ˆ , 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , ( , ) } . Wewill need the following productions: We will use the grammar to derive the arithmetic expression We begin with the production We then use the pro… \end{verbatim} ``` </details>
136. ph-64caf21c08969db601d3 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) 5) ÷ (3ˆ2), where ˆ denotes exponent. As mentioned above, we obviously want to exclude expressions such as 3 +× 6 and 3 +÷ 6 × 4 -5 . Let the set N = { S , A , B } and /Sigma1 = {+ , -, × , ÷ , ˆ , 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , ( , ) } . Wewill need the following productions: We will use the grammar to derive the arithmetic expression We begin with the production We then use the productions and to derive The pro… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{5) ÷ (3ˆ2), where ˆ denotes exponent. As mentioned above, we obviously want to exclude expressions such as 3 +× 6 and 3 +÷ 6 × 4 -5 . Let the set N = \{ S , A , B \} and /Sigma1 = \{+ , -, × , ÷ , ˆ , 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , ( , ) \} . Wewill need the following produ…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=216062 \begin{verbatim} 5) ÷ (3ˆ2), where ˆ denotes exponent. As mentioned above, we obviously want to exclude expressions such as 3 +× 6 and 3 +÷ 6 × 4 -5 . Let the set N = { S , A , B } and /Sigma1 = {+ , -, × , ÷ , ˆ , 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , ( , ) } . Wewill need the following productions: We will use the grammar to derive the arithmetic expression We begin with the production We then use the productions and to derive The pro… \end{verbatim} ``` </details>
137. ph-d351611f5d38b38f875c — automata/docling_md/AutomataTheory.md ### Plain (markdown context) xponent. As mentioned above, we obviously want to exclude expressions such as 3 +× 6 and 3 +÷ 6 × 4 -5 . Let the set N = { S , A , B } and /Sigma1 = {+ , -, × , ÷ , ˆ , 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , ( , ) } . Wewill need the following productions: We will use the grammar to derive the arithmetic expression We begin with the production We then use the productions and to derive The productions give us to derive to… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{xponent. As mentioned above, we obviously want to exclude expressions such as 3 +× 6 and 3 +÷ 6 × 4 -5 . Let the set N = \{ S , A , B \} and /Sigma1 = \{+ , -, × , ÷ , ˆ , 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , ( , ) \} . Wewill need the following productions: We will use the grammar to derive the arithmetic expression We begin with the production We then use the productions and to derive The productions give us to derive to… \end{verbatim} ```
138. ph-9d4fe3ac807f08bb2f5d — automata/docling_md/AutomataTheory.md ### Plain (markdown context) 1 = {+ , -, × , ÷ , ˆ , 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , ( , ) } . Wewill need the following productions: We will use the grammar to derive the arithmetic expression We begin with the production We then use the productions and to derive The productions give us to derive to derive The productions give us Finally we use the productions We will use the grammar to derive the arithmetic expression We begin with the production We will use the grammar to derive the arithmetic expression We begin with the production We then use the productions and to derive The productions give us to derive to derive The productions give us Finally we use the productions We will use the grammar to derive the arithmetic expression We begin with the production We then use the productions and to derive The productions give us to derive to derive The productions give us Finally we use the productions We will use the grammar to derive the arithmetic expression We begin with the production We will use the grammar to derive the arithmetic expression We begin with the production We then use the productions and to derive The productions give us to derive to derive The productions give us Finally we use the productions We will use the grammar to derive the arithmetic expression We begin with the production We then use the productions and to derive The productions give us to derive to derive The productions give us Finally we use the productions We next use the grammar to derive the arithmetic expressio… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{llowing productions: We will use the grammar to derive the arithmetic expression We begin with the production We will use the grammar to derive the arithmetic expression We begin with the production We then use the productions and to derive The productions give us to derive to derive The productions give us Finally we use the productions We next use the grammar to derive the arithmetic expressio… \end{verbatim} ```
141. ph-d096f66082c5d70bac52 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) mula-not-decoded --> We will use the grammar to derive the arithmetic expression We begin with the production We then use the productions and to derive The productions give us to derive to derive The productions give us Finally we use the productions We next use the grammar to derive the arithmetic expression We will use the grammar to derive the arithmetic expression We begin with the production We then use the producti…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=216359 \begin{verbatim} mula-not-decoded --> We will use the grammar to derive the arithmetic expression We begin with the production We then use the productions and to derive The productions give us to derive to derive The productions give us Finally we use the productions We next use the grammar to derive the arithmetic expression We begin with the production We then use the productions and to derive The productions give us to derive to derive The productions give us Finally we use the productions We next use the grammar to derive the arithmetic expression We begin with the production We then use the productions <… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{mula-not-decoded --> We begin with the production We then use the productions and to derive The productions give us to derive to derive The productions give us <!-…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=216450 \begin{verbatim} mula-not-decoded --> We begin with the production We then use the productions and to derive The productions give us to derive to derive The productions give us Finally we use the productions We next use the grammar to derive the arithmetic expression We begin with the production We then use the productions <… \end{verbatim} ``` </details>
143. ph-1bfb1e05c581f57fc13c — automata/docling_md/AutomataTheory.md ### Plain (markdown context) mula-not-decoded --> We then use the productions and to derive The productions give us to derive to derive The productions give us Finally we use the productions We next use the grammar to derive the arithmetic expression We begin with the production We then use the productions <… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{mula-not-decoded --> We then use the productions and to derive The productions give us to derive to derive The productions give us Fin…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=216510 \begin{verbatim} mula-not-decoded --> We then use the productions and to derive The productions give us to derive to derive The productions give us Finally we use the productions We next use the grammar to derive the arithmetic expression We begin with the production We then use the productions <… \end{verbatim} ``` </details>
144. ph-3a6a5714254a67ae67b7 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) rmula-not-decoded --> We then use the productions and to derive The productions give us to derive to derive The productions give us Finally we use the productions We next use the grammar to derive the arithmetic expression We begin with the production We then use the productions … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{rmula-not-decoded --> We then use the productions and to derive The productions give us to derive to derive The productions give us Finally we use the productions …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=216569 \begin{verbatim} rmula-not-decoded --> We then use the productions and to derive The productions give us to derive to derive The productions give us Finally we use the productions We next use the grammar to derive the arithmetic expression We begin with the production We then use the productions … \end{verbatim} ``` </details>
145. ph-5aa1d5eb7cd3e77983a6 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) rmula-not-decoded --> We then use the productions and to derive The productions give us to derive to derive The productions give us Finally we use the productions We next use the grammar to derive the arithmetic expression We begin with the production We then use the productions Finally we use the production… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{rmula-not-decoded --> We then use the productions and to derive The productions give us to derive to derive The productions give us Finally we use the productions …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=216599 \begin{verbatim} rmula-not-decoded --> We then use the productions and to derive The productions give us to derive to derive The productions give us Finally we use the productions We next use the grammar to derive the arithmetic expression We begin with the production We then use the productions Finally we use the production… \end{verbatim} ``` </details>
146. ph-dbb986f185b5ac3cf484 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) use the productions and to derive The productions give us to derive to derive The productions give us Finally we use the productions We next use the grammar to derive the arithmetic expression We begin with the production We then use the productions Finally we use the productions … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{use the productions and to derive The productions give us to derive to derive The productions give us Finally we use the productions We next use the grammar to der…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=216629 \begin{verbatim} use the productions and to derive The productions give us to derive to derive The productions give us Finally we use the productions We next use the grammar to derive the arithmetic expression We begin with the production We then use the productions Finally we use the productions … \end{verbatim} ``` </details>
147. ph-164f2837f74dc029dc05 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) erive The productions give us to derive to derive The productions give us Finally we use the productions We next use the grammar to derive the arithmetic expression We begin with the production We then use the productions Finally we use the productions Finally we use the productions We next use the grammar to derive the arithmetic expressio…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=216659 \begin{verbatim} erive The productions give us to derive to derive The productions give us Finally we use the productions We next use the grammar to derive the arithmetic expression We begin with the production We then use the productions Finally we use the productions Finally we use the productions We next use the grammar to derive the arithmetic expression We begin with the production We then use the productions Finally we use the productions Example 4.3 In a similar manner, we may form arithmetic e… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ctions give us Finally we use the productions We next use the grammar to derive the arithmetic expression We begin with the production…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=216721 \begin{verbatim} ctions give us Finally we use the productions We next use the grammar to derive the arithmetic expression We begin with the production We then use the productions Finally we use the productions Example 4.3 In a similar manner, we may form arithmetic e… \end{verbatim} ``` </details>
149. ph-2af1a6713c03f50c1b8a — automata/docling_md/AutomataTheory.md ### Plain (markdown context) ot-decoded --> Finally we use the productions We next use the grammar to derive the arithmetic expression We begin with the production We then use the productions Finally we use the productions Example 4.3 In a similar manner, we may form arithmetic expressions in postfix notatio… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ot-decoded --> Finally we use the productions We next use the grammar to derive the arithmetic expression We begin with the production …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=216751 \begin{verbatim} ot-decoded --> Finally we use the productions We next use the grammar to derive the arithmetic expression We begin with the production We then use the productions Finally we use the productions Example 4.3 In a similar manner, we may form arithmetic expressions in postfix notatio… \end{verbatim} ``` </details>
150. ph-2a69d7149e4ee3fe3b26 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) grammar to derive the arithmetic expression We begin with the production We then use the productions Finally we use the productions Example 4.3 In a similar manner, we may form arithmetic expressions in postfix notation. Let the set N = { S , A , B } and to derive We will need the following productions: Cons… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{grammar to derive the arithmetic expression We begin with the production We then use the productions Fina…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=216905 \begin{verbatim} grammar to derive the arithmetic expression We begin with the production We then use the productions Finally we use the productions Example 4.3 In a similar manner, we may form arithmetic expressions in postfix notation. Let the set N = { S , A , B } and to derive We will need the following productions: Cons… \end{verbatim} ``` </details>
151. ph-c7fd2fced5a0cc9a1ab1 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) n with the production We then use the productions Finally we use the productions Example 4.3 In a similar manner, we may form arithmetic expressions in postfix notation. Let the set N = { S , A , B } and to derive We will need the following productions: Consider the expression 3 2 + 4 7 +× . Since our integers are all less than ten, 3 2… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{n with the production We then use the productions Finally we use the productions We then use the productions Finally we use the productions Example 4.3 In a similar manner, we may form arithmetic expressions in postfix notation. Let the set N = { S , A , B } and to derive We will need the following productions: Consider the expression 3 2 + 4 7 +× . Since our integers are all less than ten, 3 2… \end{verbatim} ```
152. ph-b570cc1bce8a8ffd9459 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) -> Example 4.3 In a similar manner, we may form arithmetic expressions in postfix notation. Let the set N = { S , A , B } and to derive We will need the following productions: Consider the expression 3 2 + 4 7 +× . Since our integers are all less than ten, 3 2 + represents the integer symbol 3, followed by the integer symbol 2 and the + symbol. To construct this expression we begin with the production We then use the productions Example 4.3 In a similar manner, we may form arithmetic expressions in postfix notation. Let the set N = \{ S , A , B \} and to derive We will need the following productions: Consider the exp…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=217247 \begin{verbatim} -> Example 4.3 In a similar manner, we may form arithmetic expressions in postfix notation. Let the set N = { S , A , B } and to derive We will need the following productions: Consider the expression 3 2 + 4 7 +× . Since our integers are all less than ten, 3 2 + represents the integer symbol 3, followed by the integer symbol 2 and the + symbol. To construct this expression we begin with the production We then use the productions to derive We will need the following productions: Consider the expression 3 2 + 4 7 +× . Since our integers are all less than ten, 3 2 + represents the integer symbol 3, followed by the integer symbol 2 and the + symbol. To construct this expression we begin with the production We then use the productions to derive We will need the following productions: Consider the expression 3 2 + 4 7 +× . Since our integers are all less th…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=217306 \begin{verbatim} manner, we may form arithmetic expressions in postfix notation. Let the set N = { S , A , B } and to derive We will need the following productions: Consider the expression 3 2 + 4 7 +× . Since our integers are all less than ten, 3 2 + represents the integer symbol 3, followed by the integer symbol 2 and the + symbol. To construct this expression we begin with the production We then use the productions to derive We will need the following productions: Consider the expression 3 2 + 4 7 +× . Since our integers are all less than ten, 3 2 + represents the integer symbol 3, followed by the integer symbol 2 and the + symbol. To construct this expression we begin with the production We then use the productions Finally we use the pro… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{expressions in postfix notation. Let the set N = \{ S , A , B \} and to derive We will need the following productions: Consider the expression 3 2 + 4 7 +× . Since our integers are all less than ten, 3 2 + represents the in…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=217336 \begin{verbatim} expressions in postfix notation. Let the set N = { S , A , B } and to derive We will need the following productions: Consider the expression 3 2 + 4 7 +× . Since our integers are all less than ten, 3 2 + represents the integer symbol 3, followed by the integer symbol 2 and the + symbol. To construct this expression we begin with the production We then use the productions Finally we use the pro… \end{verbatim} ``` </details>
155. ph-d964d23cff70fffba317 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) on. Let the set N = { S , A , B } and to derive We will need the following productions: Consider the expression 3 2 + 4 7 +× . Since our integers are all less than ten, 3 2 + represents the integer symbol 3, followed by the integer symbol 2 and the + symbol. To construct this expression we begin with the production We then use the productions Finally we use the productions to derive We will need the following productions: Consider the expression 3 2 + 4 7 +× . Since our integers are all less than ten, 3 2 + represents the integer symbol 3, followed by t…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=217366 \begin{verbatim} on. Let the set N = { S , A , B } and to derive We will need the following productions: Consider the expression 3 2 + 4 7 +× . Since our integers are all less than ten, 3 2 + represents the integer symbol 3, followed by the integer symbol 2 and the + symbol. To construct this expression we begin with the production We then use the productions Finally we use the productions to derive We will need the following productions: Consider the expression 3 2 + 4 7 +× . Since our integers are all less than ten, 3 2 + represents the integer symbol 3, followed by the integer symbol 2 and the + symbol. To construct this expression we begin with the production We then use the productions Finally we use the productions to derive We will need the following productions: Consider the expression 3 2 + 4 7 +× . Since our integers are all less than ten, 3 2 + represents the integer symbol 3, followed by the integer symbol 2 and the + …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=217396 \begin{verbatim} B } and to derive We will need the following productions: Consider the expression 3 2 + 4 7 +× . Since our integers are all less than ten, 3 2 + represents the integer symbol 3, followed by the integer symbol 2 and the + symbol. To construct this expression we begin with the production We then use the productions Finally we use the productions Consider the expression 3 2 + 4 7 +× . Since our integers are all less than ten, 3 2 + represents the integer symbol 3, followed by the integer symbol 2 and the + symbol. To construct this expression we begin with the production We then use the productions Finally we use the productions Example 4.4 Agrammar may also be used to derive prop… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{the following productions: Consider the expression 3 2 + 4 7 +× . Since our integers are all less than ten, 3 2 + represents the integer symbol 3, followed by the integer symbol 2 and the + symbol. To construct this expression we begin with the produ…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=217458 \begin{verbatim} the following productions: Consider the expression 3 2 + 4 7 +× . Since our integers are all less than ten, 3 2 + represents the integer symbol 3, followed by the integer symbol 2 and the + symbol. To construct this expression we begin with the production We then use the productions Finally we use the productions Example 4.4 Agrammar may also be used to derive prop… \end{verbatim} ``` </details>
158. ph-bde649340b6444770548 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) !-- formula-not-decoded --> Consider the expression 3 2 + 4 7 +× . Since our integers are all less than ten, 3 2 + represents the integer symbol 3, followed by the integer symbol 2 and the + symbol. To construct this expression we begin with the production We then use the productions Finally we use the productions Example 4.4 Agrammar may also be used to derive proper sentences. These sentence… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{!-- formula-not-decoded --> Consider the expression 3 2 + 4 7 +× . Since our integers are all less than ten, 3 2 + represents the integer symbol 3, followed by the integer symbol 2 and the + symbol. To construct this expression we begin with the production Consider the expression 3 2 + 4 7 +× . Since our integers are all less than ten, 3 2 + represents the integer symbol 3, followed by the integer symbol 2 and the + symbol. To construct this expression we begin with the production We then use the productions Finally we use the productions Example 4.4 Agrammar may also be used to derive proper sentences. These sentence… \end{verbatim} ```
159. ph-45f12d17962feb11797b — automata/docling_md/AutomataTheory.md ### Plain (markdown context) he fast horse leaped over the old fence. The cowboy rode slowly into the sunset. to derive The productions give us to derive Before actually stating the grammar let us decide upon its structure. This allows us to be assured that each sentence in the grammar is a grammatically correct sentence. Each of our sentences has a noun phrase (noun p), a verb phrase (verb p), and another noun phrase. In addition the last two sentences have a preposition (prep). Therefore let the first production be In our example, the most general form of a noun phrase is an article followe… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{he fast horse leaped over the old fence. The cowboy rode slowly into the sunset. to derive The productions give us to derive Before actually stating the grammar let us decide upon its structure. This allows us to be assured that each sentence in the grammar is a grammatically co…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=218308 \begin{verbatim} he fast horse leaped over the old fence. The cowboy rode slowly into the sunset. to derive The productions give us to derive Before actually stating the grammar let us decide upon its structure. This allows us to be assured that each sentence in the grammar is a grammatically correct sentence. Each of our sentences has a noun phrase (noun p), a verb phrase (verb p), and another noun phrase. In addition the last two sentences have a preposition (prep). Therefore let the first production be In our example, the most general form of a noun phrase is an article followe… \end{verbatim} ``` </details>
160. ph-3452988dc40c0d37f6a7 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) upon its structure. This allows us to be assured that each sentence in the grammar is a grammatically correct sentence. Each of our sentences has a noun phrase (noun p), a verb phrase (verb p), and another noun phrase. In addition the last two sentences have a preposition (prep). Therefore let the first production be In our example, the most general form of a noun phrase is an article followed by an adjective and then a noun. Therefore let the next production be where 'art' represents article and 'adj' represents adjective The most gen… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{upon its structure. This allows us to be assured that each sentence in the grammar is a grammatically correct sentence. Each of our sentences has a noun phrase (noun p), a verb phrase (verb p), and another noun phrase. In addition the last two sentences have a preposition (prep)…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=218487 \begin{verbatim} upon its structure. This allows us to be assured that each sentence in the grammar is a grammatically correct sentence. Each of our sentences has a noun phrase (noun p), a verb phrase (verb p), and another noun phrase. In addition the last two sentences have a preposition (prep). Therefore let the first production be In our example, the most general form of a noun phrase is an article followed by an adjective and then a noun. Therefore let the next production be where 'art' represents article and 'adj' represents adjective The most gen… \end{verbatim} ``` </details>
161. ph-bc7a29b537fe1ad632c3 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) her noun phrase. In addition the last two sentences have a preposition (prep). Therefore let the first production be In our example, the most general form of a noun phrase is an article followed by an adjective and then a noun. Therefore let the next production be where 'art' represents article and 'adj' represents adjective The most general form of a verb phrase is a verb followed by an adverb. Therefore let the next production be where 'adv' represents adverb. At this point, we know that the terminal set … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{her noun phrase. In addition the last two sentences have a preposition (prep). Therefore let the first production be In our example, the most general form of a noun phrase is an article followed by an adjective and then a noun. Therefore let the next…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=218690 \begin{verbatim} her noun phrase. In addition the last two sentences have a preposition (prep). Therefore let the first production be In our example, the most general form of a noun phrase is an article followed by an adjective and then a noun. Therefore let the next production be where 'art' represents article and 'adj' represents adjective The most general form of a verb phrase is a verb followed by an adverb. Therefore let the next production be where 'adv' represents adverb. At this point, we know that the terminal set … \end{verbatim} ``` </details>
162. ph-a311739bc3fe590b79a7 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) ode, slowly, into, sunset } . The nonterminal set N = { S , < noun p >, < verb p >, < art > , < adj >, < noun > , < adv > , < verb > , < prep > } . We next need productions which will assign values to < art > , < adj >, < noun > , < adv > , and < verb > . In some of our sentences we do not need < art > , < adjective >, < prep > , and < adv > . To solve this problem, we include the productions By assigning these symbols to the empty set, we simply erase them when t… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ode, slowly, into, sunset \} . The nonterminal set N = \{ S , \< noun p \>, \< verb p \>, \< art \> , \< adj \>, \< noun \> , \< adv \> , \< verb \> , \< prep \> \} . We next need productions which will assign values to \< art \> , \< adj \>,…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=219385 \begin{verbatim} ode, slowly, into, sunset } . The nonterminal set N = { S , < noun p >, < verb p >, < art > , < adj >, < noun > , < adv > , < verb > , < prep > } . We next need productions which will assign values to < art > , < adj >, < noun > , < adv > , and < verb > . In some of our sentences we do not need < art > , < adjective >, < prep > , and < adv > . To solve this problem, we include the productions By assigning these symbols to the empty set, we simply erase them when t… \end{verbatim} ``` </details>
163. ph-6171cf67e52ba4611526 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) rep > } . We next need productions which will assign values to < art > , < adj >, < noun > , < adv > , and < verb > . In some of our sentences we do not need < art > , < adjective >, < prep > , and < adv > . To solve this problem, we include the productions By assigning these symbols to the empty set, we simply erase them when they are not needed. The remainder of our productions consists of the following: To derive the sentence 'Joe chased the dog,' we begin with By assigning these symbols to the empty set, we simply erase them when they are not needed. The remainder of our productions consists of the following: To derive the sentence 'Joe chased the dog,' we begin with By assigning these symbols to the empty set, we simply erase them when they are not needed. The remainder of our productions consists of the following: To derive the sentence 'Joe chased the dog,' we begin with to derive Using the production By assigning these symbo…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=219658 \begin{verbatim} j >, < noun > , < adv > , and < verb > . In some of our sentences we do not need < art > , < adjective >, < prep > , and < adv > . To solve this problem, we include the productions By assigning these symbols to the empty set, we simply erase them when they are not needed. The remainder of our productions consists of the following: To derive the sentence 'Joe chased the dog,' we begin with to derive Using the production By assigning these symbols to the empty set, we simply erase them when they are not needed. The remainder of our productions consists of the following: To derive the sentence 'Joe chased the dog,' we begin with to derive Using the production we derive By assigning these symbols to the empty set, we simply erase them…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=219699 \begin{verbatim} and < verb > . In some of our sentences we do not need < art > , < adjective >, < prep > , and < adv > . To solve this problem, we include the productions By assigning these symbols to the empty set, we simply erase them when they are not needed. The remainder of our productions consists of the following: To derive the sentence 'Joe chased the dog,' we begin with to derive Using the production we derive By assigning these symbols to the empty set, we simply erase them when they are not needed. The remainder of our productions consists of the following: To derive the sentence 'Joe chased the dog,' we begin with to derive Using the production we derive Using we derive By assigning these symbols to the empty set, we simply erase them when they are not needed. The remainder of our prod…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=219751 \begin{verbatim} not need < art > , < adjective >, < prep > , and < adv > . To solve this problem, we include the productions By assigning these symbols to the empty set, we simply erase them when they are not needed. The remainder of our productions consists of the following: To derive the sentence 'Joe chased the dog,' we begin with to derive Using the production we derive Using we derive By assigning these symbols to the empty set, we simply erase them when they are not needed. The remainder of our productions consists of the following: To derive the sentence 'Joe chased the dog,' we begin with to derive Using the production we derive Using we derive Repeati… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{gt;, \< prep \> , and \< adv \> . To solve this problem, we include the productions By assigning these symbols to the empty set, we simply erase them when they are not needed. The remainder of our productions consists of the following: By assigning these symbols to the empty set, we simply erase them when they are not needed. The remainder of our productions consists of the following: To derive the sentence 'Joe chased the dog,' we begin with to derive Using the production we derive Using we derive Repeati… \end{verbatim} ``` </details>
168. ph-842fc02beed11a2394a1 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) lve this problem, we include the productions By assigning these symbols to the empty set, we simply erase them when they are not needed. The remainder of our productions consists of the following: To derive the sentence 'Joe chased the dog,' we begin with to derive Using the production we derive Using we derive Repeating the process for the second < noun phrase … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{lve this problem, we include the productions By assigning these symbols to the empty set, we simply erase them when they are not needed. The remainder of our productions consists of the following: To derive the sentence '…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=219839 \begin{verbatim} lve this problem, we include the productions By assigning these symbols to the empty set, we simply erase them when they are not needed. The remainder of our productions consists of the following: To derive the sentence 'Joe chased the dog,' we begin with to derive Using the production we derive Using we derive Repeating the process for the second < noun phrase … \end{verbatim} ``` </details>
169. ph-174b853e1b6912c04a3e — automata/docling_md/AutomataTheory.md ### Plain (markdown context) he productions By assigning these symbols to the empty set, we simply erase them when they are not needed. The remainder of our productions consists of the following: To derive the sentence 'Joe chased the dog,' we begin with to derive Using the production we derive Using we derive Repeating the process for the second < noun phrase > , we derive By assigning these symbols to the empty set, we simply erase them when they are not needed. The remainder of our productions consists of the following: To derive the sentence 'Joe chased the dog,' we begin …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=219869 \begin{verbatim} he productions By assigning these symbols to the empty set, we simply erase them when they are not needed. The remainder of our productions consists of the following: To derive the sentence 'Joe chased the dog,' we begin with to derive Using the production we derive Using we derive Repeating the process for the second < noun phrase > , we derive To derive the sentence 'Joe chased the dog,' we begin with to derive Using the production we derive Using we derive Repeating the process for the second < noun phrase > , we derive Using we derive Using we derive Using we derive 'Joe ch… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{rase them when they are not needed. The remainder of our productions consists of the following: To derive the sentence 'Joe chased the dog,' we begin with to derive Using the production To derive the sentence 'Joe chased the dog,' we begin with to derive Using the production we derive Using we derive Repeating the process for the second < noun phrase > , we derive Using we derive Using we derive Using we derive 'Joe ch… \end{verbatim} ``` </details>
171. ph-e20c8c9635efcbfee5c4 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) nsists of the following: To derive the sentence 'Joe chased the dog,' we begin with to derive Using the production we derive Using we derive Repeating the process for the second < noun phrase > , we derive Using we derive Using we derive Using we derive 'Joe chased the dog.' To derive the sentence 'The fast horse leaped over the t… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{nsists of the following: To derive the sentence 'Joe chased the dog,' we begin with to derive Using the production we derive Using we …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=220042 \begin{verbatim} nsists of the following: To derive the sentence 'Joe chased the dog,' we begin with to derive Using the production we derive Using we derive Repeating the process for the second < noun phrase > , we derive Using we derive Using we derive Using we derive 'Joe chased the dog.' To derive the sentence 'The fast horse leaped over the t… \end{verbatim} ``` </details>
172. ph-ecbc59b1f439d4e2af84 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) ive Using the production we derive Using we derive Repeating the process for the second < noun phrase > , we derive Using we derive Using we derive Using we derive 'Joe chased the dog.' To derive the sentence 'The fast horse leaped over the tall fence,' we again begin with to derive Using the production we derive Using we derive Repeating the process for the second \< noun phrase \> , we derive Using the production we derive Using we derive Repeating the process for the second < noun phrase > , we derive Using we derive Using we derive Using we derive 'Joe chased the dog.' To derive the sentence 'The fast horse leaped over the tall fence,' we again begin with to derive we derive Using we derive Repeating the process for the second < noun phrase > , we derive Using we derive Using we derive Using we derive 'Joe chased the dog.' To derive the sentence 'The fast horse leaped over the tall fence,' we again begin with to derive … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{the production we derive Using we derive Repeating the process for the second \< noun phrase \> , we derive Using we derive Using …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=220235 \begin{verbatim} the production we derive Using we derive Repeating the process for the second < noun phrase > , we derive Using we derive Using we derive Using we derive 'Joe chased the dog.' To derive the sentence 'The fast horse leaped over the tall fence,' we again begin with to derive … \end{verbatim} ``` </details>
174. ph-fc0f65618a6669794d62 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) ot-decoded --> we derive Using we derive Repeating the process for the second < noun phrase > , we derive Using we derive Using we derive Using we derive 'Joe chased the dog.' To derive the sentence 'The fast horse leaped over the tall fence,' we again begin with to derive … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ot-decoded --> we derive Using we derive Repeating the process for the second \< noun phrase \> , we derive Using we derive Using we derive Using we derive Using we derive Repeating the process for the second < noun phrase > , we derive Using we derive Using we derive Using we derive 'Joe chased the dog.' To derive the sentence 'The fast horse leaped over the tall fence,' we again begin with to derive … \end{verbatim} ```
175. ph-6629614ec07fcd383c1e — automata/docling_md/AutomataTheory.md ### Plain (markdown context) - formula-not-decoded --> Using we derive Repeating the process for the second < noun phrase > , we derive Using we derive Using we derive Using we derive 'Joe chased the dog.' To derive the sentence 'The fast horse leaped over the tall fence,' we again begin with to derive … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{- formula-not-decoded --> Using we derive Repeating the process for the second \< noun phrase \> , we derive Using we derive Using we derive Using we derive 'J…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=220295 \begin{verbatim} - formula-not-decoded --> Using we derive Repeating the process for the second < noun phrase > , we derive Using we derive Using we derive Using we derive 'Joe chased the dog.' To derive the sentence 'The fast horse leaped over the tall fence,' we again begin with to derive … \end{verbatim} ``` </details>
176. ph-d619ba19f3fac914db8b — automata/docling_md/AutomataTheory.md ### Plain (markdown context) ng we derive Repeating the process for the second < noun phrase > , we derive Using we derive Using we derive Using we derive 'Joe chased the dog.' To derive the sentence 'The fast horse leaped over the tall fence,' we again begin with to derive Using the production Repeating the process for the second \< noun phrase \> , we derive Using we derive Using we derive Using we derive 'Joe chased the dog.' To derive…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=220325 \begin{verbatim} ng we derive Repeating the process for the second < noun phrase > , we derive Using we derive Using we derive Using we derive 'Joe chased the dog.' To derive the sentence 'The fast horse leaped over the tall fence,' we again begin with to derive Using the production Repeating the process for the second < noun phrase > , we derive Using we derive Using we derive Using we derive 'Joe chased the dog.' To derive the sentence 'The fast horse leaped over the tall fence,' we again begin with to derive Using the production we deri… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{-decoded --> Repeating the process for the second \< noun phrase \> , we derive Using we derive Using we derive Using we derive 'Joe chased the dog.' To derive the sentence 'The fast horse…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=220355 \begin{verbatim} -decoded --> Repeating the process for the second < noun phrase > , we derive Using we derive Using we derive Using we derive 'Joe chased the dog.' To derive the sentence 'The fast horse leaped over the tall fence,' we again begin with to derive Using the production we deri… \end{verbatim} ``` </details>
178. ph-f245fe7b8ae13c09b8cb — automata/docling_md/AutomataTheory.md ### Plain (markdown context) -decoded --> Repeating the process for the second < noun phrase > , we derive Using we derive Using we derive Using we derive 'Joe chased the dog.' To derive the sentence 'The fast horse leaped over the tall fence,' we again begin with to derive Using the production we derive Repeating the process for the second \< noun phrase \> , we derive Using we derive Using we derive Using we derive 'Joe chased the dog.' To derive the sentence 'The fast horse leaped over the tall fence,'…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=220385 \begin{verbatim} -decoded --> Repeating the process for the second < noun phrase > , we derive Using we derive Using we derive Using we derive 'Joe chased the dog.' To derive the sentence 'The fast horse leaped over the tall fence,' we again begin with to derive Using the production we derive Using we derive Using we derive Using we derive 'Joe chased the dog.' To derive the sentence 'The fast horse leaped over the tall fence,' we again begin with to derive Using the production we derive Using < art > → the < art > → The &l… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{lt; noun phrase \> , we derive Using we derive Using we derive Using we derive 'Joe chased the dog.' To derive the sentence 'The fast horse leaped over the tall fence,' we again begin with t…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=220437 \begin{verbatim} lt; noun phrase > , we derive Using we derive Using we derive Using we derive 'Joe chased the dog.' To derive the sentence 'The fast horse leaped over the tall fence,' we again begin with to derive Using the production we derive Using < art > → the < art > → The &l… \end{verbatim} ``` </details>
180. ph-5427a0b175d4748fcc2b — automata/docling_md/AutomataTheory.md ### Plain (markdown context) rmula-not-decoded --> Using we derive Using we derive Using we derive 'Joe chased the dog.' To derive the sentence 'The fast horse leaped over the tall fence,' we again begin with to derive Using the production we derive Using < art > → the < art > → The < adj > → fast < noun > → hors… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{rmula-not-decoded --> Using we derive Using we derive Using we derive 'Joe chased the dog.' To derive the sentence 'The fast horse leaped over the tall fence,' we again begin with to derive <!…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=220478 \begin{verbatim} rmula-not-decoded --> Using we derive Using we derive Using we derive 'Joe chased the dog.' To derive the sentence 'The fast horse leaped over the tall fence,' we again begin with to derive Using the production we derive Using < art > → the < art > → The < adj > → fast < noun > → hors… \end{verbatim} ``` </details>
181. ph-265b4f0f8d139f913871 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) ula-not-decoded --> Using the production we derive Using < art > → the < art > → The < adj > → fast < noun > → horse we derive The fast horse < verb p >< prep >< noun p > . Using we derive Using we derive Using we derive we derive Using The fast horse < adv >< verb >< prep >&lt… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ula-not-decoded --> Using the production we derive Using \< art \> → the…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=220744 \begin{verbatim} ula-not-decoded --> Using the production we derive Using < art > → the < art > → The < adj > → fast < noun > → horse we derive The fast horse < verb p >< prep >< noun p > . Using we derive Using we derive Using we derive we derive Using The fast horse < adv >< verb >< prep >&lt… \end{verbatim} ``` </details>
182. ph-a4162487467380e065ec — automata/docling_md/AutomataTheory.md ### Plain (markdown context) ula-not-decoded --> Using the production we derive Using < art > → the < art > → The < adj > → fast < noun > → horse we derive The fast horse < verb p >< prep >< noun p > . Using we derive Using we derive Using we derive we derive Using The fast horse < adv >< verb >< prep >< noun p >. Using the production we derive Using \< art \> → the \< art \> → The \< adj…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=220774 \begin{verbatim} ula-not-decoded --> Using the production we derive Using < art > → the < art > → The < adj > → fast < noun > → horse we derive The fast horse < verb p >< prep >< noun p > . Using we derive Using we derive Using we derive we derive Using The fast horse < adv >< verb >< prep >< noun p >. Using the production we derive Using < art > → the < art > → The < adj > → fast < noun > → horse we derive The fast horse < verb p >< prep >< noun p > . Using we derive Using we derive Using we derive we derive Using The fast horse < adv >< verb >< prep >< noun p >. The fast horse leaped < prep >< noun p >. The f… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{--> Using the production we derive Using \< art \> → the \< art \> → The \< adj \> → fast \< noun \> → horse we derive The fast horse \< verb p \>\< prep \>\< noun p …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=220879 \begin{verbatim} --> Using the production we derive Using < art > → the < art > → The < adj > → fast < noun > → horse we derive The fast horse < verb p >< prep >< noun p > . Using we derive Using we derive Using we derive we derive Using The fast horse < adv >< verb >< prep >< noun p >. The fast horse leaped < prep >< noun p >. The f… \end{verbatim} ``` </details>
184. ph-5c1f37332cd0f1a29dca — automata/docling_md/AutomataTheory.md ### Plain (markdown context) > we derive Using < art > → the < art > → The < adj > → fast < noun > → horse we derive The fast horse < verb p >< prep >< noun p > . Using we derive Using we derive Using we derive we derive Using The fast horse < adv >< verb >< prep >< noun p >. The fast horse leaped < prep >< noun p >. The fast horse leaped over < noun p > . Using the production Using \< art \> → the \< art \> → The \< adj \> → fast \< noun \> → horse we derive The fast horse \< verb p \>\< prep \>\< noun p \> . Using we derive Using we derive Using we derive we derive Using Using < art > → the < art > → The < adj > → fast < noun > → horse we derive The fast horse < verb p >< prep >< noun p > . Using we derive Using we derive Using we derive we derive Using The fast horse < adv >< verb >< prep >< noun p >. The fast horse leaped < prep >< noun p >. The fast horse leaped over < noun p > . Using the production The fast horse < adv >< verb >< prep >< noun p >. The fast horse leaped < prep >< noun p >. The fast horse leaped over < noun p > . Using the production The fast horse leaped over < art >< adj >< noun >. we derive The fas… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{\> → fast \< noun \> → horse we derive The fast horse \< verb p \>\< prep \>\< noun p \> . Using we derive Using we derive Using we derive we derive Using The fast horse \< adv \>\< verb \>\< p…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=221063 \begin{verbatim} > → fast < noun > → horse we derive The fast horse < verb p >< prep >< noun p > . Using we derive Using we derive Using we derive we derive Using The fast horse < adv >< verb >< prep >< noun p >. The fast horse leaped < prep >< noun p >. The fast horse leaped over < noun p > . Using the production The fast horse leaped over < art >< adj >< noun >. we derive The fas… \end{verbatim} ``` </details>
186. ph-87e5460520845892cc81 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) s A and B are called vertices or nodes . The vertex B is called the child of A. Note that a terminal at a vertex has no children. Such a vertex is called a leaf of the tree. The leaves of the tree, when read left to right, form the word generated by the tree. If A 0 → A 1 →··· → An forms a string of edges in the tree then there is a path of length n from A 0 to An. Example 4.5 In Example 4.1, we used productions to derive 3 + 2 + 4. To construct the tree, begin with the first production used ![Image](./AutomataTheory_artifacts/image_000170_31981663902e05177e805f2c… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{s A and B are called vertices or nodes . The vertex B is called the child of A. Note that a terminal at a vertex has no children. Such a vertex is called a leaf of the tree. The leaves of the tree, when read left to right, form the word generated by the tree. If A 0 → A 1 →··· →…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=222434 \begin{verbatim} s A and B are called vertices or nodes . The vertex B is called the child of A. Note that a terminal at a vertex has no children. Such a vertex is called a leaf of the tree. The leaves of the tree, when read left to right, form the word generated by the tree. If A 0 → A 1 →··· → An forms a string of edges in the tree then there is a path of length n from A 0 to An. Example 4.5 In Example 4.1, we used productions to derive 3 + 2 + 4. To construct the tree, begin with the first production used ![Image](./AutomataTheory_artifacts/image_000170_31981663902e05177e805f2c… \end{verbatim} ``` </details>
187. ph-d3afa52d446b321c11e0 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) + 2 + 4. To construct the tree, begin with the first production used ![Image](./AutomataTheory_artifacts/image_000170_31981663902e05177e805f2cb4d7f27383ad59d3f2154e51b02e31ceeed028f9.png) A + B Then use the corresponding tree ![Image](./AutomataTheory_artifacts/image_000171_f1043d9fe7637e2d30e412f9281cf31ccb07a0f9d1ab6f893a6215b29ba5b003.png) + B to form corresponding tree of the production to form the tree of the production to get the corresponding tree ![Image](./AutomataTheory_artifacts/image_000172_3c3e35a481f705225443a3d758dada607… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{+ 2 + 4. To construct the tree, begin with the first production used ![Image](./AutomataTheory\_artifacts/image\_000170\_31981663902e05177e805f2cb4d7f27383ad59d3f2154e51b02e31ceeed028f9.png) A + B Then use the corresponding tree ![Image](./AutomataTheor…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=222863 \begin{verbatim} + 2 + 4. To construct the tree, begin with the first production used ![Image](./AutomataTheory_artifacts/image_000170_31981663902e05177e805f2cb4d7f27383ad59d3f2154e51b02e31ceeed028f9.png) A + B Then use the corresponding tree ![Image](./AutomataTheory_artifacts/image_000171_f1043d9fe7637e2d30e412f9281cf31ccb07a0f9d1ab6f893a6215b29ba5b003.png) + B to form corresponding tree of the production to form the tree of the production to get the corresponding tree ![Image](./AutomataTheory_artifacts/image_000172_3c3e35a481f705225443a3d758dada607… \end{verbatim} ``` </details>
188. ph-a4f1892a35a08ef6cae8 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) 43d9fe7637e2d30e412f9281cf31ccb07a0f9d1ab6f893a6215b29ba5b003.png) + B to form corresponding tree of the production to form the tree of the production to get the corresponding tree ![Image](./AutomataTheory_artifacts/image_000172_3c3e35a481f705225443a3d758dada607e5ff67f0f5280358b7951a1168ae84e.png) + B Then use the corresponding tree in ![Image](./AutomataTheory_artifacts/image_000173_fd4940c895584fb4cd623e92ec76fe9a29c4d5b7992273cf0017919daae279af.png) A ![Image](./AutomataTheory_artifacts/image_000174_8f464c1aaa302adf8d0d32b6a0379ded… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{43d9fe7637e2d30e412f9281cf31ccb07a0f9d1ab6f893a6215b29ba5b003.png) + B to form corresponding tree of the production to form the tree of the production to get the corresponding tree ![Image](./AutomataTheory\_artifacts/image\_000172\_3c3e35a481f705225443…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=223177 \begin{verbatim} 43d9fe7637e2d30e412f9281cf31ccb07a0f9d1ab6f893a6215b29ba5b003.png) + B to form corresponding tree of the production to form the tree of the production to get the corresponding tree ![Image](./AutomataTheory_artifacts/image_000172_3c3e35a481f705225443a3d758dada607e5ff67f0f5280358b7951a1168ae84e.png) + B Then use the corresponding tree in ![Image](./AutomataTheory_artifacts/image_000173_fd4940c895584fb4cd623e92ec76fe9a29c4d5b7992273cf0017919daae279af.png) A ![Image](./AutomataTheory_artifacts/image_000174_8f464c1aaa302adf8d0d32b6a0379ded… \end{verbatim} ``` </details>
189. ph-9f9b2fe79af31d6abc0f — automata/docling_md/AutomataTheory.md ### Plain (markdown context) --> ![Image](./AutomataTheory_artifacts/image_000172_3c3e35a481f705225443a3d758dada607e5ff67f0f5280358b7951a1168ae84e.png) + B Then use the corresponding tree in ![Image](./AutomataTheory_artifacts/image_000173_fd4940c895584fb4cd623e92ec76fe9a29c4d5b7992273cf0017919daae279af.png) A ![Image](./AutomataTheory_artifacts/image_000174_8f464c1aaa302adf8d0d32b6a0379deddea07adea2a0e601e3f8088ebf7abd69.png) + Then use the corresponding trees of the next productions to form the parse tree ![Image](./AutomataTheory_artifacts/image_000175_d9d70e5d… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{--> ![Image](./AutomataTheory\_artifacts/image\_000172\_3c3e35a481f705225443a3d758dada607e5ff67f0f5280358b7951a1168ae84e.png) + B Then use the corresponding tree in ![Image](./AutomataTheory\_artifacts/image\_000173\_fd4940c895584fb4cd623e92ec76fe9a29c4d5b7992273cf0017919daae279af.png…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=223388 \begin{verbatim} --> ![Image](./AutomataTheory_artifacts/image_000172_3c3e35a481f705225443a3d758dada607e5ff67f0f5280358b7951a1168ae84e.png) + B Then use the corresponding tree in ![Image](./AutomataTheory_artifacts/image_000173_fd4940c895584fb4cd623e92ec76fe9a29c4d5b7992273cf0017919daae279af.png) A ![Image](./AutomataTheory_artifacts/image_000174_8f464c1aaa302adf8d0d32b6a0379deddea07adea2a0e601e3f8088ebf7abd69.png) + Then use the corresponding trees of the next productions to form the parse tree ![Image](./AutomataTheory_artifacts/image_000175_d9d70e5d… \end{verbatim} ``` </details>
190. ph-669f9e2f7f8bdf7ba1bf — automata/docling_md/AutomataTheory.md ### Plain (markdown context) 2273cf0017919daae279af.png) A ![Image](./AutomataTheory_artifacts/image_000174_8f464c1aaa302adf8d0d32b6a0379deddea07adea2a0e601e3f8088ebf7abd69.png) + Then use the corresponding trees of the next productions to form the parse tree ![Image](./AutomataTheory_artifacts/image_000175_d9d70e5d6b6249912379d04fd04d942bd2aca89439bd5420f1a3db3e0cc72a63.png) Example 4.6 In Example 4.3, to derive ((2 + 3) × (4 + 5)), we use the productions ## Therefore the parse tree is the tree ![Image](./AutomataTheory_artifacts/imag… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{2273cf0017919daae279af.png) A ![Image](./AutomataTheory\_artifacts/image\_000174\_8f464c1aaa302adf8d0d32b6a0379deddea07adea2a0e601e3f8088ebf7abd69.png) + Then use the corresponding trees of the next productions to form the p…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=223645 \begin{verbatim} 2273cf0017919daae279af.png) A ![Image](./AutomataTheory_artifacts/image_000174_8f464c1aaa302adf8d0d32b6a0379deddea07adea2a0e601e3f8088ebf7abd69.png) + Then use the corresponding trees of the next productions to form the parse tree ![Image](./AutomataTheory_artifacts/image_000175_d9d70e5d6b6249912379d04fd04d942bd2aca89439bd5420f1a3db3e0cc72a63.png) Example 4.6 In Example 4.3, to derive ((2 + 3) × (4 + 5)), we use the productions ## Therefore the parse tree is the tree ![Image](./AutomataTheory_artifacts/imag… \end{verbatim} ``` </details>
191. ph-58de98dc63c7f638c1ea — automata/docling_md/AutomataTheory.md ### Plain (markdown context) arse tree ![Image](./AutomataTheory_artifacts/image_000175_d9d70e5d6b6249912379d04fd04d942bd2aca89439bd5420f1a3db3e0cc72a63.png) Example 4.6 In Example 4.3, to derive ((2 + 3) × (4 + 5)), we use the productions ## Therefore the parse tree is the tree ![Image](./AutomataTheory_artifacts/image_000176_f517b05d44932b97f57aacbeaeb3471179ae3470c7481e5271996abcb82ee993.png) 4 Example 4.7 In Example 4.4, to derive the sentence 'Joe chased the dog,' using productions to get to get < art >< adj >< noun >< verb p >< pre… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{arse tree ![Image](./AutomataTheory\_artifacts/image\_000175\_d9d70e5d6b6249912379d04fd04d942bd2aca89439bd5420f1a3db3e0cc72a63.png) Example 4.6 In Example 4.3, to derive ((2 + 3) × (4 + 5)), we use the productions \#\# Therefore the parse tree is the tree…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=223931 \begin{verbatim} arse tree ![Image](./AutomataTheory_artifacts/image_000175_d9d70e5d6b6249912379d04fd04d942bd2aca89439bd5420f1a3db3e0cc72a63.png) Example 4.6 In Example 4.3, to derive ((2 + 3) × (4 + 5)), we use the productions ## Therefore the parse tree is the tree ![Image](./AutomataTheory_artifacts/image_000176_f517b05d44932b97f57aacbeaeb3471179ae3470c7481e5271996abcb82ee993.png) 4 Example 4.7 In Example 4.4, to derive the sentence 'Joe chased the dog,' using productions to get to get < art >< adj >< noun >< verb p >< pre… \end{verbatim} ``` </details>
192. ph-81e6a0f0fcc16272e9df — automata/docling_md/AutomataTheory.md ### Plain (markdown context) 4.6 In Example 4.3, to derive ((2 + 3) × (4 + 5)), we use the productions ## Therefore the parse tree is the tree ![Image](./AutomataTheory_artifacts/image_000176_f517b05d44932b97f57aacbeaeb3471179ae3470c7481e5271996abcb82ee993.png) 4 Example 4.7 In Example 4.4, to derive the sentence 'Joe chased the dog,' using productions to get to get < art >< adj >< noun >< verb p >< prep >< noun p > Using \#\# Therefore the parse tree is the tree ![Image](./AutomataTheory\_artifacts/image\_000176\_f517b05d44932b97f57aacbeaeb3471179ae3470c7481e5271996abcb82ee993.png) 4 Example 4.7 In …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=224070 \begin{verbatim} 4.6 In Example 4.3, to derive ((2 + 3) × (4 + 5)), we use the productions ## Therefore the parse tree is the tree ![Image](./AutomataTheory_artifacts/image_000176_f517b05d44932b97f57aacbeaeb3471179ae3470c7481e5271996abcb82ee993.png) 4 Example 4.7 In Example 4.4, to derive the sentence 'Joe chased the dog,' using productions to get to get < art >< adj >< noun >< verb p >< prep >< noun p > Using ## Therefore the parse tree is the tree ![Image](./AutomataTheory_artifacts/image_000176_f517b05d44932b97f57aacbeaeb3471179ae3470c7481e5271996abcb82ee993.png) 4 Example 4.7 In Example 4.4, to derive the sentence 'Joe chased the dog,' using productions to get to get < art >< adj >< noun >< verb p >< prep >< noun p > Using \#\# Therefore the parse tree is the tree ![Image](./AutomataTheory\_artifacts/image\_000176\_f517b05d44932b97f57aacbeaeb3471179ae3470c7481e5271996abcb82ee993.png) 4 Example 4.7 In Example 4.4, to derive the sen…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=224100 \begin{verbatim} ((2 + 3) × (4 + 5)), we use the productions ## Therefore the parse tree is the tree ![Image](./AutomataTheory_artifacts/image_000176_f517b05d44932b97f57aacbeaeb3471179ae3470c7481e5271996abcb82ee993.png) 4 Example 4.7 In Example 4.4, to derive the sentence 'Joe chased the dog,' using productions to get to get < art >< adj >< noun >< verb p >< prep >< noun p > Using ## Therefore the parse tree is the tree ![Image](./AutomataTheory_artifacts/image_000176_f517b05d44932b97f57aacbeaeb3471179ae3470c7481e5271996abcb82ee993.png) 4 Example 4.7 In Example 4.4, to derive the sentence 'Joe chased the dog,' using productions to get to get < art >< adj >< noun >< verb p >< prep >< noun p > Using \#\# Therefore the parse tree is the tree ![Image](./AutomataTheory\_artifacts/image\_000176\_f517b05d44932b97f57aacbeaeb3471179ae3470c7481e5271996abcb82ee993.png) 4 Example 4.7 In Example 4.4, to derive the sentence 'Joe chased the dog,' us…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=224130 \begin{verbatim} e productions ## Therefore the parse tree is the tree ![Image](./AutomataTheory_artifacts/image_000176_f517b05d44932b97f57aacbeaeb3471179ae3470c7481e5271996abcb82ee993.png) 4 Example 4.7 In Example 4.4, to derive the sentence 'Joe chased the dog,' using productions to get to get < art >< adj >< noun >< verb p >< prep >< noun p > Using ## Therefore the parse tree is the tree ![Image](./AutomataTheory_artifacts/image_000176_f517b05d44932b97f57aacbeaeb3471179ae3470c7481e5271996abcb82ee993.png) 4 Example 4.7 In Example 4.4, to derive the sentence 'Joe chased the dog,' using productions to get to get < art >< adj >< noun >< verb p >< prep >< noun p > Using we get Again using and … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{t-decoded --> \#\# Therefore the parse tree is the tree ![Image](./AutomataTheory\_artifacts/image\_000176\_f517b05d44932b97f57aacbeaeb3471179ae3470c7481e5271996abcb82ee993.png) 4 Example 4.7 In Example 4.4, to derive the sentence 'Joe chased the dog,' using productions ## Therefore the parse tree is the tree ![Image](./AutomataTheory_artifacts/image_000176_f517b05d44932b97f57aacbeaeb3471179ae3470c7481e5271996abcb82ee993.png) 4 Example 4.7 In Example 4.4, to derive the sentence 'Joe chased the dog,' using productions to get to get < art >< adj >< noun >< verb p >< prep >< noun p > Using we get Again using and … \end{verbatim} ``` </details>
196. ph-4e0fa5420cfd13f8ab85 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) e parse tree is the tree ![Image](./AutomataTheory_artifacts/image_000176_f517b05d44932b97f57aacbeaeb3471179ae3470c7481e5271996abcb82ee993.png) 4 Example 4.7 In Example 4.4, to derive the sentence 'Joe chased the dog,' using productions to get to get < art >< adj >< noun >< verb p >< prep >< noun p > Using we get Again using and we get Using to get to get…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=224190 \begin{verbatim} e parse tree is the tree ![Image](./AutomataTheory_artifacts/image_000176_f517b05d44932b97f57aacbeaeb3471179ae3470c7481e5271996abcb82ee993.png) 4 Example 4.7 In Example 4.4, to derive the sentence 'Joe chased the dog,' using productions to get to get < art >< adj >< noun >< verb p >< prep >< noun p > Using we get Again using and we get Using to get to get < art >< adj >< noun >< verb p >< prep >< noun p > Using we get Again using and we get Using we get Using we … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{age](./AutomataTheory\_artifacts/image\_000176\_f517b05d44932b97f57aacbeaeb3471179ae3470c7481e5271996abcb82ee993.png) 4 Example 4.7 In Example 4.4, to derive the sentence 'Joe chased the dog,' using productions to get to get \< art \>\< adj \>\&l…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=224220 \begin{verbatim} age](./AutomataTheory_artifacts/image_000176_f517b05d44932b97f57aacbeaeb3471179ae3470c7481e5271996abcb82ee993.png) 4 Example 4.7 In Example 4.4, to derive the sentence 'Joe chased the dog,' using productions to get to get < art >< adj >< noun >< verb p >< prep >< noun p > Using we get Again using and we get Using we get Using we … \end{verbatim} ``` </details>
198. ph-2d057dd03765cacc56a5 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) eb3471179ae3470c7481e5271996abcb82ee993.png) 4 Example 4.7 In Example 4.4, to derive the sentence 'Joe chased the dog,' using productions to get to get < art >< adj >< noun >< verb p >< prep >< noun p > Using we get Again using and we get Using we get Using we get Using < noun > → Joe < noun > → dog < verb > → c… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{eb3471179ae3470c7481e5271996abcb82ee993.png) 4 Example 4.7 In Example 4.4, to derive the sentence 'Joe chased the dog,' using productions to get to get \< art \>\< adj \>\< noun \>\< verb p \>\< prep \>\< noun p \> Using to get to get < art >< adj >< noun >< verb p >< prep >< noun p > Using we get Again using and we get Using we get Using we get Using < noun > → Joe < noun > → dog < verb > → c… \end{verbatim} ```
199. ph-c8f485d9512b4f6e73b5 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) we get Again using and we get Using we get Using we get Using < noun > → Joe < noun > → dog < verb > → chased art → the we have the correspondence tree for 'Joe chased the dog.' Example 4.8 In Example 4.4, to derive the sentence 'The large dog leaped over the old fence,' we use productions ![Image](./AutomataTheory_artifacts/image_000177_9702dc98d1cc7f8e99244cb0a3aeff1c3c9174cfb87e07d4422bfd6d71aa573d.png) we get Again using and we get Using we get Using we get Using \< noun \> → Joe \< noun \> → dog \< verb \> → chased art → the we have the correspondence tree for 'Joe chased the dog.' Example 4.8 In Example 4.…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=224718 \begin{verbatim} we get Again using and we get Using we get Using we get Using < noun > → Joe < noun > → dog < verb > → chased art → the we have the correspondence tree for 'Joe chased the dog.' Example 4.8 In Example 4.4, to derive the sentence 'The large dog leaped over the old fence,' we use productions ![Image](./AutomataTheory_artifacts/image_000177_9702dc98d1cc7f8e99244cb0a3aeff1c3c9174cfb87e07d4422bfd6d71aa573d.png) we get Using we get Using < noun > → Joe < noun > → dog < verb > → chased art → the we have the correspondence tree for 'Joe chased the dog.' Example 4.8 In Example 4.4, to derive the sentence 'The large dog leaped over the old fence,' we use productions ![Image](./AutomataTheory_artifacts/image_000177_9702dc98d1cc7f8e99244cb0a3aeff1c3c9174cfb87e07d4422bfd6d71aa573d.png) ## Thus the parse tree is … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{we get Again using and we get Using we get Using we get Using \< noun \> → Joe \< noun \> → dog \< verb \> → chased art → the we have the correspondence tree for 'Joe chased the dog.' Example 4.8 In Example 4.4, to derive the sentence 'Th…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=224748 \begin{verbatim} we get Again using and we get Using we get Using we get Using < noun > → Joe < noun > → dog < verb > → chased art → the we have the correspondence tree for 'Joe chased the dog.' Example 4.8 In Example 4.4, to derive the sentence 'The large dog leaped over the old fence,' we use productions ![Image](./AutomataTheory_artifacts/image_000177_9702dc98d1cc7f8e99244cb0a3aeff1c3c9174cfb87e07d4422bfd6d71aa573d.png) ## Thus the parse tree is … \end{verbatim} ``` </details>
201. ph-5b4aaf16990ffc63f42f — automata/docling_md/AutomataTheory.md ### Plain (markdown context) get Using we get Using we get Using < noun > → Joe < noun > → dog < verb > → chased art → the we have the correspondence tree for 'Joe chased the dog.' Example 4.8 In Example 4.4, to derive the sentence 'The large dog leaped over the old fence,' we use productions ![Image](./AutomataTheory_artifacts/image_000177_9702dc98d1cc7f8e99244cb0a3aeff1c3c9174cfb87e07d4422bfd6d71aa573d.png) ## Thus the parse tree is ![Image](./AutomataTheory_… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{get Using we get Using we get Using \< noun \> → Joe \< noun \> → dog \< verb \> → chased art → the we have the correspondence tree for 'Joe chased the dog.' Example 4.8 In Example 4.4, to derive the sentence 'The large dog leaped over th…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=224778 \begin{verbatim} get Using we get Using we get Using < noun > → Joe < noun > → dog < verb > → chased art → the we have the correspondence tree for 'Joe chased the dog.' Example 4.8 In Example 4.4, to derive the sentence 'The large dog leaped over the old fence,' we use productions ![Image](./AutomataTheory_artifacts/image_000177_9702dc98d1cc7f8e99244cb0a3aeff1c3c9174cfb87e07d4422bfd6d71aa573d.png) ## Thus the parse tree is ![Image](./AutomataTheory_… \end{verbatim} ``` </details>
202. ph-336e495c6b0f16e8b0c1 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) ecoded --> we get Using we get Using < noun > → Joe < noun > → dog < verb > → chased art → the we have the correspondence tree for 'Joe chased the dog.' Example 4.8 In Example 4.4, to derive the sentence 'The large dog leaped over the old fence,' we use productions ![Image](./AutomataTheory_artifacts/image_000177_9702dc98d1cc7f8e99244cb0a3aeff1c3c9174cfb87e07d4422bfd6d71aa573d.png) ## Thus the parse tree is ![Image](./AutomataTheory_artifacts/image_000178_cc06d… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ecoded --> we get Using we get Using \< noun \> → Joe \< noun \> → dog \< verb \> → chased art → the we have the correspondence tree for 'Joe chased the dog.' Example 4.8 In Example 4.4, to derive the sentence 'The large dog leaped over the old fence,' we use product…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=224808 \begin{verbatim} ecoded --> we get Using we get Using < noun > → Joe < noun > → dog < verb > → chased art → the we have the correspondence tree for 'Joe chased the dog.' Example 4.8 In Example 4.4, to derive the sentence 'The large dog leaped over the old fence,' we use productions ![Image](./AutomataTheory_artifacts/image_000177_9702dc98d1cc7f8e99244cb0a3aeff1c3c9174cfb87e07d4422bfd6d71aa573d.png) ## Thus the parse tree is ![Image](./AutomataTheory_artifacts/image_000178_cc06d… \end{verbatim} ``` </details>
203. ph-ae21815d51239540757e — automata/docling_md/AutomataTheory.md ### Plain (markdown context) can only be used when a is on the left-hand side of A and b is on the right-hand side. It therefore cannot be used whenever A appears and so it is dependent on the context in which A appears. Such a grammar is called a context-sensitive grammar . In the following examples, we consider context-free grammars which generate more abstract languages: Example 4.9 Let /Gamma1 = ( N , /Sigma1 , S , P ) be the grammar defined by N = { S , A , B } , /Sigma1 = { a , b } and P be the set of productions Using the production S → AB , we derive AB . Next using the productions A … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{can only be used when a is on the left-hand side of A and b is on the right-hand side. It therefore cannot be used whenever A appears and so it is dependent on the context in which A appears. Such a grammar is called a context-sensitive grammar . In the following examples, we co…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=225964 \begin{verbatim} can only be used when a is on the left-hand side of A and b is on the right-hand side. It therefore cannot be used whenever A appears and so it is dependent on the context in which A appears. Such a grammar is called a context-sensitive grammar . In the following examples, we consider context-free grammars which generate more abstract languages: Example 4.9 Let /Gamma1 = ( N , /Sigma1 , S , P ) be the grammar defined by N = { S , A , B } , /Sigma1 = { a , b } and P be the set of productions Using the production S → AB , we derive AB . Next using the productions A … \end{verbatim} ``` </details>
204. ph-16e70d0e7e6792a9769d — automata/docling_md/AutomataTheory.md ### Plain (markdown context) the context in which A appears. Such a grammar is called a context-sensitive grammar . In the following examples, we consider context-free grammars which generate more abstract languages: Example 4.9 Let /Gamma1 = ( N , /Sigma1 , S , P ) be the grammar defined by N = { S , A , B } , /Sigma1 = { a , b } and P be the set of productions Using the production S → AB , we derive AB . Next using the productions A → a and B → λ , we derive a . If we use the productions in order, we derive b . We can also generate aabbb , aaaa , aaab , and bbbb… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{the context in which A appears. Such a grammar is called a context-sensitive grammar . In the following examples, we consider context-free grammars which generate more abstract languages: Example 4.9 Let /Gamma1 = ( N , /Sigma1 , S , P ) be the grammar defined by N = \{ S , A , B…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=226125 \begin{verbatim} the context in which A appears. Such a grammar is called a context-sensitive grammar . In the following examples, we consider context-free grammars which generate more abstract languages: Example 4.9 Let /Gamma1 = ( N , /Sigma1 , S , P ) be the grammar defined by N = { S , A , B } , /Sigma1 = { a , b } and P be the set of productions Using the production S → AB , we derive AB . Next using the productions A → a and B → λ , we derive a . If we use the productions in order, we derive b . We can also generate aabbb , aaaa , aaab , and bbbb… \end{verbatim} ``` </details>
205. ph-c5f874f3a9baaa8f8233 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) e derive AB . Next using the productions A → a and B → λ , we derive a . If we use the productions in order, we derive b . We can also generate aabbb , aaaa , aaab , and bbbbb . In fact, we can generate a m b n for all nonnegative integers m , n . Hence the expression for the language generated by /Gamma1 is a ∗ b ∗ . Example 4.10 Let /Gamma1 ′ = ( N , /Sigma1 , S , P ) be the grammar defined by N = { S , A } , /Sigma1 = { a , b } and P be the set of productions Using the productions S → aAb and A → λ we derive ab . Using the productio… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{e derive AB . Next using the productions A → a and B → λ , we derive a . If we use the productions in order, we derive b . We can also generate aabbb , aaaa , aaab , and bbbbb . In fact, we can generate a m b n for all nonnegative integers m , n . He…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=226525 \begin{verbatim} e derive AB . Next using the productions A → a and B → λ , we derive a . If we use the productions in order, we derive b . We can also generate aabbb , aaaa , aaab , and bbbbb . In fact, we can generate a m b n for all nonnegative integers m , n . Hence the expression for the language generated by /Gamma1 is a ∗ b ∗ . Example 4.10 Let /Gamma1 ′ = ( N , /Sigma1 , S , P ) be the grammar defined by N = { S , A } , /Sigma1 = { a , b } and P be the set of productions Using the productions S → aAb and A → λ we derive ab . Using the productio… \end{verbatim} ``` </details>
206. ph-da61def48abffcb7057b — automata/docling_md/AutomataTheory.md ### Plain (markdown context) mula-not-decoded --> in order, we derive b . We can also generate aabbb , aaaa , aaab , and bbbbb . In fact, we can generate a m b n for all nonnegative integers m , n . Hence the expression for the language generated by /Gamma1 is a ∗ b ∗ . Example 4.10 Let /Gamma1 ′ = ( N , /Sigma1 , S , P ) be the grammar defined by N = { S , A } , /Sigma1 = { a , b } and P be the set of productions Using the productions S → aAb and A → λ we derive ab . Using the productions in order, we derive aabb or a 2 b 2 . Using the productions in order, we derive b . We can also generate aabbb , aaaa , aaab , and bbbbb . In fact, we can generate a m b n for all nonnegative integers m , n . Hence the expression for the language generated by /Gamma1 is a ∗ b ∗ . Example 4.10 Let /Gamma1 ′ = ( N , /S…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=226633 \begin{verbatim} mula-not-decoded --> in order, we derive b . We can also generate aabbb , aaaa , aaab , and bbbbb . In fact, we can generate a m b n for all nonnegative integers m , n . Hence the expression for the language generated by /Gamma1 is a ∗ b ∗ . Example 4.10 Let /Gamma1 ′ = ( N , /Sigma1 , S , P ) be the grammar defined by N = { S , A } , /Sigma1 = { a , b } and P be the set of productions Using the productions S → aAb and A → λ we derive ab . Using the productions in order, we derive aabb or a 2 b 2 . Using the productions Using the productions S → aAb and A → λ we derive ab . Using the productions in order, we derive aabb or a 2 b 2 . Using the productions in order, we derive aaabbb or a 3 b 3 . It is easily seen that the language … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{d bbbbb . In fact, we can generate a m b n for all nonnegative integers m , n . Hence the expression for the language generated by /Gamma1 is a ∗ b ∗ . Example 4.10 Let /Gamma1 ′ = ( N , /Sigma1 , S , P ) be the grammar defined by N = \{ S , A \} , /Sigma1 = \{ a , b \} and P be the…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=226724 \begin{verbatim} d bbbbb . In fact, we can generate a m b n for all nonnegative integers m , n . Hence the expression for the language generated by /Gamma1 is a ∗ b ∗ . Example 4.10 Let /Gamma1 ′ = ( N , /Sigma1 , S , P ) be the grammar defined by N = { S , A } , /Sigma1 = { a , b } and P be the set of productions Using the productions S → aAb and A → λ we derive ab . Using the productions in order, we derive aabb or a 2 b 2 . Using the productions in order, we derive aaabbb or a 3 b 3 . It is easily seen that the language … \end{verbatim} ``` </details>
208. ph-1960cca8c3e377136bfe — automata/docling_md/AutomataTheory.md ### Plain (markdown context) -> in order, we derive aabb or a 2 b 2 . Using the productions in order, we derive aaabbb or a 3 b 3 . It is easily seen that the language generated by /Gamma1 ′ is { a n b n : n is a positive integer } . Note that this is not the same as a ∗ b ∗ since this would also include a m b n where m and n are not equal. Example 4.11 Let /Gamma1 ′′ = ( N , /Sigma1 , S , P ) be the grammar defined by N = { S , A , B } , /Sigma1 = { a , b } and P be the set of productions It can be shown that the expression for the language generated by /Gamma1 ′… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{-> in order, we derive aabb or a 2 b 2 . Using the productions in order, we derive aaabbb or a 3 b 3 . It is easily seen that the language generated by /Gamma1 ′ is \{ a n b n : n is a positive integer \} . Note that this is not the same as a ∗ b ∗ sin…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=227159 \begin{verbatim} -> in order, we derive aabb or a 2 b 2 . Using the productions in order, we derive aaabbb or a 3 b 3 . It is easily seen that the language generated by /Gamma1 ′ is { a n b n : n is a positive integer } . Note that this is not the same as a ∗ b ∗ since this would also include a m b n where m and n are not equal. Example 4.11 Let /Gamma1 ′′ = ( N , /Sigma1 , S , P ) be the grammar defined by N = { S , A , B } , /Sigma1 = { a , b } and P be the set of productions It can be shown that the expression for the language generated by /Gamma1 ′… \end{verbatim} ``` </details>
209. ph-591e27328ced69215d9a — automata/docling_md/AutomataTheory.md ### Plain (markdown context) 4.5 A context-free grammar /Gamma1 = ( N , /Sigma1 , S , P ) is called a regular grammar if every production p ∈ P has the form n → w where w is the empty word λ or the string w contains at most one nonterminal symbol and it occurs at the end of the string if at all. Therefore w could be of the form aacA , ab , λ or bA , where a , b , and c are terminals and A is a nonterminal. However, w could not be of the form aAb , aAB , or Aa . The production n → abcA could be replaced by the productions Also it is possible w could contain no terminal and one nonterminal so w… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{4.5 A context-free grammar /Gamma1 = ( N , /Sigma1 , S , P ) is called a regular grammar if every production p ∈ P has the form n → w where w is the empty word λ or the string w contains at most one nonterminal symbol and it occurs at the end of the string if at all. Therefore w…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=228436 \begin{verbatim} 4.5 A context-free grammar /Gamma1 = ( N , /Sigma1 , S , P ) is called a regular grammar if every production p ∈ P has the form n → w where w is the empty word λ or the string w contains at most one nonterminal symbol and it occurs at the end of the string if at all. Therefore w could be of the form aacA , ab , λ or bA , where a , b , and c are terminals and A is a nonterminal. However, w could not be of the form aAb , aAB , or Aa . The production n → abcA could be replaced by the productions Also it is possible w could contain no terminal and one nonterminal so w… \end{verbatim} ``` </details>
210. ph-3056c7d040a50c6cbd8b — automata/docling_md/AutomataTheory.md ### Plain (markdown context) where a , b , and c are terminals and A is a nonterminal. However, w could not be of the form aAb , aAB , or Aa . The production n → abcA could be replaced by the productions Also it is possible w could contain no terminal and one nonterminal so we have B → C , but if this is followed by C → t D , where t is a terminal, then we can combine the two productions to get B → t D . Hence it is no restriction to require each production to be one of the following forms: where A , B , and C are nonterminal elements, and a and b are terminal ele… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{where a , b , and c are terminals and A is a nonterminal. However, w could not be of the form aAb , aAB , or Aa . The production n → abcA could be replaced by the productions Also it is possible w could contain no terminal and one nonterminal so we h…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=228760 \begin{verbatim} where a , b , and c are terminals and A is a nonterminal. However, w could not be of the form aAb , aAB , or Aa . The production n → abcA could be replaced by the productions Also it is possible w could contain no terminal and one nonterminal so we have B → C , but if this is followed by C → t D , where t is a terminal, then we can combine the two productions to get B → t D . Hence it is no restriction to require each production to be one of the following forms: where A , B , and C are nonterminal elements, and a and b are terminal ele… \end{verbatim} ``` </details>
211. ph-b39593dfd9217d4a0de3 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) the string w has the form xY, x or λ where x ∈ /Sigma1 and Y ∈ N. Theorem 4.1 A language is generated by a linear regular grammar if and only if it is generated by a regular grammar. Proof Obviously every language that is generated by a linear regular grammar is generated by a regular grammar. To show every regular grammar is generated by a linear regular grammar, we divide the proof into two parts. We first show the language of a regular grammar can be generated by productions of the forms where A , B , C , and D are nonterminals and a , b are terminals. Let /Gam… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{the string w has the form xY, x or λ where x ∈ /Sigma1 and Y ∈ N. Theorem 4.1 A language is generated by a linear regular grammar if and only if it is generated by a regular grammar. Proof Obviously every language that is generated by a linear regular grammar is generated by a r…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=229592 \begin{verbatim} the string w has the form xY, x or λ where x ∈ /Sigma1 and Y ∈ N. Theorem 4.1 A language is generated by a linear regular grammar if and only if it is generated by a regular grammar. Proof Obviously every language that is generated by a linear regular grammar is generated by a regular grammar. To show every regular grammar is generated by a linear regular grammar, we divide the proof into two parts. We first show the language of a regular grammar can be generated by productions of the forms where A , B , C , and D are nonterminals and a , b are terminals. Let /Gam… \end{verbatim} ``` </details>
212. ph-fdeb21dfdb8431739d67 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) n -1 B . So any word of L will be created by the grammar /Gamma1 ′ . Conversely if A ⇒ ∗ a 1 a 2 a 3 . . . an -1 B is formed by productions A → a 1 A 1 , A 1 → a 2 A 2 , . . . , An -1 → an -2 An -2 , An → an -1 B , then there must be a production A → a 1 a 2 a 3 . . . an -1 B in /Gamma1 since the symbols A 1 , A 2 , . . . , An are symbols which appear only in forming A → a 1 a 2 a 3 . . . an -1 B . Hence we can now assume that a regular grammar can be formed using only productions of the form where A , B , C , and D are nonterminals and a , b are terminals. We wan… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{n -1 B . So any word of L will be created by the grammar /Gamma1 ′ . Conversely if A ⇒ ∗ a 1 a 2 a 3 . . . an -1 B is formed by productions A → a 1 A 1 , A 1 → a 2 A 2 , . . . , An -1 → an -2 An -2 , An → an -1 B , then there must be a production A → a 1 a 2 a 3 . . . an -1 B in…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=230672 \begin{verbatim} n -1 B . So any word of L will be created by the grammar /Gamma1 ′ . Conversely if A ⇒ ∗ a 1 a 2 a 3 . . . an -1 B is formed by productions A → a 1 A 1 , A 1 → a 2 A 2 , . . . , An -1 → an -2 An -2 , An → an -1 B , then there must be a production A → a 1 a 2 a 3 . . . an -1 B in /Gamma1 since the symbols A 1 , A 2 , . . . , An are symbols which appear only in forming A → a 1 a 2 a 3 . . . an -1 B . Hence we can now assume that a regular grammar can be formed using only productions of the form where A , B , C , and D are nonterminals and a , b are terminals. We wan… \end{verbatim} ``` </details>
213. ph-5b079f147ade23b1ee5f — automata/docling_md/AutomataTheory.md ### Plain (markdown context) where A , B , C , and D are nonterminals and a , b are terminals. We want to show that we can form a regular grammar without productions of the form C → D where C and D are both nonterminals. Call this a 1-production. Let /Gamma1 be a regular grammar formed using the productions above and L be the language generated by /Gamma1 . Assume that we have productions of the form above. Let /Gamma1 ′′ be the grammar with all 1-productions deleted and insert the production occurred in L . If where A , B , C , and D are nonterminals and a , b are terminals. We want to show that we can form a regular grammar without productions of the form C → D where C and D are both nonterminals. Call this a 1-production. Let /Gamma1 be a regular grammar…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=231172 \begin{verbatim} where A , B , C , and D are nonterminals and a , b are terminals. We want to show that we can form a regular grammar without productions of the form C → D where C and D are both nonterminals. Call this a 1-production. Let /Gamma1 be a regular grammar formed using the productions above and L be the language generated by /Gamma1 . Assume that we have productions of the form above. Let /Gamma1 ′′ be the grammar with all 1-productions deleted and insert the production occurred in L . If occurred in L . If occurred in L , insert t… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{where A , B , C , and D are nonterminals and a , b are terminals. We want to show that we can form a regular grammar without productions of the form C → D where C and D are both nonterminals. Call this a 1-production. Let /Gamma1 be a regular grammar formed using the productions…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=231202 \begin{verbatim} where A , B , C , and D are nonterminals and a , b are terminals. We want to show that we can form a regular grammar without productions of the form C → D where C and D are both nonterminals. Call this a 1-production. Let /Gamma1 be a regular grammar formed using the productions above and L be the language generated by /Gamma1 . Assume that we have productions of the form above. Let /Gamma1 ′′ be the grammar with all 1-productions deleted and insert the production occurred in L . If occurred in L , insert t… \end{verbatim} ``` </details>
215. ph-8f1f01a612228d7ac6d7 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) are terminals. We want to show that we can form a regular grammar without productions of the form C → D where C and D are both nonterminals. Call this a 1-production. Let /Gamma1 be a regular grammar formed using the productions above and L be the language generated by /Gamma1 . Assume that we have productions of the form above. Let /Gamma1 ′′ be the grammar with all 1-productions deleted and insert the production occurred in L . If occurred in L , insert the production A 1 → b in L ′′ . If occurred in L . If occurred in L , insert the production A 1 → b in L ′′ . If occurred in L . If occurred in L , insert the production A 1 → b in L ′′ . If occurred in L , insert the production A 1 → λ in L ′′ . Let L ′′ be the langua… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{e form C → D where C and D are both nonterminals. Call this a 1-production. Let /Gamma1 be a regular grammar formed using the productions above and L be the language generated by /Gamma1 . Assume that we have productions of the form above. Let /Gamma1 ′′ be the grammar with all …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=231344 \begin{verbatim} e form C → D where C and D are both nonterminals. Call this a 1-production. Let /Gamma1 be a regular grammar formed using the productions above and L be the language generated by /Gamma1 . Assume that we have productions of the form above. Let /Gamma1 ′′ be the grammar with all 1-productions deleted and insert the production occurred in L . If occurred in L , insert the production A 1 → b in L ′′ . If occurred in L , insert the production A 1 → λ in L ′′ . Let L ′′ be the langua… \end{verbatim} ``` </details>
217. ph-10153b4b35cb0221293d — automata/docling_md/AutomataTheory.md ### Plain (markdown context) adding letters to words rather than removing them. Suppose that we consider the string that has been read rather than the string left to read. In the example above, at state s 1, we have read a . At state s 2 we have read ab . At state s 3 we have read abb , and at state s 4 we have read abbc . Thus at each state we are adding a letter. Consider the grammar /Gamma1 = ( N , /Sigma1 , s 0 , P ), where N = { s 0 , s 1 , s 2 , s 3 , s 4 } , /Sigma1 = { a , b , c } , and P is the set of productions where we have a production s 4 → λ only if s 4 is a terminal state. It … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{adding letters to words rather than removing them. Suppose that we consider the string that has been read rather than the string left to read. In the example above, at state s 1, we have read a . At state s 2 we have read ab . At state s 3 we have read abb , and at state s 4 we …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=234050 \begin{verbatim} adding letters to words rather than removing them. Suppose that we consider the string that has been read rather than the string left to read. In the example above, at state s 1, we have read a . At state s 2 we have read ab . At state s 3 we have read abb , and at state s 4 we have read abbc . Thus at each state we are adding a letter. Consider the grammar /Gamma1 = ( N , /Sigma1 , s 0 , P ), where N = { s 0 , s 1 , s 2 , s 3 , s 4 } , /Sigma1 = { a , b , c } , and P is the set of productions where we have a production s 4 → λ only if s 4 is a terminal state. It … \end{verbatim} ``` </details>
218. ph-d0ec663151135a0c3700 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) productions B → at and t → λ . Obviously this does not change the language of the grammar. Let M = ( /Sigma1 , Q , s 0 , T , F ) be the automaton in which Q is the set of nonterminals together with the additional nonterminal t , s 0 = S . The set F is defined by F ( a , A ) = B if and only if A → aB is in P . The state B ∈ T if B → λ . Example 4.14 Let /Gamma1 = ( N , /Sigma1 , S , P ) be the grammar defined by N = { S , A , B , C } , /Sigma1 = { a , b , c } , and P be the set of productions The corresponding automaton is ![Image](./AutomataTheory_artifacts/image_… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{productions B → at and t → λ . Obviously this does not change the language of the grammar. Let M = ( /Sigma1 , Q , s 0 , T , F ) be the automaton in which Q is the set of nonterminals together with the additional nonterminal t , s 0 = S . The set F is defined by F ( a , A ) = B …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=238912 \begin{verbatim} productions B → at and t → λ . Obviously this does not change the language of the grammar. Let M = ( /Sigma1 , Q , s 0 , T , F ) be the automaton in which Q is the set of nonterminals together with the additional nonterminal t , s 0 = S . The set F is defined by F ( a , A ) = B if and only if A → aB is in P . The state B ∈ T if B → λ . Example 4.14 Let /Gamma1 = ( N , /Sigma1 , S , P ) be the grammar defined by N = { S , A , B , C } , /Sigma1 = { a , b , c } , and P be the set of productions The corresponding automaton is ![Image](./AutomataTheory_artifacts/image_… \end{verbatim} ``` </details>
219. ph-901c28f19b29bfc6b965 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) B → λ . Example 4.14 Let /Gamma1 = ( N , /Sigma1 , S , P ) be the grammar defined by N = { S , A , B , C } , /Sigma1 = { a , b , c } , and P be the set of productions The corresponding automaton is ![Image](./AutomataTheory_artifacts/image_000181_ac32653a06fc360c3b9dc744f85b0a235f681dfcc5c31676a83b1622c8ac2b74.png) Example 4.15 Let /Gamma1 = ( N , T , S , P ) be the grammar defined by N = { S , A , B , C } , T = { a , b , c } , and P be the set of productions The corresponding automaton is ![Image](./AutomataTheory_artifacts/image_0001… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{B → λ . Example 4.14 Let /Gamma1 = ( N , /Sigma1 , S , P ) be the grammar defined by N = \{ S , A , B , C \} , /Sigma1 = \{ a , b , c \} , and P be the set of productions The corresponding automaton is ![Image](./AutomataTheory\_artifacts/image\_000181\_ac3…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=239242 \begin{verbatim} B → λ . Example 4.14 Let /Gamma1 = ( N , /Sigma1 , S , P ) be the grammar defined by N = { S , A , B , C } , /Sigma1 = { a , b , c } , and P be the set of productions The corresponding automaton is ![Image](./AutomataTheory_artifacts/image_000181_ac32653a06fc360c3b9dc744f85b0a235f681dfcc5c31676a83b1622c8ac2b74.png) Example 4.15 Let /Gamma1 = ( N , T , S , P ) be the grammar defined by N = { S , A , B , C } , T = { a , b , c } , and P be the set of productions The corresponding automaton is ![Image](./AutomataTheory_artifacts/image_0001… \end{verbatim} ``` </details>
220. ph-2e8e9aa6026712f6eb43 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) ammar in Example 4.9, construct a parse tree for abbb . - (2) Using the grammar in Example 4.10, construct a parse tree for aaabbb . - (3) Using the grammar in Example 4.11, construct a parse tree for babaab . - (4) In Example 4.4, derive the statement 'The cowboy rode slowly into the sunset' and construct the correspondence parse tree. - (5) Find the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B } , T = { a , b } and the set of productions P given by - (6) Find the language generated by the grammar /Gamma1 = ( N , T , S … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ammar in Example 4.9, construct a parse tree for abbb . - (2) Using the grammar in Example 4.10, construct a parse tree for aaabbb . - (3) Using the grammar in Example 4.11, construct a parse tree for babaab . - (4) In Example 4.4, derive the statement 'The cowboy rode slowly in…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=241739 \begin{verbatim} ammar in Example 4.9, construct a parse tree for abbb . - (2) Using the grammar in Example 4.10, construct a parse tree for aaabbb . - (3) Using the grammar in Example 4.11, construct a parse tree for babaab . - (4) In Example 4.4, derive the statement 'The cowboy rode slowly into the sunset' and construct the correspondence parse tree. - (5) Find the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B } , T = { a , b } and the set of productions P given by - (6) Find the language generated by the grammar /Gamma1 = ( N , T , S … \end{verbatim} ``` </details>
221. ph-ae0c2db378ed28139f4b — automata/docling_md/AutomataTheory.md ### Plain (markdown context) tree for babaab . - (4) In Example 4.4, derive the statement 'The cowboy rode slowly into the sunset' and construct the correspondence parse tree. - (5) Find the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B } , T = { a , b } and the set of productions P given by - (6) Find the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B } , T = { a , b } and the set of productions P given by - (7) Find the language generated by the grammar /Gamma1 = ( N , T , S , P… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{tree for babaab . - (4) In Example 4.4, derive the statement 'The cowboy rode slowly into the sunset' and construct the correspondence parse tree. - (5) Find the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = \{ S , A , B \} , T = \{ a , b \} and the se…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=241930 \begin{verbatim} tree for babaab . - (4) In Example 4.4, derive the statement 'The cowboy rode slowly into the sunset' and construct the correspondence parse tree. - (5) Find the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B } , T = { a , b } and the set of productions P given by - (6) Find the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B } , T = { a , b } and the set of productions P given by - (7) Find the language generated by the grammar /Gamma1 = ( N , T , S , P… \end{verbatim} ``` </details>
222. ph-5e9fe99682dce3d24574 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) ammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B } , T = { a , b } and the set of productions P given by - (6) Find the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B } , T = { a , b } and the set of productions P given by - (7) Find the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B } , T = { a , b } and the set of productions P given by - (8) Find the language generated by the grammar /Gamma1 = ( N , T , S , P … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ammar /Gamma1 = ( N , T , S , P ) defined by N = \{ S , A , B \} , T = \{ a , b \} and the set of productions P given by - (6) Find the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = \{ S , A , B \} , T = \{ a , b \} and the set…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=242121 \begin{verbatim} ammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B } , T = { a , b } and the set of productions P given by - (6) Find the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B } , T = { a , b } and the set of productions P given by - (7) Find the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B } , T = { a , b } and the set of productions P given by - (8) Find the language generated by the grammar /Gamma1 = ( N , T , S , P … \end{verbatim} ``` </details>
223. ph-0fc1055f62c461517f2f — automata/docling_md/AutomataTheory.md ### Plain (markdown context) r /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B } , T = { a , b } and the set of productions P given by - (7) Find the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B } , T = { a , b } and the set of productions P given by - (8) Find the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B , C } , T = { a , b } and the set of productions P given by - (9) Find the grammar which generates the language ww r where w is a strin… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{r /Gamma1 = ( N , T , S , P ) defined by N = \{ S , A , B \} , T = \{ a , b \} and the set of productions P given by - (7) Find the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = \{ S , A , B \} , T = \{ a , b \} and the set of …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=242316 \begin{verbatim} r /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B } , T = { a , b } and the set of productions P given by - (7) Find the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B } , T = { a , b } and the set of productions P given by - (8) Find the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B , C } , T = { a , b } and the set of productions P given by - (9) Find the grammar which generates the language ww r where w is a strin… \end{verbatim} ``` </details>
224. ph-24f06ac775af71cbc593 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) 22) Construct a grammar which generates the language expressed by ((aa ∗ b) ∨ bb ∗ a)ac ∗ . - (21) Construct a grammar which generates the language expressed by (a ∨ b) ∗ (aa ∨ bb)(a ∨ b) ∗ . - (23) Construct a grammar to generate arithmetic expressions for positive integers less than ten in prefix notation. - (24) Find an automaton which accepts the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B } , T = { a , b } and the set of productions P given by - (25) Find an automaton which accepts the language generated by the gra… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{22) Construct a grammar which generates the language expressed by ((aa ∗ b) ∨ bb ∗ a)ac ∗ . - (21) Construct a grammar which generates the language expressed by (a ∨ b) ∗ (aa ∨ bb)(a ∨ b) ∗ . - (23) Construct a grammar to generate arithmetic expressions for positive integers les…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=244104 \begin{verbatim} 22) Construct a grammar which generates the language expressed by ((aa ∗ b) ∨ bb ∗ a)ac ∗ . - (21) Construct a grammar which generates the language expressed by (a ∨ b) ∗ (aa ∨ bb)(a ∨ b) ∗ . - (23) Construct a grammar to generate arithmetic expressions for positive integers less than ten in prefix notation. - (24) Find an automaton which accepts the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B } , T = { a , b } and the set of productions P given by - (25) Find an automaton which accepts the language generated by the gra… \end{verbatim} ``` </details>
225. ph-0d362652614dc5efb2d7 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) to generate arithmetic expressions for positive integers less than ten in prefix notation. - (24) Find an automaton which accepts the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B } , T = { a , b } and the set of productions P given by - (25) Find an automaton which accepts the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B } , T = { a , b } and the set of productions P given by - (26) Find an automaton which accepts the language generated by the gramm… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{to generate arithmetic expressions for positive integers less than ten in prefix notation. - (24) Find an automaton which accepts the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = \{ S , A , B \} , T = \{ a , b \} and the set of productions P given by …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=244323 \begin{verbatim} to generate arithmetic expressions for positive integers less than ten in prefix notation. - (24) Find an automaton which accepts the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B } , T = { a , b } and the set of productions P given by - (25) Find an automaton which accepts the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B } , T = { a , b } and the set of productions P given by - (26) Find an automaton which accepts the language generated by the gramm… \end{verbatim} ``` </details>
226. ph-d7ceeb5ade227e719455 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) } , T = { a , b } and the set of productions P given by - (25) Find an automaton which accepts the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B } , T = { a , b } and the set of productions P given by - (26) Find an automaton which accepts the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B , C } , T = { a , b } and the set of productions P given by - (27) Find an automaton which accepts the language generated by the grammar… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{\} , T = \{ a , b \} and the set of productions P given by - (25) Find an automaton which accepts the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = \{ S , A , B \} , T = \{ a , b \} and the set of productions P given by - (25) Find an automaton which accepts the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B } , T = { a , b } and the set of productions P given by - (26) Find an automaton which accepts the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B , C } , T = { a , b } and the set of productions P given by - (27) Find an automaton which accepts the language generated by the grammar… \end{verbatim} ```
227. ph-f7f39d19e52239d8359f — automata/docling_md/AutomataTheory.md ### Plain (markdown context) T = { a , b } and the set of productions P given by - (26) Find an automaton which accepts the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B , C } , T = { a , b } and the set of productions P given by - (27) Find an automaton which accepts the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B , C } , T = { a , b } and the set of productions P given by - (28) Find an automaton which accepts the language generated by the grammar… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{T = \{ a , b \} and the set of productions P given by - (26) Find an automaton which accepts the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = \{ S , A , B , C \} , T = \{ a , b \} and the set of productions P given by - (26) Find an automaton which accepts the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B , C } , T = { a , b } and the set of productions P given by - (27) Find an automaton which accepts the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B , C } , T = { a , b } and the set of productions P given by - (28) Find an automaton which accepts the language generated by the grammar… \end{verbatim} ```
228. ph-6c48d09231165cd60444 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) T = { a , b } and the set of productions P given by - (27) Find an automaton which accepts the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B , C } , T = { a , b } and the set of productions P given by - (28) Find an automaton which accepts the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B , C } , T = { a , b } and the set of productions P given by - (29) Find an automaton which accepts the language generated by the grammar… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{T = \{ a , b \} and the set of productions P given by - (27) Find an automaton which accepts the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = \{ S , A , B , C \} , T = \{ a , b \} and the set of productions P given by - (27) Find an automaton which accepts the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B , C } , T = { a , b } and the set of productions P given by - (28) Find an automaton which accepts the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B , C } , T = { a , b } and the set of productions P given by - (29) Find an automaton which accepts the language generated by the grammar… \end{verbatim} ```
229. ph-569fb05f72e95248f0d3 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) T = { a , b } and the set of productions P given by - (28) Find an automaton which accepts the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B , C } , T = { a , b } and the set of productions P given by - (29) Find an automaton which accepts the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B , C } , T = { a , b } and the set of productions P given by - (30) Construct a grammar which generates the language accepted by the auto… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{T = \{ a , b \} and the set of productions P given by - (28) Find an automaton which accepts the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = \{ S , A , B , C \} , T = \{ a , b \} and the set of productions P given by - (28) Find an automaton which accepts the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B , C } , T = { a , b } and the set of productions P given by - (29) Find an automaton which accepts the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B , C } , T = { a , b } and the set of productions P given by - (30) Construct a grammar which generates the language accepted by the auto… \end{verbatim} ```
230. ph-0e0df5bd94c417780713 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) n - (36) Construct a grammar which generates the language accepted by the automaton ![Image](./AutomataTheory_artifacts/image_000188_5ba4c1145345b9e4f95b7318fabe020415ce2e81fc410d3e76b4c5fcf05435e5.png) ![Image](./AutomataTheory_artifacts/image_000189_509a5bc5dfb461de92a1976ff376d17f9cac1f8cef51044298d0e63707d471c8.png) ## 4.2 Chomsky normal form and Greibach normal form Definition 4.8 A context-free grammar /Gamma1 is in Chomsky normal form if each of its productions is either of the form where A, B, and C are nonterminals and a is a … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{n - (36) Construct a grammar which generates the language accepted by the automaton ![Image](./AutomataTheory\_artifacts/image\_000188\_5ba4c1145345b9e4f95b7318fabe020415ce2e81fc410d3e76b4c5fcf05435e5.png) ![Image](./AutomataTheory\_artifacts/image\_000189\_509a5bc5dfb461de92a1976ff37…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=246844 \begin{verbatim} n - (36) Construct a grammar which generates the language accepted by the automaton ![Image](./AutomataTheory_artifacts/image_000188_5ba4c1145345b9e4f95b7318fabe020415ce2e81fc410d3e76b4c5fcf05435e5.png) ![Image](./AutomataTheory_artifacts/image_000189_509a5bc5dfb461de92a1976ff376d17f9cac1f8cef51044298d0e63707d471c8.png) ## 4.2 Chomsky normal form and Greibach normal form Definition 4.8 A context-free grammar /Gamma1 is in Chomsky normal form if each of its productions is either of the form where A, B, and C are nonterminals and a is a … \end{verbatim} ``` </details>
231. ph-98427d4d828a8c7ca1e1 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) hich generates the language accepted by the automaton ![Image](./AutomataTheory_artifacts/image_000188_5ba4c1145345b9e4f95b7318fabe020415ce2e81fc410d3e76b4c5fcf05435e5.png) ![Image](./AutomataTheory_artifacts/image_000189_509a5bc5dfb461de92a1976ff376d17f9cac1f8cef51044298d0e63707d471c8.png) ## 4.2 Chomsky normal form and Greibach normal form Definition 4.8 A context-free grammar /Gamma1 is in Chomsky normal form if each of its productions is either of the form where A, B, and C are nonterminals and a is a terminal. Definition 4.9 A con… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{hich generates the language accepted by the automaton ![Image](./AutomataTheory\_artifacts/image\_000188\_5ba4c1145345b9e4f95b7318fabe020415ce2e81fc410d3e76b4c5fcf05435e5.png) ![Image](./AutomataTheory\_artifacts/image\_000189\_509a5bc5dfb461de92a1976ff376d17f9cac1f8cef51044298d0e6370…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=246874 \begin{verbatim} hich generates the language accepted by the automaton ![Image](./AutomataTheory_artifacts/image_000188_5ba4c1145345b9e4f95b7318fabe020415ce2e81fc410d3e76b4c5fcf05435e5.png) ![Image](./AutomataTheory_artifacts/image_000189_509a5bc5dfb461de92a1976ff376d17f9cac1f8cef51044298d0e63707d471c8.png) ## 4.2 Chomsky normal form and Greibach normal form Definition 4.8 A context-free grammar /Gamma1 is in Chomsky normal form if each of its productions is either of the form where A, B, and C are nonterminals and a is a terminal. Definition 4.9 A con… \end{verbatim} ``` </details>
232. ph-3fe6a8e56b01c65f32f4 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) tifacts/image_000189_509a5bc5dfb461de92a1976ff376d17f9cac1f8cef51044298d0e63707d471c8.png) ## 4.2 Chomsky normal form and Greibach normal form Definition 4.8 A context-free grammar /Gamma1 is in Chomsky normal form if each of its productions is either of the form where A, B, and C are nonterminals and a is a terminal. Definition 4.9 A context-free grammar /Gamma1 is in Greibach normal form if each of its productions is of the form where a is a terminal and W is a possibly empty string of nonterminals. Wesha… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{tifacts/image\_000189\_509a5bc5dfb461de92a1976ff376d17f9cac1f8cef51044298d0e63707d471c8.png) \#\# 4.2 Chomsky normal form and Greibach normal form Definition 4.8 A context-free grammar /Gamma1 is in Chomsky normal form if each of its productions is either of the form where A, B, and C are nonterminals and a is a terminal. Definition 4.9 A context-free grammar /Gamma1 is in Greibach normal form if each of its productions is of the form where a is a terminal and W is a possibly empty string of nonterminals. Wesha… \end{verbatim} ```
233. ph-c600e23dc627fe1143df — automata/docling_md/AutomataTheory.md ### Plain (markdown context) ∈ ( N ∪ T ) ∗ , be a derivation in /Gamma1 with n steps, then W can be expressed as W 1 W 2 where U ⇒ ∗ W 1 , V ⇒ ∗ W 2 are derivations in /Gamma1 , both containing at most n steps. Proof The proof of this lemma uses induction on the number of steps in the production. Assume there is one step. Then only one nonterminal is replaced using a production. Assume it is the production A → w B w ′ . Either A is in the string U or is in the string V . Without loss of generality assume it is in U , so so letting W 1 … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{∈ ( N ∪ T ) ∗ , be a derivation in /Gamma1 with n steps, then W can be expressed as W 1 W 2 where U ⇒ ∗ W 1 , V ⇒ ∗ W 2 are derivations in /Gamma1 , both containing at most n steps. Proof The proof of this lemma uses induction on the number of steps in the production. Assume the…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=248551 \begin{verbatim} ∈ ( N ∪ T ) ∗ , be a derivation in /Gamma1 with n steps, then W can be expressed as W 1 W 2 where U ⇒ ∗ W 1 , V ⇒ ∗ W 2 are derivations in /Gamma1 , both containing at most n steps. Proof The proof of this lemma uses induction on the number of steps in the production. Assume there is one step. Then only one nonterminal is replaced using a production. Assume it is the production A → w B w ′ . Either A is in the string U or is in the string V . Without loss of generality assume it is in U , so so letting W 1 … \end{verbatim} ``` </details>
234. ph-fe11cb9c15aabd945aa7 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) on in /Gamma1 with n steps, then W can be expressed as W 1 W 2 where U ⇒ ∗ W 1 , V ⇒ ∗ W 2 are derivations in /Gamma1 , both containing at most n steps. Proof The proof of this lemma uses induction on the number of steps in the production. Assume there is one step. Then only one nonterminal is replaced using a production. Assume it is the production A → w B w ′ . Either A is in the string U or is in the string V . Without loss of generality assume it is in U , so so letting W 1 = U ′ and W 2 = V we are done… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{on in /Gamma1 with n steps, then W can be expressed as W 1 W 2 where U ⇒ ∗ W 1 , V ⇒ ∗ W 2 are derivations in /Gamma1 , both containing at most n steps. Proof The proof of this lemma uses induction on the number of steps in the production. Assume there is one step. Then only one…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=248581 \begin{verbatim} on in /Gamma1 with n steps, then W can be expressed as W 1 W 2 where U ⇒ ∗ W 1 , V ⇒ ∗ W 2 are derivations in /Gamma1 , both containing at most n steps. Proof The proof of this lemma uses induction on the number of steps in the production. Assume there is one step. Then only one nonterminal is replaced using a production. Assume it is the production A → w B w ′ . Either A is in the string U or is in the string V . Without loss of generality assume it is in U , so so letting W 1 = U ′ and W 2 = V we are done… \end{verbatim} ``` </details>
235. ph-a39fb4b6d3fb0983775f — automata/docling_md/AutomataTheory.md ### Plain (markdown context) en W can be expressed as W 1 W 2 where U ⇒ ∗ W 1 , V ⇒ ∗ W 2 are derivations in /Gamma1 , both containing at most n steps. Proof The proof of this lemma uses induction on the number of steps in the production. Assume there is one step. Then only one nonterminal is replaced using a production. Assume it is the production A → w B w ′ . Either A is in the string U or is in the string V . Without loss of generality assume it is in U , so so letting W 1 = U ′ and W 2 = V we are done. Assume the lemma is true for… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{en W can be expressed as W 1 W 2 where U ⇒ ∗ W 1 , V ⇒ ∗ W 2 are derivations in /Gamma1 , both containing at most n steps. Proof The proof of this lemma uses induction on the number of steps in the production. Assume there is one step. Then only one nonterminal is replaced using…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=248611 \begin{verbatim} en W can be expressed as W 1 W 2 where U ⇒ ∗ W 1 , V ⇒ ∗ W 2 are derivations in /Gamma1 , both containing at most n steps. Proof The proof of this lemma uses induction on the number of steps in the production. Assume there is one step. Then only one nonterminal is replaced using a production. Assume it is the production A → w B w ′ . Either A is in the string U or is in the string V . Without loss of generality assume it is in U , so so letting W 1 = U ′ and W 2 = V we are done. Assume the lemma is true for… \end{verbatim} ``` </details>
236. ph-b52bed5d25452cbacded — automata/docling_md/AutomataTheory.md ### Plain (markdown context) W 1 = U ′ and W 2 = V we are done. Assume the lemma is true for all derivations with less than k steps. Assume UV ⇒ ∗ W contains k steps . As above assume the first step is UV ⇒ U ′ V where U ⇒ U ′ . Note that U ′ V ⇒ ∗ W uses only k -1steps. By induction there are derivations U ′ ⇒ ∗ W 1 , V ⇒ ∗ W 2 containing at most k steps. Therefore U ⇒ U ′ , U ′ ⇒ ∗ W 1 , V ⇒ ∗ W 2 are the required derivations. /square One of the results of this lemma is that we can get from UV ⇒ ∗ W by the derivations where W = W 1 W 2 since if X α ⇒ X β is a derivation, then so are X α V ⇒… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{W 1 = U ′ and W 2 = V we are done. Assume the lemma is true for all derivations with less than k steps. Assume UV ⇒ ∗ W contains k steps . As above assume the first step is UV ⇒ U ′ V where U ⇒ U ′ . Note that U ′ V ⇒ ∗ W uses only k -1steps. By induction there are derivations U…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=249152 \begin{verbatim} W 1 = U ′ and W 2 = V we are done. Assume the lemma is true for all derivations with less than k steps. Assume UV ⇒ ∗ W contains k steps . As above assume the first step is UV ⇒ U ′ V where U ⇒ U ′ . Note that U ′ V ⇒ ∗ W uses only k -1steps. By induction there are derivations U ′ ⇒ ∗ W 1 , V ⇒ ∗ W 2 containing at most k steps. Therefore U ⇒ U ′ , U ′ ⇒ ∗ W 1 , V ⇒ ∗ W 2 are the required derivations. /square One of the results of this lemma is that we can get from UV ⇒ ∗ W by the derivations where W = W 1 W 2 since if X α ⇒ X β is a derivation, then so are X α V ⇒… \end{verbatim} ``` </details>
237. ph-90430218142c47ef516d — automata/docling_md/AutomataTheory.md ### Plain (markdown context) n /Gamma1 can be replaced with the original productions in /Gamma1 used to define it. To show L ⊆ L ′ , let w ∈ L . Using induction on the number of steps in the derivation, we show that if S ⇒ ∗ w using productions in P , then S ⇒ ∗ w using productions in P ′ . If n = 1 then S ⇒ w is obviously a production in P ′ since w /negationslash= λ . Assume n = k and S ⇒ ∗ w is a derivation in /Gamma1 containing k steps. Let S ⇒ A 1 A 2 A 3 . . . Am be the first derivation where Ai ∈ N ∪ T . Therefore By Lemma 4.2, there exist derivations Ai ⇒ ∗ w i in /Gamma1 , for 1 ≤ i … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{n /Gamma1 can be replaced with the original productions in /Gamma1 used to define it. To show L ⊆ L ′ , let w ∈ L . Using induction on the number of steps in the derivation, we show that if S ⇒ ∗ w using productions in P , then S ⇒ ∗ w using productions in P ′ . If n = 1 then S …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=253087 \begin{verbatim} n /Gamma1 can be replaced with the original productions in /Gamma1 used to define it. To show L ⊆ L ′ , let w ∈ L . Using induction on the number of steps in the derivation, we show that if S ⇒ ∗ w using productions in P , then S ⇒ ∗ w using productions in P ′ . If n = 1 then S ⇒ w is obviously a production in P ′ since w /negationslash= λ . Assume n = k and S ⇒ ∗ w is a derivation in /Gamma1 containing k steps. Let S ⇒ A 1 A 2 A 3 . . . Am be the first derivation where Ai ∈ N ∪ T . Therefore By Lemma 4.2, there exist derivations Ai ⇒ ∗ w i in /Gamma1 , for 1 ≤ i … \end{verbatim} ``` </details>
238. ph-bf65346d5869efbe018d — automata/docling_md/AutomataTheory.md ### Plain (markdown context) , then remove the trivial productions and include A 1 → B . Conversely, assume S ⇒ ∗ w occurs in /Gamma1 . We use induction on the number of trivial derivations to show that there is a derivation S ⇒ ∗ w in /Gamma1 ′ . Obviously if there is no trivial production then the derivation is in /Gamma1 ′ . Assume there are k trivial productions in the derivation. Assume that the derivation is a leftmost derivation of w . Assume S ⇒ ∗ w has the form By construction, /Gamma1 ( L ′ ) ⊆ /Gamma1 ( L ). where A 1 → A 2 → A 3 →··· → Am is the last sequence of trivial production… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{, then remove the trivial productions and include A 1 → B . Conversely, assume S ⇒ ∗ w occurs in /Gamma1 . We use induction on the number of trivial derivations to show that there is a derivation S ⇒ ∗ w in /Gamma1 ′ . Obviously if there is no trivial production then the derivat…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=255752 \begin{verbatim} , then remove the trivial productions and include A 1 → B . Conversely, assume S ⇒ ∗ w occurs in /Gamma1 . We use induction on the number of trivial derivations to show that there is a derivation S ⇒ ∗ w in /Gamma1 ′ . Obviously if there is no trivial production then the derivation is in /Gamma1 ′ . Assume there are k trivial productions in the derivation. Assume that the derivation is a leftmost derivation of w . Assume S ⇒ ∗ w has the form By construction, /Gamma1 ( L ′ ) ⊆ /Gamma1 ( L ). where A 1 → A 2 → A 3 →··· → Am is the last sequence of trivial production… \end{verbatim} ``` </details>
239. ph-a75b87f2e97d6dae1217 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) amma1 ′ . Obviously if there is no trivial production then the derivation is in /Gamma1 ′ . Assume there are k trivial productions in the derivation. Assume that the derivation is a leftmost derivation of w . Assume S ⇒ ∗ w has the form By construction, /Gamma1 ( L ′ ) ⊆ /Gamma1 ( L ). where A 1 → A 2 → A 3 →··· → Am is the last sequence of trivial productions in the derivation, and Am → w ′ . Then there are derivations V 1 ⇒ ∗ w 1 , V 2 ⇒ ∗ w 2 in /Gamma1 , and has less trivial productions and all productions are in /Gamma1 ∪ /Gamma1 … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{amma1 ′ . Obviously if there is no trivial production then the derivation is in /Gamma1 ′ . Assume there are k trivial productions in the derivation. Assume that the derivation is a leftmost derivation of w . Assume S ⇒ ∗ w has the form By construction, /Gamma1 ( L ′ ) ⊆ /Gamma1…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=255963 \begin{verbatim} amma1 ′ . Obviously if there is no trivial production then the derivation is in /Gamma1 ′ . Assume there are k trivial productions in the derivation. Assume that the derivation is a leftmost derivation of w . Assume S ⇒ ∗ w has the form By construction, /Gamma1 ( L ′ ) ⊆ /Gamma1 ( L ). where A 1 → A 2 → A 3 →··· → Am is the last sequence of trivial productions in the derivation, and Am → w ′ . Then there are derivations V 1 ⇒ ∗ w 1 , V 2 ⇒ ∗ w 2 in /Gamma1 , and has less trivial productions and all productions are in /Gamma1 ∪ /Gamma1 … \end{verbatim} ``` </details>
240. ph-5530b1c20d13be33c6f0 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) , . . . , Am with A ′ 1 , A ′ 2 , A ′ 3 , . . . , A ′ m where A ′ i = Ai if Ai is a nonterminal and A ′ i = Xai if Ai is a terminal. Thus if we have V 1 a 1 V 2 a 2 V 3 a 3 . . . VnanVn + 1 where Vi ∈ N ∗ and ai is a terminal, replace it with V 1 Xa 1 V 2 Xa 2 V 3 Xa 3 . . . Vn Xan Vn + 1 and add productions Xai → ai for 1 ≤ i ≤ n . Let /Gamma1 ′ = ( N , T , S , P ′ ) be the new grammar formed. We need to show that /Gamma1 ( L ′ ) = /Gamma1 ( L ). Clearly /Gamma1 ( L ) ⊆ /Gamma1 ( L ′ ) since in /Gamma1 can be replaced by in /Gamma1 ′ … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{, . . . , Am with A ′ 1 , A ′ 2 , A ′ 3 , . . . , A ′ m where A ′ i = Ai if Ai is a nonterminal and A ′ i = Xai if Ai is a terminal. Thus if we have V 1 a 1 V 2 a 2 V 3 a 3 . . . VnanVn + 1 where Vi ∈ N ∗ and ai is a terminal, replace it with V 1 Xa 1 V 2 Xa 2 V 3 Xa 3 . . . Vn …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=257506 \begin{verbatim} , . . . , Am with A ′ 1 , A ′ 2 , A ′ 3 , . . . , A ′ m where A ′ i = Ai if Ai is a nonterminal and A ′ i = Xai if Ai is a terminal. Thus if we have V 1 a 1 V 2 a 2 V 3 a 3 . . . VnanVn + 1 where Vi ∈ N ∗ and ai is a terminal, replace it with V 1 Xa 1 V 2 Xa 2 V 3 Xa 3 . . . Vn Xan Vn + 1 and add productions Xai → ai for 1 ≤ i ≤ n . Let /Gamma1 ′ = ( N , T , S , P ′ ) be the new grammar formed. We need to show that /Gamma1 ( L ′ ) = /Gamma1 ( L ). Clearly /Gamma1 ( L ) ⊆ /Gamma1 ( L ′ ) since in /Gamma1 can be replaced by in /Gamma1 ′ … \end{verbatim} ``` </details>
241. ph-7c8b69c82f924da3ac23 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) e A ′ i = Ai if Ai is a nonterminal and A ′ i = Xai if Ai is a terminal. Thus if we have V 1 a 1 V 2 a 2 V 3 a 3 . . . VnanVn + 1 where Vi ∈ N ∗ and ai is a terminal, replace it with V 1 Xa 1 V 2 Xa 2 V 3 Xa 3 . . . Vn Xan Vn + 1 and add productions Xai → ai for 1 ≤ i ≤ n . Let /Gamma1 ′ = ( N , T , S , P ′ ) be the new grammar formed. We need to show that /Gamma1 ( L ′ ) = /Gamma1 ( L ). Clearly /Gamma1 ( L ) ⊆ /Gamma1 ( L ′ ) since in /Gamma1 can be replaced by in /Gamma1 ′ . Conversely assume that in the derivation in /Gamma1 can be replaced by in /Gamma1 ′ . Conversely assume that in the derivation in /Gamma1 can be replaced by in /Gamma1 ′ . Conversely assume that in the derivation where U ⇒ ∗ w 1 and A ⇒ ∗ w and Vi ⇒ v i are productions in /Gamma1 where <!-… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{V 1 a 1 V 2 a 2 V 3 a 3 . . . VnanVn + 1 where Vi ∈ N ∗ and ai is a terminal, replace it with V 1 Xa 1 V 2 Xa 2 V 3 Xa 3 . . . Vn Xan Vn + 1 and add productions Xai → ai for 1 ≤ i ≤ n . Let /Gamma1 ′ = ( N , T , S , P ′ ) be the new grammar formed. We need to show that /Gamma1 (…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=257655 \begin{verbatim} V 1 a 1 V 2 a 2 V 3 a 3 . . . VnanVn + 1 where Vi ∈ N ∗ and ai is a terminal, replace it with V 1 Xa 1 V 2 Xa 2 V 3 Xa 3 . . . Vn Xan Vn + 1 and add productions Xai → ai for 1 ≤ i ≤ n . Let /Gamma1 ′ = ( N , T , S , P ′ ) be the new grammar formed. We need to show that /Gamma1 ( L ′ ) = /Gamma1 ( L ). Clearly /Gamma1 ( L ) ⊆ /Gamma1 ( L ′ ) since in /Gamma1 can be replaced by in /Gamma1 ′ . Conversely assume that in the derivation where U ⇒ ∗ w 1 and A ⇒ ∗ w and Vi ⇒ v i are productions in /Gamma1 where <!-… \end{verbatim} ``` </details>
243. ph-2b5ac42ef7bcb953a6a4 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) 2 Xa 2 V 3 Xa 3 . . . Vn Xan Vn + 1 and add productions Xai → ai for 1 ≤ i ≤ n . Let /Gamma1 ′ = ( N , T , S , P ′ ) be the new grammar formed. We need to show that /Gamma1 ( L ′ ) = /Gamma1 ( L ). Clearly /Gamma1 ( L ) ⊆ /Gamma1 ( L ′ ) since in /Gamma1 can be replaced by in /Gamma1 ′ . Conversely assume that in the derivation where U ⇒ ∗ w 1 and A ⇒ ∗ w and Vi ⇒ v i are productions in /Gamma1 where is a production which is in /Gamma1 ′ and not in /Gamma1 and the derivation is … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{2 Xa 2 V 3 Xa 3 . . . Vn Xan Vn + 1 and add productions Xai → ai for 1 ≤ i ≤ n . Let /Gamma1 ′ = ( N , T , S , P ′ ) be the new grammar formed. We need to show that /Gamma1 ( L ′ ) = /Gamma1 ( L ). Clearly /Gamma1 ( L ) ⊆ /Gamma1 ( L ′ ) since in /Ga…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=257761 \begin{verbatim} 2 Xa 2 V 3 Xa 3 . . . Vn Xan Vn + 1 and add productions Xai → ai for 1 ≤ i ≤ n . Let /Gamma1 ′ = ( N , T , S , P ′ ) be the new grammar formed. We need to show that /Gamma1 ( L ′ ) = /Gamma1 ( L ). Clearly /Gamma1 ( L ) ⊆ /Gamma1 ( L ′ ) since in /Gamma1 can be replaced by in /Gamma1 ′ . Conversely assume that in the derivation where U ⇒ ∗ w 1 and A ⇒ ∗ w and Vi ⇒ v i are productions in /Gamma1 where is a production which is in /Gamma1 ′ and not in /Gamma1 and the derivation is … \end{verbatim} ``` </details>
244. ph-2a1d16b47101bd3b0744 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) P ′ ) be the new grammar formed. We need to show that /Gamma1 ( L ′ ) = /Gamma1 ( L ). Clearly /Gamma1 ( L ) ⊆ /Gamma1 ( L ′ ) since in /Gamma1 can be replaced by in /Gamma1 ′ . Conversely assume that in the derivation where U ⇒ ∗ w 1 and A ⇒ ∗ w and Vi ⇒ v i are productions in /Gamma1 where is a production which is in /Gamma1 ′ and not in /Gamma1 and the derivation is This may be replaced by … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{P ′ ) be the new grammar formed. We need to show that /Gamma1 ( L ′ ) = /Gamma1 ( L ). Clearly /Gamma1 ( L ) ⊆ /Gamma1 ( L ′ ) since in /Gamma1 can be replaced by in /Gamma1 ′ . Conversely assume that in the derivation <!…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=257871 \begin{verbatim} P ′ ) be the new grammar formed. We need to show that /Gamma1 ( L ′ ) = /Gamma1 ( L ). Clearly /Gamma1 ( L ) ⊆ /Gamma1 ( L ′ ) since in /Gamma1 can be replaced by in /Gamma1 ′ . Conversely assume that in the derivation where U ⇒ ∗ w 1 and A ⇒ ∗ w and Vi ⇒ v i are productions in /Gamma1 where is a production which is in /Gamma1 ′ and not in /Gamma1 and the derivation is This may be replaced by … \end{verbatim} ``` </details>
245. ph-9f1880d8f16464495485 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) ed. We need to show that /Gamma1 ( L ′ ) = /Gamma1 ( L ). Clearly /Gamma1 ( L ) ⊆ /Gamma1 ( L ′ ) since in /Gamma1 can be replaced by in /Gamma1 ′ . Conversely assume that in the derivation where U ⇒ ∗ w 1 and A ⇒ ∗ w and Vi ⇒ v i are productions in /Gamma1 where is a production which is in /Gamma1 ′ and not in /Gamma1 and the derivation is This may be replaced by We have a derivation for S ⇒ … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ed. We need to show that /Gamma1 ( L ′ ) = /Gamma1 ( L ). Clearly /Gamma1 ( L ) ⊆ /Gamma1 ( L ′ ) since in /Gamma1 can be replaced by in /Gamma1 ′ . Conversely assume that in the derivation wh…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=257901 \begin{verbatim} ed. We need to show that /Gamma1 ( L ′ ) = /Gamma1 ( L ). Clearly /Gamma1 ( L ) ⊆ /Gamma1 ( L ′ ) since in /Gamma1 can be replaced by in /Gamma1 ′ . Conversely assume that in the derivation where U ⇒ ∗ w 1 and A ⇒ ∗ w and Vi ⇒ v i are productions in /Gamma1 where is a production which is in /Gamma1 ′ and not in /Gamma1 and the derivation is This may be replaced by We have a derivation for S ⇒ … \end{verbatim} ``` </details>
246. ph-17c0050d6a2e5e338579 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) ). Clearly /Gamma1 ( L ) ⊆ /Gamma1 ( L ′ ) since in /Gamma1 can be replaced by in /Gamma1 ′ . Conversely assume that in the derivation where U ⇒ ∗ w 1 and A ⇒ ∗ w and Vi ⇒ v i are productions in /Gamma1 where is a production which is in /Gamma1 ′ and not in /Gamma1 and the derivation is This may be replaced by We have a derivation for S ⇒ ∗ w 1 ww 2 in /Gamma1 . Hence /Gamma1 ′ ⊆ /Gamma1 . /sq… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{). Clearly /Gamma1 ( L ) ⊆ /Gamma1 ( L ′ ) since in /Gamma1 can be replaced by in /Gamma1 ′ . Conversely assume that in the derivation where U ⇒ ∗ w 1 and A ⇒ ∗ w and Vi ⇒ v i are productions …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=257956 \begin{verbatim} ). Clearly /Gamma1 ( L ) ⊆ /Gamma1 ( L ′ ) since in /Gamma1 can be replaced by in /Gamma1 ′ . Conversely assume that in the derivation where U ⇒ ∗ w 1 and A ⇒ ∗ w and Vi ⇒ v i are productions in /Gamma1 where is a production which is in /Gamma1 ′ and not in /Gamma1 and the derivation is This may be replaced by We have a derivation for S ⇒ ∗ w 1 ww 2 in /Gamma1 . Hence /Gamma1 ′ ⊆ /Gamma1 . /sq… \end{verbatim} ``` </details>
247. ph-e43b8580c28ef95f8d55 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) replaced by We have a derivation for S ⇒ ∗ w 1 ww 2 in /Gamma1 . Hence /Gamma1 ′ ⊆ /Gamma1 . /square From the above lemmas we are now able to prove that a context-free grammar /Gamma1 whose language does not contain the empty word can be expressed in Chomsky normal form . Lemma 4.7 If /Gamma1 ( L ) , the language generated by /Gamma1 = ( N , T , S , P ) , does not contain the empty word, then there exists a grammar in which every production has either the form where A, B, and C are nonterminals and a is a t… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{replaced by We have a derivation for S ⇒ ∗ w 1 ww 2 in /Gamma1 . Hence /Gamma1 ′ ⊆ /Gamma1 . /square From the above lemmas we are now able to prove that a context-free grammar /Gamma1 whose language does not contain the empty word can be expressed in…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=258443 \begin{verbatim} replaced by We have a derivation for S ⇒ ∗ w 1 ww 2 in /Gamma1 . Hence /Gamma1 ′ ⊆ /Gamma1 . /square From the above lemmas we are now able to prove that a context-free grammar /Gamma1 whose language does not contain the empty word can be expressed in Chomsky normal form . Lemma 4.7 If /Gamma1 ( L ) , the language generated by /Gamma1 = ( N , T , S , P ) , does not contain the empty word, then there exists a grammar in which every production has either the form where A, B, and C are nonterminals and a is a t… \end{verbatim} ``` </details>
248. ph-3910116f434a450f72e8 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) decoded --> We have a derivation for S ⇒ ∗ w 1 ww 2 in /Gamma1 . Hence /Gamma1 ′ ⊆ /Gamma1 . /square From the above lemmas we are now able to prove that a context-free grammar /Gamma1 whose language does not contain the empty word can be expressed in Chomsky normal form . Lemma 4.7 If /Gamma1 ( L ) , the language generated by /Gamma1 = ( N , T , S , P ) , does not contain the empty word, then there exists a grammar in which every production has either the form where A, B, and C are nonterminals and a is a terminal such that /Gamma1 ( L… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{decoded --> We have a derivation for S ⇒ ∗ w 1 ww 2 in /Gamma1 . Hence /Gamma1 ′ ⊆ /Gamma1 . /square From the above lemmas we are now able to prove that a context-free grammar /Gamma1 whose language does not contain the empty word can be expressed in Chomsky normal form . Lemma …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=258473 \begin{verbatim} decoded --> We have a derivation for S ⇒ ∗ w 1 ww 2 in /Gamma1 . Hence /Gamma1 ′ ⊆ /Gamma1 . /square From the above lemmas we are now able to prove that a context-free grammar /Gamma1 whose language does not contain the empty word can be expressed in Chomsky normal form . Lemma 4.7 If /Gamma1 ( L ) , the language generated by /Gamma1 = ( N , T , S , P ) , does not contain the empty word, then there exists a grammar in which every production has either the form where A, B, and C are nonterminals and a is a terminal such that /Gamma1 ( L… \end{verbatim} ``` </details>
249. ph-7436ac7dd4946586a5a1 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) Gamma1 ( L ) = /Gamma1 ( L ′ ) . Proof By the previous lemma, in which every production has either the form A → A 1 A 2 A 3 . . . Am where A , A 1 , A 2 , A 3 , . . . , Am are nonterminals or A → a where A is a nonterminal and a is a terminal. We construct a new grammar by replacing every production of the form A → A 1 A 2 A 3 . . . Am by the set of productions A → A 1 X 1 , X 1 → A 2 X 2 , . . . , Xm -2 → Am -1 Am , where each replacement of a production in /Gamma1 uses a new set of symbols. is a derivation in /Gamma1 ′ , /Gamma1 ( L ) ⊆ /Gamma1 ( L ′ ). Converse… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{Gamma1 ( L ) = /Gamma1 ( L ′ ) . Proof By the previous lemma, in which every production has either the form A → A 1 A 2 A 3 . . . Am where A , A 1 , A 2 , A 3 , . . . , Am are nonterminals or A → a where A is a nonterminal and a is a terminal. We construct a new grammar by repla…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=259069 \begin{verbatim} Gamma1 ( L ) = /Gamma1 ( L ′ ) . Proof By the previous lemma, in which every production has either the form A → A 1 A 2 A 3 . . . Am where A , A 1 , A 2 , A 3 , . . . , Am are nonterminals or A → a where A is a nonterminal and a is a terminal. We construct a new grammar by replacing every production of the form A → A 1 A 2 A 3 . . . Am by the set of productions A → A 1 X 1 , X 1 → A 2 X 2 , . . . , Xm -2 → Am -1 Am , where each replacement of a production in /Gamma1 uses a new set of symbols. is a derivation in /Gamma1 ′ , /Gamma1 ( L ) ⊆ /Gamma1 ( L ′ ). Converse… \end{verbatim} ``` </details>
250. ph-5eb3e4649aec3b9d4e7e — automata/docling_md/AutomataTheory.md ### Plain (markdown context) is a derivation in /Gamma1 ′ , /Gamma1 ( L ) ⊆ /Gamma1 ( L ′ ). Conversely, if S ⇒ ∗ w in /Gamma1 ′ contains no productions which are not in /Gamma1 , then w ∈ /Gamma1 ( L ). If it does, let Wm bethelast term in the derivation containing a symbol in /Gamma1 ′ which is not in /Gamma1 so we have Wm ⇒ Wm + 1 ⇒ ∗ w and Wm ⇒ Wm + 1 has the form U ′ Xm -2 V ⇒ UAm -1 AmV . Therefore the derivation uses the set or of productions A → A 1 X 1 , X 1 → A 2 X 2 , . . . , Xm -2 → Am -1 Am and has the form where U ′ = UA ′ 1 A ′ 2 A ′ 3 · · · A ′ m -2 and Ai ⇒ ∗ Ai is a derivati… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{is a derivation in /Gamma1 ′ , /Gamma1 ( L ) ⊆ /Gamma1 ( L ′ ). Conversely, if S ⇒ ∗ w in /Gamma1 ′ contains no productions which are not in /Gamma1 , then w ∈ /Gamma1 ( L ). If it does, let Wm bethelast term in the derivation containing a symbol in /Gamma1 ′ which is not in /Ga…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=259599 \begin{verbatim} is a derivation in /Gamma1 ′ , /Gamma1 ( L ) ⊆ /Gamma1 ( L ′ ). Conversely, if S ⇒ ∗ w in /Gamma1 ′ contains no productions which are not in /Gamma1 , then w ∈ /Gamma1 ( L ). If it does, let Wm bethelast term in the derivation containing a symbol in /Gamma1 ′ which is not in /Gamma1 so we have Wm ⇒ Wm + 1 ⇒ ∗ w and Wm ⇒ Wm + 1 has the form U ′ Xm -2 V ⇒ UAm -1 AmV . Therefore the derivation uses the set or of productions A → A 1 X 1 , X 1 → A 2 X 2 , . . . , Xm -2 → Am -1 Am and has the form where U ′ = UA ′ 1 A ′ 2 A ′ 3 · · · A ′ m -2 and Ai ⇒ ∗ Ai is a derivati… \end{verbatim} ``` </details>
251. ph-7544f7dd826ee070f19e — automata/docling_md/AutomataTheory.md ### Plain (markdown context) ons, we have no bound on how many derivations may occur before the first terminal symbol appears at the left of the string. For example, using the production A → Aa , we can generate the string Aa n for arbitrary n , using n derivations without beginning a string with a terminal symbol. We can eliminate this particular problem by eliminating the productions of the form A → Aa . This is called elimination of left recursion . In a grammar /Gamma1 with no λ productions or trivial productions, let be productions in which the right-hand side of the production begins wi… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ons, we have no bound on how many derivations may occur before the first terminal symbol appears at the left of the string. For example, using the production A → Aa , we can generate the string Aa n for arbitrary n , using n derivations without beginning a string with a terminal…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=261723 \begin{verbatim} ons, we have no bound on how many derivations may occur before the first terminal symbol appears at the left of the string. For example, using the production A → Aa , we can generate the string Aa n for arbitrary n , using n derivations without beginning a string with a terminal symbol. We can eliminate this particular problem by eliminating the productions of the form A → Aa . This is called elimination of left recursion . In a grammar /Gamma1 with no λ productions or trivial productions, let be productions in which the right-hand side of the production begins wi… \end{verbatim} ``` </details>
252. ph-027218ecd54b2cd8d619 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) e string. For example, using the production A → Aa , we can generate the string Aa n for arbitrary n , using n derivations without beginning a string with a terminal symbol. We can eliminate this particular problem by eliminating the productions of the form A → Aa . This is called elimination of left recursion . In a grammar /Gamma1 with no λ productions or trivial productions, let be productions in which the right-hand side of the production begins with an A and be productions in which the right-hand side of the production does not be… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{e string. For example, using the production A → Aa , we can generate the string Aa n for arbitrary n , using n derivations without beginning a string with a terminal symbol. We can eliminate this particular problem by eliminating the productions of the form A → Aa . This is call…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=261837 \begin{verbatim} e string. For example, using the production A → Aa , we can generate the string Aa n for arbitrary n , using n derivations without beginning a string with a terminal symbol. We can eliminate this particular problem by eliminating the productions of the form A → Aa . This is called elimination of left recursion . In a grammar /Gamma1 with no λ productions or trivial productions, let be productions in which the right-hand side of the production begins with an A and be productions in which the right-hand side of the production does not be… \end{verbatim} ``` </details>
253. ph-4efada13017aafee8ec7 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) Vi A ′ and A ′ → Vi . - (2) Form productions A → Ui A ′ for 1 ≤ i ≤ n . Lemma 4.8 /Gamma1 ( L ) = /Gamma1 ′ ( L ) . Proof Let a derivation beginning with A have the form assuming A → Ui A ′ for 1 ≤ i ≤ n and we have A ⇒ AV (1) ⇒ AV (2) V (1) ⇒ ∗ AV ( k ) . . . V (2) V (1) ⇒ U ( i ) V ( k ) . . . V (2) V (1) where V ( j ) ∈ { V 1 , V 2 , . . . , Vn } for all 1 ≤ j ≤ k and U ( i ) ∈ { U 1 , U 2 , . . . , Um } . Therefore using leftmost derivation, any production containing A will have the form But can be replaced by But can be replaced by But can be replaced by Placing w on the left and W… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{oductions A → Ui A ′ for 1 ≤ i ≤ n . Lemma 4.8 /Gamma1 ( L ) = /Gamma1 ′ ( L ) . Proof Let a derivation beginning with A have the form assuming A → Ui A ′ for 1 ≤ i ≤ n and we have A ⇒ AV (1) ⇒ AV (2) V (1) ⇒ ∗ AV ( k ) . . . V (2) V (1) ⇒ U ( i ) V ( k ) . . . V (2) V (1) where…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=262700 \begin{verbatim} oductions A → Ui A ′ for 1 ≤ i ≤ n . Lemma 4.8 /Gamma1 ( L ) = /Gamma1 ′ ( L ) . Proof Let a derivation beginning with A have the form assuming A → Ui A ′ for 1 ≤ i ≤ n and we have A ⇒ AV (1) ⇒ AV (2) V (1) ⇒ ∗ AV ( k ) . . . V (2) V (1) ⇒ U ( i ) V ( k ) . . . V (2) V (1) where V ( j ) ∈ { V 1 , V 2 , . . . , Vn } for all 1 ≤ j ≤ k and U ( i ) ∈ { U 1 , U 2 , . . . , Um } . Therefore using leftmost derivation, any production containing A will have the form But can be replaced by Placing w on the left and W… \end{verbatim} ``` </details>
255. ph-eed6a86e57982a737021 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) amma1 ( L ) = /Gamma1 ′ ( L ) . Proof Let a derivation beginning with A have the form assuming A → Ui A ′ for 1 ≤ i ≤ n and we have A ⇒ AV (1) ⇒ AV (2) V (1) ⇒ ∗ AV ( k ) . . . V (2) V (1) ⇒ U ( i ) V ( k ) . . . V (2) V (1) where V ( j ) ∈ { V 1 , V 2 , . . . , Vn } for all 1 ≤ j ≤ k and U ( i ) ∈ { U 1 , U 2 , . . . , Um } . Therefore using leftmost derivation, any production containing A will have the form But can be replaced by Placing w on the left and W on the right of each term, we have, w AW ⇒ ∗ w U… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{amma1 ( L ) = /Gamma1 ′ ( L ) . Proof Let a derivation beginning with A have the form assuming A → Ui A ′ for 1 ≤ i ≤ n and we have A ⇒ AV (1) ⇒ AV (2) V (1) ⇒ ∗ AV ( k ) . . . V (2) V (1) ⇒ U ( i ) V ( k ) . . . V (2) V (1) where V ( j ) ∈ \{ V 1 , V 2 , . . . , Vn \} for all 1 ≤…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=262750 \begin{verbatim} amma1 ( L ) = /Gamma1 ′ ( L ) . Proof Let a derivation beginning with A have the form assuming A → Ui A ′ for 1 ≤ i ≤ n and we have A ⇒ AV (1) ⇒ AV (2) V (1) ⇒ ∗ AV ( k ) . . . V (2) V (1) ⇒ U ( i ) V ( k ) . . . V (2) V (1) where V ( j ) ∈ { V 1 , V 2 , . . . , Vn } for all 1 ≤ j ≤ k and U ( i ) ∈ { U 1 , U 2 , . . . , Um } . Therefore using leftmost derivation, any production containing A will have the form But can be replaced by Placing w on the left and W on the right of each term, we have, w AW ⇒ ∗ w U… \end{verbatim} ``` </details>
256. ph-b0b05fce2e786562c671 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) ree grammar can be expressed in Greibach normal form can be proved by first expressing the grammar in Chomsky normal form. We shall not do so however so that the development for Chomsky normal form may be omitted if desired. Using the above lemmas, we are about to take a giant leap toward proving that every context-free grammar can be expressed in Greibach normal form. Lemma 4.10 Any context-free grammar which does not generate λ can be expressed so that each of its productions is of the form where a is a terminal and W is a string which is empty or consists of a … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ree grammar can be expressed in Greibach normal form can be proved by first expressing the grammar in Chomsky normal form. We shall not do so however so that the development for Chomsky normal form may be omitted if desired. Using the above lemmas, we are about to take a giant l…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=264091 \begin{verbatim} ree grammar can be expressed in Greibach normal form can be proved by first expressing the grammar in Chomsky normal form. We shall not do so however so that the development for Chomsky normal form may be omitted if desired. Using the above lemmas, we are about to take a giant leap toward proving that every context-free grammar can be expressed in Greibach normal form. Lemma 4.10 Any context-free grammar which does not generate λ can be expressed so that each of its productions is of the form where a is a terminal and W is a string which is empty or consists of a … \end{verbatim} ``` </details>
257. ph-c4d0124d4b662829b398 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) m. Lemma 4.10 Any context-free grammar which does not generate λ can be expressed so that each of its productions is of the form where a is a terminal and W is a string which is empty or consists of a string of terminals and/or nonterminals. Proof We first order the nonterminals beginning with S , the start symbol. For simplicity, let the nonterminals be A 1 , A 2 , A 3 , . . . , Am . Our first goal is to change every production so that it is either in the form where a is a terminal and W is a string which is empty or consists of a str… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{m. Lemma 4.10 Any context-free grammar which does not generate λ can be expressed so that each of its productions is of the form where a is a terminal and W is a string which is empty or consists of a string of terminals and/or nonterminals. Proof We…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=264460 \begin{verbatim} m. Lemma 4.10 Any context-free grammar which does not generate λ can be expressed so that each of its productions is of the form where a is a terminal and W is a string which is empty or consists of a string of terminals and/or nonterminals. Proof We first order the nonterminals beginning with S , the start symbol. For simplicity, let the nonterminals be A 1 , A 2 , A 3 , . . . , Am . Our first goal is to change every production so that it is either in the form where a is a terminal and W is a string which is empty or consists of a str… \end{verbatim} ``` </details>
258. ph-382518091f9e4e04cfc3 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) where a is a terminal and W is a string which is empty or consists of a string of terminals and/or nonterminals. Proof We first order the nonterminals beginning with S , the start symbol. For simplicity, let the nonterminals be A 1 , A 2 , A 3 , . . . , Am . Our first goal is to change every production so that it is either in the form where a is a terminal and W is a string which is empty or consists of a string of terminals and/or nonterminals, or in the form where i < j and Y consists of a string of terminals and/or nonterminals. … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{where a is a terminal and W is a string which is empty or consists of a string of terminals and/or nonterminals. Proof We first order the nonterminals beginning with S , the start symbol. For simplicity, let the nonterminals be A 1 , A 2 , A 3 , . . . , Am . Our first goal is to…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=264619 \begin{verbatim} where a is a terminal and W is a string which is empty or consists of a string of terminals and/or nonterminals. Proof We first order the nonterminals beginning with S , the start symbol. For simplicity, let the nonterminals be A 1 , A 2 , A 3 , . . . , Am . Our first goal is to change every production so that it is either in the form where a is a terminal and W is a string which is empty or consists of a string of terminals and/or nonterminals, or in the form where i < j and Y consists of a string of terminals and/or nonterminals. … \end{verbatim} ``` </details>
259. ph-986e764098e52924b3df — automata/docling_md/AutomataTheory.md ### Plain (markdown context) y Ai where i < k . We now prove the statement for i = k . In each case where Ak → AjY is a production for k > j , use the procedure in Lemma 4.9 to eliminate Aj . When Ak → AkY is a production, use the process of elimination of left recursion to remove Ak from the right-hand side. Therefore by induction we have every production so that it is either in the form where a is a terminal and W is a string which is empty or consists of a string of terminals and/or nonterminals, or in the form where i < j and Y consists of a string of terminals and/or nonterminal… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{y Ai where i \< k . We now prove the statement for i = k . In each case where Ak → AjY is a production for k \> j , use the procedure in Lemma 4.9 to eliminate Aj . When Ak → AkY is a production, use the process of elimination of left recursion to remove Ak from the right-ha…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=265707 \begin{verbatim} y Ai where i < k . We now prove the statement for i = k . In each case where Ak → AjY is a production for k > j , use the procedure in Lemma 4.9 to eliminate Aj . When Ak → AkY is a production, use the process of elimination of left recursion to remove Ak from the right-hand side. Therefore by induction we have every production so that it is either in the form where a is a terminal and W is a string which is empty or consists of a string of terminals and/or nonterminals, or in the form where i < j and Y consists of a string of terminals and/or nonterminal… \end{verbatim} ``` </details>
260. ph-20fe283db17ceec6bace — automata/docling_md/AutomataTheory.md ### Plain (markdown context) nterminals. Anyproduction with Am onthe left-hand side must have the form Am → aW since there is no nonterminal larger than Am . If there is a production of the form Am -1 → AmW ′ , use the procedures in Lemma 4.9 to eliminate Am . The result is a production of the form Am → bW ′′ . Assume k is the largest value so Ak → AjY is a production where k < j . Again using the procedures in Lemma 4.9 to eliminate Aj , we have a procedure of the form Ak → aW . When the process is completed, we have where a is a terminal and W is a string which is empty or consists of a … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{nterminals. Anyproduction with Am onthe left-hand side must have the form Am → aW since there is no nonterminal larger than Am . If there is a production of the form Am -1 → AmW ′ , use the procedures in Lemma 4.9 to eliminate Am . The result is a production of the form Am → bW …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=266301 \begin{verbatim} nterminals. Anyproduction with Am onthe left-hand side must have the form Am → aW since there is no nonterminal larger than Am . If there is a production of the form Am -1 → AmW ′ , use the procedures in Lemma 4.9 to eliminate Am . The result is a production of the form Am → bW ′′ . Assume k is the largest value so Ak → AjY is a production where k < j . Again using the procedures in Lemma 4.9 to eliminate Aj , we have a procedure of the form Ak → aW . When the process is completed, we have where a is a terminal and W is a string which is empty or consists of a … \end{verbatim} ``` </details>
261. ph-ec2256515df85e8b8342 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) i , it is impossible to have a production of the form Bi → Bj W . Therefore productions with Bi on the left have the form Bi → aW or Bi → Aj W . Repeating the process above we can change these to the form Bi → aW , and the lemma is proved. /square Theorem 4.5 Every context-free grammar whose language does not contain λ can be expressed in Greibach normal form. Proof Weoutline the proof. The details are left to the reader. Since we already know that every production can be written in the form where a is a terminal and W is a string which is empty or consists of a s… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{i , it is impossible to have a production of the form Bi → Bj W . Therefore productions with Bi on the left have the form Bi → aW or Bi → Aj W . Repeating the process above we can change these to the form Bi → aW , and the lemma is proved. /square Theorem 4.5 Every context-free …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=267076 \begin{verbatim} i , it is impossible to have a production of the form Bi → Bj W . Therefore productions with Bi on the left have the form Bi → aW or Bi → Aj W . Repeating the process above we can change these to the form Bi → aW , and the lemma is proved. /square Theorem 4.5 Every context-free grammar whose language does not contain λ can be expressed in Greibach normal form. Proof Weoutline the proof. The details are left to the reader. Since we already know that every production can be written in the form where a is a terminal and W is a string which is empty or consists of a s… \end{verbatim} ``` </details>
262. ph-03d415245e767d7f7918 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) /Gamma1 with B on the left. Let /Gamma1 ′ be the grammar with production A → UBV removed and the productions A → UWiV for 1 ≤ i ≤ m added, then /Gamma1 ( L ) = /Gamma1 ′ ( L ),' prove /Gamma1 ( L ) ⊆ /Gamma1 ′ ( L ). - (2) Prove Theorem 4.5 'Every context-free grammar can be expressed in Greibach normal form.' - (3) Complete the proof of Lemma 4.8. - (4) Let /Gamma1 ′′ = ( N , /Sigma1 , S , P ) be the grammar defined by N = { S , A , B } , /Sigma1 = { a , b } , and P be the set of productions Express this grammar in Chomsky normal form. - (5) Express the previous … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{/Gamma1 with B on the left. Let /Gamma1 ′ be the grammar with production A → UBV removed and the productions A → UWiV for 1 ≤ i ≤ m added, then /Gamma1 ( L ) = /Gamma1 ′ ( L ),' prove /Gamma1 ( L ) ⊆ /Gamma1 ′ ( L ). - (2) Prove Theorem 4.5 'Every context-free grammar can be exp…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=268221 \begin{verbatim} /Gamma1 with B on the left. Let /Gamma1 ′ be the grammar with production A → UBV removed and the productions A → UWiV for 1 ≤ i ≤ m added, then /Gamma1 ( L ) = /Gamma1 ′ ( L ),' prove /Gamma1 ( L ) ⊆ /Gamma1 ′ ( L ). - (2) Prove Theorem 4.5 'Every context-free grammar can be expressed in Greibach normal form.' - (3) Complete the proof of Lemma 4.8. - (4) Let /Gamma1 ′′ = ( N , /Sigma1 , S , P ) be the grammar defined by N = { S , A , B } , /Sigma1 = { a , b } , and P be the set of productions Express this grammar in Chomsky normal form. - (5) Express the previous … \end{verbatim} ``` </details>
263. ph-48c2b5837f51aa62a5cc — automata/docling_md/AutomataTheory.md ### Plain (markdown context) ext-free grammar can be expressed in Greibach normal form.' - (3) Complete the proof of Lemma 4.8. - (4) Let /Gamma1 ′′ = ( N , /Sigma1 , S , P ) be the grammar defined by N = { S , A , B } , /Sigma1 = { a , b } , and P be the set of productions Express this grammar in Chomsky normal form. - (5) Express the previous grammar in Greibach normal form. - (6) Let /Gamma1 = ( N , T , S , P ) be the grammar with N = { S } , T = { a , b } , and P contain the productions Express this grammar in Chomsky normal form. - (7) Express the previous gr… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ext-free grammar can be expressed in Greibach normal form.' - (3) Complete the proof of Lemma 4.8. - (4) Let /Gamma1 ′′ = ( N , /Sigma1 , S , P ) be the grammar defined by N = \{ S , A , B \} , /Sigma1 = \{ a , b \} , and P be the set of productions Expr…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=268474 \begin{verbatim} ext-free grammar can be expressed in Greibach normal form.' - (3) Complete the proof of Lemma 4.8. - (4) Let /Gamma1 ′′ = ( N , /Sigma1 , S , P ) be the grammar defined by N = { S , A , B } , /Sigma1 = { a , b } , and P be the set of productions Express this grammar in Chomsky normal form. - (5) Express the previous grammar in Greibach normal form. - (6) Let /Gamma1 = ( N , T , S , P ) be the grammar with N = { S } , T = { a , b } , and P contain the productions Express this grammar in Chomsky normal form. - (7) Express the previous gr… \end{verbatim} ``` </details>
264. ph-f14ac9d12ff627d371f0 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) this grammar in Chomsky normal form. - (5) Express the previous grammar in Greibach normal form. - (6) Let /Gamma1 = ( N , T , S , P ) be the grammar with N = { S } , T = { a , b } , and P contain the productions Express this grammar in Chomsky normal form. - (7) Express the previous grammar in Greibach normal form. - (8) Let /Gamma1 ′′ = ( N , /Sigma1 , S , P ) be the grammar defined by N = { S , A , B } , /Sigma1 = { a , b } , and P be the set of productions Express this grammar in Chomsky normal form. - (9) Express the previous gram… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{this grammar in Chomsky normal form. - (5) Express the previous grammar in Greibach normal form. - (6) Let /Gamma1 = ( N , T , S , P ) be the grammar with N = \{ S \} , T = \{ a , b \} , and P contain the productions Express this grammar in Chomsky norma…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=268758 \begin{verbatim} this grammar in Chomsky normal form. - (5) Express the previous grammar in Greibach normal form. - (6) Let /Gamma1 = ( N , T , S , P ) be the grammar with N = { S } , T = { a , b } , and P contain the productions Express this grammar in Chomsky normal form. - (7) Express the previous grammar in Greibach normal form. - (8) Let /Gamma1 ′′ = ( N , /Sigma1 , S , P ) be the grammar defined by N = { S , A , B } , /Sigma1 = { a , b } , and P be the set of productions Express this grammar in Chomsky normal form. - (9) Express the previous gram… \end{verbatim} ``` </details>
265. ph-268f9f833b691456a778 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) e first part of the production with the second part, so does the PDA. The stack then resembles the strings derived in the grammar except that the terminals on the left of the derived string (top of the stack) are then removed as they occur in the stack and compared with the letters on the tape. As before a word is accepted if the word has been read and the stack is empty. Example 4.17 Let /Gamma1 = ( N , T , S , P ) be the grammar with N = { S } , T = { a , b } , and P contain the productions which generates the language { ww R : w ∈ T ∗ } . This has the PDA ![Ima… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{e first part of the production with the second part, so does the PDA. The stack then resembles the strings derived in the grammar except that the terminals on the left of the derived string (top of the stack) are then removed as they occur in the stack and compared with the lett…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=272890 \begin{verbatim} e first part of the production with the second part, so does the PDA. The stack then resembles the strings derived in the grammar except that the terminals on the left of the derived string (top of the stack) are then removed as they occur in the stack and compared with the letters on the tape. As before a word is accepted if the word has been read and the stack is empty. Example 4.17 Let /Gamma1 = ( N , T , S , P ) be the grammar with N = { S } , T = { a , b } , and P contain the productions which generates the language { ww R : w ∈ T ∗ } . This has the PDA ![Ima… \end{verbatim} ``` </details>
266. ph-7102a852a200a342b522 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) | | t | pop a | Sa | abba | t | read b | a | a | | t | read a | Sa | bba | t | pop a | a | λ | | t | pop S | a | bba | t | read a | λ | λ | | t | push bSb | bSba | bba | t | accept | λ | λ | Example 4.18 Let /Gamma1 = ( N , T , S , P ) be the grammar with N = { S } , T = { a , b } , and P contain the productions which generates the language { w : w ∈ A ∗ and contains an even number of a s and an even number of b s. } . This has the PDA ![Image](./AutomataTheory_artifacts/image_000195_70fd125189dbb28a3982406b175df13baf1d6fd0ab24cca159fd284d49179241.png) Consider th… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{| | t | pop a | Sa | abba | t | read b | a | a | | t | read a | Sa | bba | t | pop a | a | λ | | t | pop S | a | bba | t | read a | λ | λ | | t | push bSb | bSba | bba | t | accept | λ | λ | Example 4.18 Let /Gamma1 = ( N , T , S , P ) be the grammar with N = \{ S \} , T = \{ a , b…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=274230 \begin{verbatim} | | t | pop a | Sa | abba | t | read b | a | a | | t | read a | Sa | bba | t | pop a | a | λ | | t | pop S | a | bba | t | read a | λ | λ | | t | push bSb | bSba | bba | t | accept | λ | λ | Example 4.18 Let /Gamma1 = ( N , T , S , P ) be the grammar with N = { S } , T = { a , b } , and P contain the productions which generates the language { w : w ∈ A ∗ and contains an even number of a s and an even number of b s. } . This has the PDA ![Image](./AutomataTheory_artifacts/image_000195_70fd125189dbb28a3982406b175df13baf1d6fd0ab24cca159fd284d49179241.png) Consider th… \end{verbatim} ``` </details>
267. ph-fb9422197d21e01036aa — automata/docling_md/AutomataTheory.md ### Plain (markdown context) n condition ( ab , s , BaaB ) gives notation ( ab , s , BaaB ) /turnstileleft ( b , t , aaB ) and the transition (( a , s , λ ) , ( t , D )) when the PDA is in condition ( ab , s , BaaB ) gives notation ( ab , s , BaaB ) /turnstileleft ( b , t , DBaaB ). We say that ( s , u , V ) /turnstileleft ∗ ( t , v, W ) if the PDA can be changed from ( s , u , V ) to ( t , v, W ) in a finite number of transitions. Lemma 4.11 Let /Gamma1 ( L ) be a context-free language. There exists a PDA that accepts L where /Sigma1 is a finite alphabet, Q is a finite set of states, s is th… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{n condition ( ab , s , BaaB ) gives notation ( ab , s , BaaB ) /turnstileleft ( b , t , aaB ) and the transition (( a , s , λ ) , ( t , D )) when the PDA is in condition ( ab , s , BaaB ) gives notation ( ab , s , BaaB ) /turnstileleft ( b , t , DBaaB ). We say that ( s , u , V …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=277001 \begin{verbatim} n condition ( ab , s , BaaB ) gives notation ( ab , s , BaaB ) /turnstileleft ( b , t , aaB ) and the transition (( a , s , λ ) , ( t , D )) when the PDA is in condition ( ab , s , BaaB ) gives notation ( ab , s , BaaB ) /turnstileleft ( b , t , DBaaB ). We say that ( s , u , V ) /turnstileleft ∗ ( t , v, W ) if the PDA can be changed from ( s , u , V ) to ( t , v, W ) in a finite number of transitions. Lemma 4.11 Let /Gamma1 ( L ) be a context-free language. There exists a PDA that accepts L where /Sigma1 is a finite alphabet, Q is a finite set of states, s is th… \end{verbatim} ``` </details>
268. ph-d575f1f0cc1b8389a548 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) hat ( s , u , V ) /turnstileleft ∗ ( t , v, W ) if the PDA can be changed from ( s , u , V ) to ( t , v, W ) in a finite number of transitions. Lemma 4.11 Let /Gamma1 ( L ) be a context-free language. There exists a PDA that accepts L where /Sigma1 is a finite alphabet, Q is a finite set of states, s is the initial or starting state, I is a finite of stack symbols, ϒ is the transition relation, and F is the set of acceptance states. The relation ϒ is a subset of Proof As previously mentioned, we shall assume that the PDA has two states… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{hat ( s , u , V ) /turnstileleft ∗ ( t , v, W ) if the PDA can be changed from ( s , u , V ) to ( t , v, W ) in a finite number of transitions. Lemma 4.11 Let /Gamma1 ( L ) be a context-free language. There exists a PDA that accepts L where /Sigma1 i…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=277264 \begin{verbatim} hat ( s , u , V ) /turnstileleft ∗ ( t , v, W ) if the PDA can be changed from ( s , u , V ) to ( t , v, W ) in a finite number of transitions. Lemma 4.11 Let /Gamma1 ( L ) be a context-free language. There exists a PDA that accepts L where /Sigma1 is a finite alphabet, Q is a finite set of states, s is the initial or starting state, I is a finite of stack symbols, ϒ is the transition relation, and F is the set of acceptance states. The relation ϒ is a subset of Proof As previously mentioned, we shall assume that the PDA has two states… \end{verbatim} ``` </details>
269. ph-4c2f76a4551f74eca865 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) ins with a nonterminal or is empty, then ( t , α, S ) /turnstileleft ∗ ( t , λ, β ). Hence if S ⇒ ∗ α in /Gamma1 where α ∈ T ∗ , then ( s , α, λ ) /turnstileleft ( t , α, S ) /turnstileleft ∗ ( t , λ, λ ), and α is accepted by the PDA. We prove this using induction on the length of the derivation. Suppose n = 0, but then we have S ⇒ ∗ S , so α = λ , β = S , and ( t , λ, S ) /turnstileleft ∗ ( t , λ, S ) gives us ( t , α, S ) /turnstileleft ∗ ( t , λ, β ). Now assume S ⇒ ∗ γ in k + 1 steps. Say Then there is a first nonterminal B in the string mk and a production B… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ins with a nonterminal or is empty, then ( t , α, S ) /turnstileleft ∗ ( t , λ, β ). Hence if S ⇒ ∗ α in /Gamma1 where α ∈ T ∗ , then ( s , α, λ ) /turnstileleft ( t , α, S ) /turnstileleft ∗ ( t , λ, λ ), and α is accepted by the PDA. We prove this using induction on the length…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=280214 \begin{verbatim} ins with a nonterminal or is empty, then ( t , α, S ) /turnstileleft ∗ ( t , λ, β ). Hence if S ⇒ ∗ α in /Gamma1 where α ∈ T ∗ , then ( s , α, λ ) /turnstileleft ( t , α, S ) /turnstileleft ∗ ( t , λ, λ ), and α is accepted by the PDA. We prove this using induction on the length of the derivation. Suppose n = 0, but then we have S ⇒ ∗ S , so α = λ , β = S , and ( t , λ, S ) /turnstileleft ∗ ( t , λ, S ) gives us ( t , α, S ) /turnstileleft ∗ ( t , λ, β ). Now assume S ⇒ ∗ γ in k + 1 steps. Say Then there is a first nonterminal B in the string mk and a production B… \end{verbatim} ``` </details>
270. ph-f58819b7279bcc8b765d — automata/docling_md/AutomataTheory.md ### Plain (markdown context) ut read in passing from state p to state q and leaving the stack as it was in state p . The productions consist of the following four types. - (1) For each q T , the production S s , λ, q . - (2) For each transition (( p , a , B ) , ( q , D )) ∈ ϒ , where B , D ∈ I ∪ { λ } , the productions p , B , t a q , D , t for all t Q . 3. ∈ →〈 〉 4. 〈 〉 → 〈 〉 ∈ 5. { } 1 2 Bn ∈ - (3) For each transition (( p , a , D ) , ( q , B 1 B 2 . . . Bn )) ∈ ϒ , where D ∈ I ∪ λ , B , B , . . . , C , the productions for all q 1 , q 2 , . . . , qn -1 , t ∈ Q . - (4) For each q ∈ Q , the p… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ut read in passing from state p to state q and leaving the stack as it was in state p . The productions consist of the following four types. - (1) For each q T , the production S s , λ, q . - (2) For each transition (( p , a , B ) , ( q , D )) ∈ ϒ , where B , D ∈ I ∪ \{ λ \} , the…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=283400 \begin{verbatim} ut read in passing from state p to state q and leaving the stack as it was in state p . The productions consist of the following four types. - (1) For each q T , the production S s , λ, q . - (2) For each transition (( p , a , B ) , ( q , D )) ∈ ϒ , where B , D ∈ I ∪ { λ } , the productions p , B , t a q , D , t for all t Q . 3. ∈ →〈 〉 4. 〈 〉 → 〈 〉 ∈ 5. { } 1 2 Bn ∈ - (3) For each transition (( p , a , D ) , ( q , B 1 B 2 . . . Bn )) ∈ ϒ , where D ∈ I ∪ λ , B , B , . . . , C , the productions for all q 1 , q 2 , . . . , qn -1 , t ∈ Q . - (4) For each q ∈ Q , the p… \end{verbatim} ``` </details>
271. ph-004a015a4a3eacf15a40 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) h other nonterminals. Hence a word in the language of the grammar cannot be generated without these productions. Lemma 4.12 A language M ( L ) accepted by a pushdown automaton M = ( /Sigma1 , Q , s , I , ϒ, F ) , is a context-free language. Proof Using the grammar /Gamma1 = ( N , /Sigma1 , S , P ), where the nonterminals and productions are described above, we show that /Gamma1 generates the same language as accepted by M . We first show that for p , q ∈ Q , B ∈ I ∪ { λ } and w ∈ A ∗ , that Thus for t ∈ Q , 〈 s , λ, t 〉 ⇒ ∗ w if and only if ( s , w, λ ) /turnstile… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{h other nonterminals. Hence a word in the language of the grammar cannot be generated without these productions. Lemma 4.12 A language M ( L ) accepted by a pushdown automaton M = ( /Sigma1 , Q , s , I , ϒ, F ) , is a context-free language. Proof Using the grammar /Gamma1 = ( N …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=285477 \begin{verbatim} h other nonterminals. Hence a word in the language of the grammar cannot be generated without these productions. Lemma 4.12 A language M ( L ) accepted by a pushdown automaton M = ( /Sigma1 , Q , s , I , ϒ, F ) , is a context-free language. Proof Using the grammar /Gamma1 = ( N , /Sigma1 , S , P ), where the nonterminals and productions are described above, we show that /Gamma1 generates the same language as accepted by M . We first show that for p , q ∈ Q , B ∈ I ∪ { λ } and w ∈ A ∗ , that Thus for t ∈ Q , 〈 s , λ, t 〉 ⇒ ∗ w if and only if ( s , w, λ ) /turnstile… \end{verbatim} ``` </details>
272. ph-83c38e5da7b3c5681516 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) nstileleft ∗ ( p , λ, λ ) which is obvious. Assume n = k > 1, then the first production can only be of type (2) or type (3). If it is type (2), we have 〈 p , B , q 〉 → a 〈 r , D , q 〉 for p , r ∈ Q , where (( p , a , B ) , ( r , D )) ∈ ϒ . Henceletting w = a v , ( p , w, B ) /turnstileleft ( q , v, D ) and by induction if 〈 r , D , q 〉 ⇒ ∗ v then ( q , v, D ) /turnstileleft ∗ ( q , λ, λ ). Therefore ( p , w, B ) /turnstileleft ∗ ( q , λ, λ ). If the first production is of type (3), we have and (( p , a , B ) , ( q , B 1 B 2 . . . Bn )) ∈ ϒ . So if w = a v , ( p… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{nstileleft ∗ ( p , λ, λ ) which is obvious. Assume n = k \> 1, then the first production can only be of type (2) or type (3). If it is type (2), we have 〈 p , B , q 〉 → a 〈 r , D , q 〉 for p , r ∈ Q , where (( p , a , B ) , ( r , D )) ∈ ϒ . Henceletting w = a v , ( p , w, B ) …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=286542 \begin{verbatim} nstileleft ∗ ( p , λ, λ ) which is obvious. Assume n = k > 1, then the first production can only be of type (2) or type (3). If it is type (2), we have 〈 p , B , q 〉 → a 〈 r , D , q 〉 for p , r ∈ Q , where (( p , a , B ) , ( r , D )) ∈ ϒ . Henceletting w = a v , ( p , w, B ) /turnstileleft ( q , v, D ) and by induction if 〈 r , D , q 〉 ⇒ ∗ v then ( q , v, D ) /turnstileleft ∗ ( q , λ, λ ). Therefore ( p , w, B ) /turnstileleft ∗ ( q , λ, λ ). If the first production is of type (3), we have and (( p , a , B ) , ( q , B 1 B 2 . . . Bn )) ∈ ϒ . So if w = a v , ( p… \end{verbatim} ``` </details>
273. ph-ae12280ca1811da4e959 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) ileleft ∗ ( q , λ, λ ). Therefore ( p , w, B ) /turnstileleft ∗ ( q , λ, λ ). If the first production is of type (3), we have and (( p , a , B ) , ( q , B 1 B 2 . . . Bn )) ∈ ϒ . So if w = a v , ( p , w, B ) /turnstileleft ( q , v, B 1 B 2 . . . Bn ). For convenience of notation, let q = qn . Let 〈 qi -1 , Bi , qi 〉 ⇒ ∗ ui so that w = au 1 u 2 . . . un and v = u 1 u 2 . . . un . By induction, ( qi -1 , ui , Bi ) /turnstileleft ∗ ( qi , λ, λ ). Therefore we have so that ( p , w, B ) ( q , λ, λ ). We now show that if ( p , w, B ) /turnst… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ileleft ∗ ( q , λ, λ ). Therefore ( p , w, B ) /turnstileleft ∗ ( q , λ, λ ). If the first production is of type (3), we have and (( p , a , B ) , ( q , B 1 B 2 . . . Bn )) ∈ ϒ . So if w = a v , ( p , w, B ) /turnstileleft ( q , v, B 1 B 2 . . . Bn )…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=286914 \begin{verbatim} ileleft ∗ ( q , λ, λ ). Therefore ( p , w, B ) /turnstileleft ∗ ( q , λ, λ ). If the first production is of type (3), we have and (( p , a , B ) , ( q , B 1 B 2 . . . Bn )) ∈ ϒ . So if w = a v , ( p , w, B ) /turnstileleft ( q , v, B 1 B 2 . . . Bn ). For convenience of notation, let q = qn . Let 〈 qi -1 , Bi , qi 〉 ⇒ ∗ ui so that w = au 1 u 2 . . . un and v = u 1 u 2 . . . un . By induction, ( qi -1 , ui , Bi ) /turnstileleft ∗ ( qi , λ, λ ). Therefore we have so that ( p , w, B ) ( q , λ, λ ). We now show that if ( p , w, B ) /turnst… \end{verbatim} ``` </details>
274. ph-d4f6d127bda1ef16eb25 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) o that ( p , w, B ) ( q , λ, λ ). We now show that if ( p , w, B ) /turnstileleft ∗ ( q , λ, λ ) then 〈 p , B , q 〉 ⇒ ∗ w . We use induction on the number of steps in ( p , w, B ) /turnstileleft ∗ ( q , λ, λ ). If there are 0 steps, then p = q and w = B = λ . This corresponds to 〈 p , λ, p 〉 ⇒ λ which is one of the productions. Therefore the statement is true for 0 steps. /turnstileleft ∗ Assume ( p , w, B ) /turnstileleft ∗ ( q , λ, λ ) in k + 1 steps. First assume that we have w = a v and where (( p , a , B ) , ( r , D )) ∈ ϒ , and B , D ∈ I ∪ { λ } , giving pro… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{o that ( p , w, B ) ( q , λ, λ ). We now show that if ( p , w, B ) /turnstileleft ∗ ( q , λ, λ ) then 〈 p , B , q 〉 ⇒ ∗ w . We use induction on the number of steps in ( p , w, B ) /turnstileleft ∗ ( q , λ, λ ). If there are 0 steps, then p = q and w = B = λ . This corresponds to…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=287445 \begin{verbatim} o that ( p , w, B ) ( q , λ, λ ). We now show that if ( p , w, B ) /turnstileleft ∗ ( q , λ, λ ) then 〈 p , B , q 〉 ⇒ ∗ w . We use induction on the number of steps in ( p , w, B ) /turnstileleft ∗ ( q , λ, λ ). If there are 0 steps, then p = q and w = B = λ . This corresponds to 〈 p , λ, p 〉 ⇒ λ which is one of the productions. Therefore the statement is true for 0 steps. /turnstileleft ∗ Assume ( p , w, B ) /turnstileleft ∗ ( q , λ, λ ) in k + 1 steps. First assume that we have w = a v and where (( p , a , B ) , ( r , D )) ∈ ϒ , and B , D ∈ I ∪ { λ } , giving pro… \end{verbatim} ``` </details>
275. ph-fcef3319e7c15c02d0c9 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) ∗ ( q , λ, λ ) in k + 1 steps. First assume that we have w = a v and where (( p , a , B ) , ( r , D )) ∈ ϒ , and B , D ∈ I ∪ { λ } , giving productions 〈 p , B , q 〉 → a 〈 r , D , q 〉 . Since ( q , v, D ) /turnstileleft ∗ ( q , λ, λ ) by induction hypothesis, 〈 r , D , q 〉 ⇒ ∗ v . Therefore 〈 p , B , q 〉 ⇒ a 〈 r , D , q 〉 ⇒ ∗ a v = w and we are finished. Next assume w = a v and the first step is ( p , w, B ) /turnstileleft ( q , v, B 1 B 2 . . . Bn ) so we have and each Bi is eventually removed from the sta… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{∗ ( q , λ, λ ) in k + 1 steps. First assume that we have w = a v and where (( p , a , B ) , ( r , D )) ∈ ϒ , and B , D ∈ I ∪ \{ λ \} , giving productions 〈 p , B , q 〉 → a 〈 r , D , q 〉 . Since ( q , v, D ) /turnstileleft ∗ ( q , λ, λ ) by induction hy…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=287874 \begin{verbatim} ∗ ( q , λ, λ ) in k + 1 steps. First assume that we have w = a v and where (( p , a , B ) , ( r , D )) ∈ ϒ , and B , D ∈ I ∪ { λ } , giving productions 〈 p , B , q 〉 → a 〈 r , D , q 〉 . Since ( q , v, D ) /turnstileleft ∗ ( q , λ, λ ) by induction hypothesis, 〈 r , D , q 〉 ⇒ ∗ v . Therefore 〈 p , B , q 〉 ⇒ a 〈 r , D , q 〉 ⇒ ∗ a v = w and we are finished. Next assume w = a v and the first step is ( p , w, B ) /turnstileleft ( q , v, B 1 B 2 . . . Bn ) so we have and each Bi is eventually removed from the sta… \end{verbatim} ``` </details>
276. ph-4989090d5cb3a7088f69 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) . First assume that we have w = a v and where (( p , a , B ) , ( r , D )) ∈ ϒ , and B , D ∈ I ∪ { λ } , giving productions 〈 p , B , q 〉 → a 〈 r , D , q 〉 . Since ( q , v, D ) /turnstileleft ∗ ( q , λ, λ ) by induction hypothesis, 〈 r , D , q 〉 ⇒ ∗ v . Therefore 〈 p , B , q 〉 ⇒ a 〈 r , D , q 〉 ⇒ ∗ a v = w and we are finished. Next assume w = a v and the first step is ( p , w, B ) /turnstileleft ( q , v, B 1 B 2 . . . Bn ) so we have and each Bi is eventually removed from the stack in order so that there are… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{. First assume that we have w = a v and where (( p , a , B ) , ( r , D )) ∈ ϒ , and B , D ∈ I ∪ \{ λ \} , giving productions 〈 p , B , q 〉 → a 〈 r , D , q 〉 . Since ( q , v, D ) /turnstileleft ∗ ( q , λ, λ ) by induction hypothesis, 〈 r , D , q 〉 ⇒ ∗ v…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=287904 \begin{verbatim} . First assume that we have w = a v and where (( p , a , B ) , ( r , D )) ∈ ϒ , and B , D ∈ I ∪ { λ } , giving productions 〈 p , B , q 〉 → a 〈 r , D , q 〉 . Since ( q , v, D ) /turnstileleft ∗ ( q , λ, λ ) by induction hypothesis, 〈 r , D , q 〉 ⇒ ∗ v . Therefore 〈 p , B , q 〉 ⇒ a 〈 r , D , q 〉 ⇒ ∗ a v = w and we are finished. Next assume w = a v and the first step is ( p , w, B ) /turnstileleft ( q , v, B 1 B 2 . . . Bn ) so we have and each Bi is eventually removed from the stack in order so that there are… \end{verbatim} ``` </details>
277. ph-5225fa7ff9a905c0462e — automata/docling_md/AutomataTheory.md ### Plain (markdown context) v, D ) /turnstileleft ∗ ( q , λ, λ ) by induction hypothesis, 〈 r , D , q 〉 ⇒ ∗ v . Therefore 〈 p , B , q 〉 ⇒ a 〈 r , D , q 〉 ⇒ ∗ a v = w and we are finished. Next assume w = a v and the first step is ( p , w, B ) /turnstileleft ( q , v, B 1 B 2 . . . Bn ) so we have and each Bi is eventually removed from the stack in order so that there are states q , q , . . . , where q and v v v . . . v v such that 1 2 qn -1 qn qn = = 1 2 n -1 n By the induction hypothesis, 〈 qi -1 , Bi , qi 〉 ⇒ ∗ v i . But since the pro… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{v, D ) /turnstileleft ∗ ( q , λ, λ ) by induction hypothesis, 〈 r , D , q 〉 ⇒ ∗ v . Therefore 〈 p , B , q 〉 ⇒ a 〈 r , D , q 〉 ⇒ ∗ a v = w and we are finished. Next assume w = a v and the first step is ( p , w, B ) /turnstileleft ( q , v, B 1 B 2 . . . Bn ) so we have and each Bi is eventually removed from the stack in order so that there are states q , q , . . . , where q and v v v . . . v v such that 1 2 qn -1 qn qn = = 1 2 n -1 n By the induction hypothesis, 〈 qi -1 , Bi , qi 〉 ⇒ ∗ v i . But since the pro… \end{verbatim} ```
278. ph-1963a1891d878d1bd382 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) ∗ a v = w and we are finished. Next assume w = a v and the first step is ( p , w, B ) /turnstileleft ( q , v, B 1 B 2 . . . Bn ) so we have and each Bi is eventually removed from the stack in order so that there are states q , q , . . . , where q and v v v . . . v v such that 1 2 qn -1 qn qn = = 1 2 n -1 n By the induction hypothesis, 〈 qi -1 , Bi , qi 〉 ⇒ ∗ v i . But since the production is type (3), so that 〈 p , B , q 〉 ⇒ ∗ w . /square Theorem 4.6 Alanguage is context-free if… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{∗ a v = w and we are finished. Next assume w = a v and the first step is ( p , w, B ) /turnstileleft ( q , v, B 1 B 2 . . . Bn ) so we have and each Bi is eventually removed from the stack in order so that there are state…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=288232 \begin{verbatim} ∗ a v = w and we are finished. Next assume w = a v and the first step is ( p , w, B ) /turnstileleft ( q , v, B 1 B 2 . . . Bn ) so we have and each Bi is eventually removed from the stack in order so that there are states q , q , . . . , where q and v v v . . . v v such that 1 2 qn -1 qn qn = = 1 2 n -1 n By the induction hypothesis, 〈 qi -1 , Bi , qi 〉 ⇒ ∗ v i . But since the production is type (3), so that 〈 p , B , q 〉 ⇒ ∗ w . /square Theorem 4.6 Alanguage is context-free if… \end{verbatim} ``` </details>
279. ph-10c622cfab6d4341370e — automata/docling_md/AutomataTheory.md ### Plain (markdown context) - formula-not-decoded --> By the induction hypothesis, 〈 qi -1 , Bi , qi 〉 ⇒ ∗ v i . But since the production is type (3), so that 〈 p , B , q 〉 ⇒ ∗ w . /square Theorem 4.6 Alanguage is context-free if and only if it is accepted by a PDA. ## Exercises - (1) Construct a pushdown automaton which reads the same language as the grammar /Gamma1 = ( N , /Sigma1 , S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by - (2) Construct a pushdown automaton which reads the same language as generate… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{- formula-not-decoded --> By the induction hypothesis, 〈 qi -1 , Bi , qi 〉 ⇒ ∗ v i . But since the production is type (3), so that 〈 p , B , q 〉 ⇒ ∗ w . /square Theorem 4.6 Alanguage is context-free if and only if it is accepted by a PDA. \#\# Exercise…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=288607 \begin{verbatim} - formula-not-decoded --> By the induction hypothesis, 〈 qi -1 , Bi , qi 〉 ⇒ ∗ v i . But since the production is type (3), so that 〈 p , B , q 〉 ⇒ ∗ w . /square Theorem 4.6 Alanguage is context-free if and only if it is accepted by a PDA. ## Exercises - (1) Construct a pushdown automaton which reads the same language as the grammar /Gamma1 = ( N , /Sigma1 , S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by - (2) Construct a pushdown automaton which reads the same language as generate… \end{verbatim} ``` </details>
280. ph-adf352b684c18242dc3d — automata/docling_md/AutomataTheory.md ### Plain (markdown context) ccepted by a PDA. ## Exercises - (1) Construct a pushdown automaton which reads the same language as the grammar /Gamma1 = ( N , /Sigma1 , S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by - (2) Construct a pushdown automaton which reads the same language as generated by the grammar /Gamma1 = ( N , /Sigma1 , S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by - (3) Construct a pushdown automaton which reads the same language as gener… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ccepted by a PDA. \#\# Exercises - (1) Construct a pushdown automaton which reads the same language as the grammar /Gamma1 = ( N , /Sigma1 , S , P ) defined by N = \{ S , A , B \} , /Sigma1 = \{ a , b , c \} , and the set of productions P given by - (2) Co…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=288862 \begin{verbatim} ccepted by a PDA. ## Exercises - (1) Construct a pushdown automaton which reads the same language as the grammar /Gamma1 = ( N , /Sigma1 , S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by - (2) Construct a pushdown automaton which reads the same language as generated by the grammar /Gamma1 = ( N , /Sigma1 , S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by - (3) Construct a pushdown automaton which reads the same language as gener… \end{verbatim} ``` </details>
281. ph-b97f5cb312c1c8173db7 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) -not-decoded --> - (2) Construct a pushdown automaton which reads the same language as generated by the grammar /Gamma1 = ( N , /Sigma1 , S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by - (3) Construct a pushdown automaton which reads the same language as generated by the grammar /Gamma1 = ( N , /Sigma1 , S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c , } , and the set of productions P given by - (4) Construct a pushdown automaton which reads the same language as gene… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{-not-decoded --> - (2) Construct a pushdown automaton which reads the same language as generated by the grammar /Gamma1 = ( N , /Sigma1 , S , P ) defined by N = \{ S , A , B \} , /Sigma1 = \{ a , b , c \} , and the set of productions P given by - (3) Con…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=289119 \begin{verbatim} -not-decoded --> - (2) Construct a pushdown automaton which reads the same language as generated by the grammar /Gamma1 = ( N , /Sigma1 , S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by - (3) Construct a pushdown automaton which reads the same language as generated by the grammar /Gamma1 = ( N , /Sigma1 , S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c , } , and the set of productions P given by - (4) Construct a pushdown automaton which reads the same language as gene… \end{verbatim} ``` </details>
282. ph-c8b9ec7e6a4137f7b84f — automata/docling_md/AutomataTheory.md ### Plain (markdown context) -not-decoded --> - (3) Construct a pushdown automaton which reads the same language as generated by the grammar /Gamma1 = ( N , /Sigma1 , S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c , } , and the set of productions P given by - (4) Construct a pushdown automaton which reads the same language as generated by the grammar /Gamma1 = ( N , /Sigma1 , S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by - (5) Construct a pushdown automaton which reads the same language as gene… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{-not-decoded --> - (3) Construct a pushdown automaton which reads the same language as generated by the grammar /Gamma1 = ( N , /Sigma1 , S , P ) defined by N = \{ S , A , B \} , /Sigma1 = \{ a , b , c , \} , and the set of productions P given by - (4) C…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=289374 \begin{verbatim} -not-decoded --> - (3) Construct a pushdown automaton which reads the same language as generated by the grammar /Gamma1 = ( N , /Sigma1 , S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c , } , and the set of productions P given by - (4) Construct a pushdown automaton which reads the same language as generated by the grammar /Gamma1 = ( N , /Sigma1 , S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by - (5) Construct a pushdown automaton which reads the same language as gene… \end{verbatim} ``` </details>
283. ph-dfc7348afd50ced2c8be — automata/docling_md/AutomataTheory.md ### Plain (markdown context) ot-decoded --> - (4) Construct a pushdown automaton which reads the same language as generated by the grammar /Gamma1 = ( N , /Sigma1 , S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by - (5) Construct a pushdown automaton which reads the same language as generated by the grammar /Gamma1 = ( N , /Sigma1 , S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c , d } , and the set of productions P given by - (6) Construct a grammar which generates the language read by the pushdow… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ot-decoded --> - (4) Construct a pushdown automaton which reads the same language as generated by the grammar /Gamma1 = ( N , /Sigma1 , S , P ) defined by N = \{ S , A , B \} , /Sigma1 = \{ a , b , c \} , and the set of productions P given by - (5) Const…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=289633 \begin{verbatim} ot-decoded --> - (4) Construct a pushdown automaton which reads the same language as generated by the grammar /Gamma1 = ( N , /Sigma1 , S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by - (5) Construct a pushdown automaton which reads the same language as generated by the grammar /Gamma1 = ( N , /Sigma1 , S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c , d } , and the set of productions P given by - (6) Construct a grammar which generates the language read by the pushdow… \end{verbatim} ``` </details>
284. ph-725313c712bdee48cc5f — automata/docling_md/AutomataTheory.md ### Plain (markdown context) e the grammar is in Greibach normal form. - (11) Construct a pushdown automaton that reads the same language as the grammar /Gamma1 = ( N , /Sigma1 , S , P ) defined by N = { S , B } ∪ /Sigma1 , /Sigma1 = { a , b , c } , and the set of productions P given by ![Image](./AutomataTheory_artifacts/image_000202_e9b0569031c5d3dd136abfc521bc0580ef1b2fd1ec6dc42aac4a9642c31c9a9e.png) ![Image](./AutomataTheory_artifacts/image_000203_49abd552d81bd1620b2260dcd62a38991bfd7a0dc38b7a7819b097408548b7fe.png) - (12) Construct a pushdown automaton that reads the same language as the… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{e the grammar is in Greibach normal form. - (11) Construct a pushdown automaton that reads the same language as the grammar /Gamma1 = ( N , /Sigma1 , S , P ) defined by N = \{ S , B \} ∪ /Sigma1 , /Sigma1 = \{ a , b , c \} , and the set of productions P given by ![Image](./AutomataT…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=290847 \begin{verbatim} e the grammar is in Greibach normal form. - (11) Construct a pushdown automaton that reads the same language as the grammar /Gamma1 = ( N , /Sigma1 , S , P ) defined by N = { S , B } ∪ /Sigma1 , /Sigma1 = { a , b , c } , and the set of productions P given by ![Image](./AutomataTheory_artifacts/image_000202_e9b0569031c5d3dd136abfc521bc0580ef1b2fd1ec6dc42aac4a9642c31c9a9e.png) ![Image](./AutomataTheory_artifacts/image_000203_49abd552d81bd1620b2260dcd62a38991bfd7a0dc38b7a7819b097408548b7fe.png) - (12) Construct a pushdown automaton that reads the same language as the… \end{verbatim} ``` </details>
285. ph-eef34c4c242b64d7193f — automata/docling_md/AutomataTheory.md ### Plain (markdown context) tions P given by ![Image](./AutomataTheory_artifacts/image_000202_e9b0569031c5d3dd136abfc521bc0580ef1b2fd1ec6dc42aac4a9642c31c9a9e.png) ![Image](./AutomataTheory_artifacts/image_000203_49abd552d81bd1620b2260dcd62a38991bfd7a0dc38b7a7819b097408548b7fe.png) - (12) Construct a pushdown automaton that reads the same language as the grammar /Gamma1 = ( N , /Sigma1 , S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by - (13) Construct a pushdown automaton that reads the same language as the g… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{tions P given by ![Image](./AutomataTheory\_artifacts/image\_000202\_e9b0569031c5d3dd136abfc521bc0580ef1b2fd1ec6dc42aac4a9642c31c9a9e.png) ![Image](./AutomataTheory\_artifacts/image\_000203\_49abd552d81bd1620b2260dcd62a38991bfd7a0dc38b7a7819b097408548b7fe.png) - (12) Construct a pushdown automaton that reads the same language as the grammar /Gamma1 = ( N , /Sigma1 , S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by - (13) Construct a pushdown automaton that reads the same language as the g… \end{verbatim} ```
286. ph-c2984ce05aaabfe94168 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) 19b097408548b7fe.png) - (12) Construct a pushdown automaton that reads the same language as the grammar /Gamma1 = ( N , /Sigma1 , S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by - (13) Construct a pushdown automaton that reads the same language as the grammar /Gamma1 = ( N , ϒ, S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by - (14) Construct a pushdown automaton that reads the same language as the g… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{19b097408548b7fe.png) - (12) Construct a pushdown automaton that reads the same language as the grammar /Gamma1 = ( N , /Sigma1 , S , P ) defined by N = \{ S , A , B \} , /Sigma1 = \{ a , b , c \} , and the set of productions P given by - (12) Construct a pushdown automaton that reads the same language as the grammar /Gamma1 = ( N , /Sigma1 , S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by - (13) Construct a pushdown automaton that reads the same language as the grammar /Gamma1 = ( N , ϒ, S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by - (14) Construct a pushdown automaton that reads the same language as the g… \end{verbatim} ```
287. ph-8f86b2b16aa93526dacd — automata/docling_md/AutomataTheory.md ### Plain (markdown context) the set of productions P given by - (13) Construct a pushdown automaton that reads the same language as the grammar /Gamma1 = ( N , ϒ, S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by - (14) Construct a pushdown automaton that reads the same language as the grammar /Gamma1 = ( N , ϒ, S , P ) defined by N = { S , A , B } , ϒ = { a , b , c } , and the set of productions P given by - (15) Construct a pushdown automaton that reads the same language as the gr… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{the set of productions P given by - (13) Construct a pushdown automaton that reads the same language as the grammar /Gamma1 = ( N , ϒ, S , P ) defined by N = \{ S , A , B \} , /Sigma1 = \{ a , b , c \} , and the set of productions P given by - (13) Construct a pushdown automaton that reads the same language as the grammar /Gamma1 = ( N , ϒ, S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by - (14) Construct a pushdown automaton that reads the same language as the grammar /Gamma1 = ( N , ϒ, S , P ) defined by N = { S , A , B } , ϒ = { a , b , c } , and the set of productions P given by - (15) Construct a pushdown automaton that reads the same language as the gr… \end{verbatim} ```
288. ph-29ef0d75fa32481eb507 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) nd the set of productions P given by - (14) Construct a pushdown automaton that reads the same language as the grammar /Gamma1 = ( N , ϒ, S , P ) defined by N = { S , A , B } , ϒ = { a , b , c } , and the set of productions P given by - (15) Construct a pushdown automaton that reads the same language as the grammar /Gamma1 = ( N , ϒ, S , P ) defined by N = { S , A , B } , ϒ = { a , b , c , d } , and the set of productions P given by ## 4.4 The Pumping Lemma and decidability Just as we were able to show that… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{nd the set of productions P given by - (14) Construct a pushdown automaton that reads the same language as the grammar /Gamma1 = ( N , ϒ, S , P ) defined by N = \{ S , A , B \} , ϒ = \{ a , b , c \} , and the set of productions P given by - (14) Construct a pushdown automaton that reads the same language as the grammar /Gamma1 = ( N , ϒ, S , P ) defined by N = { S , A , B } , ϒ = { a , b , c } , and the set of productions P given by - (15) Construct a pushdown automaton that reads the same language as the grammar /Gamma1 = ( N , ϒ, S , P ) defined by N = { S , A , B } , ϒ = { a , b , c , d } , and the set of productions P given by ## 4.4 The Pumping Lemma and decidability Just as we were able to show that… \end{verbatim} ```
289. ph-3dcfd9d06074ad03a24b — automata/docling_md/AutomataTheory.md ### Plain (markdown context) . Let M = 2 p . Assume there is a word w in L with length greater than or equal to M . Then by the previous theorem, the derivation tree has height greater than p . Therefore there is a path S →··· → a where a is a letter in the derivation tree with length greater than p and a is a letter of w . Since there are only p productions, some nonterminal occurs more than once on the left-hand side of a production. Let C be the first nonterminal to occur the second time. Therefore we have a derivation where α ⇒ ∗ x , β ⇒ ∗ y , C ⇒ ∗ uC v and C ⇒ w . But using these deriva… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{. Let M = 2 p . Assume there is a word w in L with length greater than or equal to M . Then by the previous theorem, the derivation tree has height greater than p . Therefore there is a path S →··· → a where a is a letter in the derivation tree with length greater than p and a i…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=294116 \begin{verbatim} . Let M = 2 p . Assume there is a word w in L with length greater than or equal to M . Then by the previous theorem, the derivation tree has height greater than p . Therefore there is a path S →··· → a where a is a letter in the derivation tree with length greater than p and a is a letter of w . Since there are only p productions, some nonterminal occurs more than once on the left-hand side of a production. Let C be the first nonterminal to occur the second time. Therefore we have a derivation where α ⇒ ∗ x , β ⇒ ∗ y , C ⇒ ∗ uC v and C ⇒ w . But using these deriva… \end{verbatim} ``` </details>
290. ph-a505b92c9171a13bfaea — automata/docling_md/AutomataTheory.md ### Plain (markdown context) 1 S 1 · · · S 1 . Using leftmost derivations we can derive S ⇒ ∗ S 1 S 1 S 1 · · · S 1 ⇒ ∗ w 1 S 1 S 1 · · · S 1 ⇒ ∗ w 1 w 2 S 1 · · · S 1 ⇒ ∗ w 1 w 2 · · · w n in /Gamma1 . Hence L ∗ 1 = L , the language generated by /Gamma1 . /square Theorem 4.9 The set of context-free languages is not closed under the operations of intersection and complement. Proof The sets { a n b n c m : m , n ≥ 0 } and { a n b m c m : m , n ≥ 0 } are contextfree. The first is generated by the grammar with productions The second is generated by the grammar with productions The second is generated by the grammar with productions The second is generated by the grammar with productions However, the intersection is the language L = { a m b m a m : m ≥ 0 } , whic… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{⇒ ∗ w 1 S 1 S 1 · · · S 1 ⇒ ∗ w 1 w 2 S 1 · · · S 1 ⇒ ∗ w 1 w 2 · · · w n in /Gamma1 . Hence L ∗ 1 = L , the language generated by /Gamma1 . /square Theorem 4.9 The set of context-free languages is not closed under the operations of intersection and complement. Proof The sets \{ …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=299146 \begin{verbatim} ⇒ ∗ w 1 S 1 S 1 · · · S 1 ⇒ ∗ w 1 w 2 S 1 · · · S 1 ⇒ ∗ w 1 w 2 · · · w n in /Gamma1 . Hence L ∗ 1 = L , the language generated by /Gamma1 . /square Theorem 4.9 The set of context-free languages is not closed under the operations of intersection and complement. Proof The sets { a n b n c m : m , n ≥ 0 } and { a n b m c m : m , n ≥ 0 } are contextfree. The first is generated by the grammar with productions The second is generated by the grammar with productions However, the intersection is the language L = { a m b m a m : m ≥ 0 } , whic… \end{verbatim} ``` </details>
292. ph-c81a2593594e7d844a42 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) generated by a context-free grammar is finite or infinite. Proof Since it is possible to determine whether a word is in the language of a context-free grammar, simply try all words with length between 2 p and 2 p + 1 to see if one of them is in the context-free grammar. If one is, the grammar is infinite. If not the grammar is finite. /square ## Exercises - (1) Let grammar /Gamma1 = ( N , ϒ, S , P ) be defined by N = { S , A , B } , ϒ = { a , b , c } , and the set of productions P given by Let L be the language generated by /Gamma1 . Find the grammar that generate… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{generated by a context-free grammar is finite or infinite. Proof Since it is possible to determine whether a word is in the language of a context-free grammar, simply try all words with length between 2 p and 2 p + 1 to see if one of them is in the context-free grammar. If one i…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=305198 \begin{verbatim} generated by a context-free grammar is finite or infinite. Proof Since it is possible to determine whether a word is in the language of a context-free grammar, simply try all words with length between 2 p and 2 p + 1 to see if one of them is in the context-free grammar. If one is, the grammar is infinite. If not the grammar is finite. /square ## Exercises - (1) Let grammar /Gamma1 = ( N , ϒ, S , P ) be defined by N = { S , A , B } , ϒ = { a , b , c } , and the set of productions P given by Let L be the language generated by /Gamma1 . Find the grammar that generate… \end{verbatim} ``` </details>
293. ph-dd9585d988a9571e092a — automata/docling_md/AutomataTheory.md ### Plain (markdown context) inite. If not the grammar is finite. /square ## Exercises - (1) Let grammar /Gamma1 = ( N , ϒ, S , P ) be defined by N = { S , A , B } , ϒ = { a , b , c } , and the set of productions P given by Let L be the language generated by /Gamma1 . Find the grammar that generates L ∗ . - (2) Let L 1 be the language generated by the grammar /Gamma1 1 = ( N , /Sigma1 , S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by and L 2 be the language generated by the grammar /Gamma1 2 = ( N , /Sigma1 , … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{inite. If not the grammar is finite. /square \#\# Exercises - (1) Let grammar /Gamma1 = ( N , ϒ, S , P ) be defined by N = \{ S , A , B \} , ϒ = \{ a , b , c \} , and the set of productions P given by Let L be the language generated by /Gamma1 . Find the g…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=305500 \begin{verbatim} inite. If not the grammar is finite. /square ## Exercises - (1) Let grammar /Gamma1 = ( N , ϒ, S , P ) be defined by N = { S , A , B } , ϒ = { a , b , c } , and the set of productions P given by Let L be the language generated by /Gamma1 . Find the grammar that generates L ∗ . - (2) Let L 1 be the language generated by the grammar /Gamma1 1 = ( N , /Sigma1 , S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by and L 2 be the language generated by the grammar /Gamma1 2 = ( N , /Sigma1 , … \end{verbatim} ``` </details>
294. ph-3c50ef6f0ff98da5a51c — automata/docling_md/AutomataTheory.md ### Plain (markdown context) not-decoded --> Let L be the language generated by /Gamma1 . Find the grammar that generates L ∗ . - (2) Let L 1 be the language generated by the grammar /Gamma1 1 = ( N , /Sigma1 , S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by and L 2 be the language generated by the grammar /Gamma1 2 = ( N , /Sigma1 , S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by Find the grammar that generates L 1 L 2 . - (3) Let L 1 be the language gene… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{not-decoded --> Let L be the language generated by /Gamma1 . Find the grammar that generates L ∗ . - (2) Let L 1 be the language generated by the grammar /Gamma1 1 = ( N , /Sigma1 , S , P ) defined by N = \{ S , A , B \} , /Sigma1 = \{ a , b , c \} , and the set of productions P giv…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=305711 \begin{verbatim} not-decoded --> Let L be the language generated by /Gamma1 . Find the grammar that generates L ∗ . - (2) Let L 1 be the language generated by the grammar /Gamma1 1 = ( N , /Sigma1 , S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by and L 2 be the language generated by the grammar /Gamma1 2 = ( N , /Sigma1 , S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by Find the grammar that generates L 1 L 2 . - (3) Let L 1 be the language gene… \end{verbatim} ``` </details>
295. ph-d2337a4043de93746a4a — automata/docling_md/AutomataTheory.md ### Plain (markdown context) the set of productions P given by and L 2 be the language generated by the grammar /Gamma1 2 = ( N , /Sigma1 , S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by Find the grammar that generates L 1 L 2 . - (3) Let L 1 be the language generated by the grammar /Gamma1 1 = ( N , ϒ, S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by and L 2 be the language generated by the grammar /Gamma1 2 = ( N , ϒ, S , P )… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{the set of productions P given by and L 2 be the language generated by the grammar /Gamma1 2 = ( N , /Sigma1 , S , P ) defined by N = \{ S , A , B \} , /Sigma1 = \{ a , b , c \} , and the set of productions P given by Find th…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=305964 \begin{verbatim} the set of productions P given by and L 2 be the language generated by the grammar /Gamma1 2 = ( N , /Sigma1 , S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by Find the grammar that generates L 1 L 2 . - (3) Let L 1 be the language generated by the grammar /Gamma1 1 = ( N , ϒ, S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by and L 2 be the language generated by the grammar /Gamma1 2 = ( N , ϒ, S , P )… \end{verbatim} ``` </details>
296. ph-91faf3704bb661e63232 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) b , c } , and the set of productions P given by Find the grammar that generates L 1 L 2 . - (3) Let L 1 be the language generated by the grammar /Gamma1 1 = ( N , ϒ, S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by and L 2 be the language generated by the grammar /Gamma1 2 = ( N , ϒ, S , P ) defined by N = { S , A , B } , ϒ = { a , b , c } , and the set of productions P given by Find the grammar that generates L 1 ∪ L 2 . Determine whether the following … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{b , c \} , and the set of productions P given by Find the grammar that generates L 1 L 2 . - (3) Let L 1 be the language generated by the grammar /Gamma1 1 = ( N , ϒ, S , P ) defined by N = \{ S , A , B \} , /Sigma1 = \{ a , b , c \} , and the set of prod…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=306162 \begin{verbatim} b , c } , and the set of productions P given by Find the grammar that generates L 1 L 2 . - (3) Let L 1 be the language generated by the grammar /Gamma1 1 = ( N , ϒ, S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by and L 2 be the language generated by the grammar /Gamma1 2 = ( N , ϒ, S , P ) defined by N = { S , A , B } , ϒ = { a , b , c } , and the set of productions P given by Find the grammar that generates L 1 ∪ L 2 . Determine whether the following … \end{verbatim} ``` </details>
297. ph-de7816b59d28dfe1dac0 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) t of the tape symbols. As input, readingablankissimplyreadingtheabsenceofanyofthetapesymbols.Printing a blank is considered to be erasing the symbol currently in that square. We use # for blank. The Turing machine shown below is in state s 1 and is reading letter a . ![Image](./AutomataTheory_artifacts/image_000204_93b8824c7887d5a3cd67e9a38e9de031df722093638a54d1b81993447d1cf49e.png) More formally we have the following definition. Definition 5.1 A deterministic Turing machine is a quintuple where Q is the set of states, /Gamma1 is a finite set of tape symbols, whi… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{t of the tape symbols. As input, readingablankissimplyreadingtheabsenceofanyofthetapesymbols.Printing a blank is considered to be erasing the symbol currently in that square. We use \# for blank. The Turing machine shown below is in state s 1 and is reading letter a . ![Image](./…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=310013 \begin{verbatim} t of the tape symbols. As input, readingablankissimplyreadingtheabsenceofanyofthetapesymbols.Printing a blank is considered to be erasing the symbol currently in that square. We use # for blank. The Turing machine shown below is in state s 1 and is reading letter a . ![Image](./AutomataTheory_artifacts/image_000204_93b8824c7887d5a3cd67e9a38e9de031df722093638a54d1b81993447d1cf49e.png) More formally we have the following definition. Definition 5.1 A deterministic Turing machine is a quintuple where Q is the set of states, /Gamma1 is a finite set of tape symbols, whi… \end{verbatim} ``` </details>
298. ph-a5d3b2cb897ada98412f — automata/docling_md/AutomataTheory.md ### Plain (markdown context) set of tape symbols, which includes the alphabet and #, s 0 is the starting state, h is the halt state, and δ is a function from Q × /Gamma1 to Q × /Gamma1 × N where N consists of L which indicates a movement on the tape one position to the left, R which indicates a movement on the tape one position to the right, and # which indicates that no movement takes place. Just like any computer, a Turing machine has a program or set of rules which tell the machine what to do. An example of a rule is This rules says that if the machine is in st… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{set of tape symbols, which includes the alphabet and \#, s 0 is the starting state, h is the halt state, and δ is a function from Q × /Gamma1 to Q × /Gamma1 × N where N consists of L which indicates a movement on the tape one position to the left, R which indicates a movement on …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=310592 \begin{verbatim} set of tape symbols, which includes the alphabet and #, s 0 is the starting state, h is the halt state, and δ is a function from Q × /Gamma1 to Q × /Gamma1 × N where N consists of L which indicates a movement on the tape one position to the left, R which indicates a movement on the tape one position to the right, and # which indicates that no movement takes place. Just like any computer, a Turing machine has a program or set of rules which tell the machine what to do. An example of a rule is This rules says that if the machine is in st… \end{verbatim} ``` </details>
299. ph-dd05f5137be94463c734 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) cludes the alphabet and #, s 0 is the starting state, h is the halt state, and δ is a function from Q × /Gamma1 to Q × /Gamma1 × N where N consists of L which indicates a movement on the tape one position to the left, R which indicates a movement on the tape one position to the right, and # which indicates that no movement takes place. Just like any computer, a Turing machine has a program or set of rules which tell the machine what to do. An example of a rule is This rules says that if the machine is in state s 1 and reads the letter … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{cludes the alphabet and \#, s 0 is the starting state, h is the halt state, and δ is a function from Q × /Gamma1 to Q × /Gamma1 × N where N consists of L which indicates a movement on the tape one position to the left, R which indicates a movement on the tape one position to the …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=310622 \begin{verbatim} cludes the alphabet and #, s 0 is the starting state, h is the halt state, and δ is a function from Q × /Gamma1 to Q × /Gamma1 × N where N consists of L which indicates a movement on the tape one position to the left, R which indicates a movement on the tape one position to the right, and # which indicates that no movement takes place. Just like any computer, a Turing machine has a program or set of rules which tell the machine what to do. An example of a rule is This rules says that if the machine is in state s 1 and reads the letter … \end{verbatim} ``` </details>
300. ph-ee352ebc823150539c47 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) indicates a movement on the tape one position to the right, and # which indicates that no movement takes place. Just like any computer, a Turing machine has a program or set of rules which tell the machine what to do. An example of a rule is This rules says that if the machine is in state s 1 and reads the letter a , it is to change to state s 2 , print the letter b in place of the letter a and move one square to the left. The rule which we shall denote as says that if the machine is in state s 1 and reads … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{indicates a movement on the tape one position to the right, and \# which indicates that no movement takes place. Just like any computer, a Turing machine has a program or set of rules which tell the machine what to do. An example of a rule is This rules says that if the machine is in state s 1 and reads the letter a , it is to change to state s 2 , print the letter b in place of the letter a and move one square to the left. The rule which we shall denote as says that if the machine is in state s 1 and reads … \end{verbatim} ```
301. ph-65839ebdc1ea6378f0a2 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) e what to do. An example of a rule is This rules says that if the machine is in state s 1 and reads the letter a , it is to change to state s 2 , print the letter b in place of the letter a and move one square to the left. The rule which we shall denote as says that if the machine is in state s 1 and reads the letter a , it changes to state s 2 , erases the a and moves one square to the right. The rule says that if the machine is in state s 1 and reads a blank then it halts, and… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{e what to do. An example of a rule is This rules says that if the machine is in state s 1 and reads the letter a , it is to change to state s 2 , print the letter b in place of the letter a and move one square to the left…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=311053 \begin{verbatim} e what to do. An example of a rule is This rules says that if the machine is in state s 1 and reads the letter a , it is to change to state s 2 , print the letter b in place of the letter a and move one square to the left. The rule which we shall denote as says that if the machine is in state s 1 and reads the letter a , it changes to state s 2 , erases the a and moves one square to the right. The rule says that if the machine is in state s 1 and reads a blank then it halts, and… \end{verbatim} ``` </details>
302. ph-fff73e2a21fbcf67faa5 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) he tape to the left or to the right. Some definitions allow a machine either to print a letter or to move the head, but not both. Thus it requires two separate rules to print a letter and move the position on the tape. Weshall begin with a program that simply moves the position of the machine on the tape from the beginning to the end of a string. The alphabet is /Sigma1 = { a , b } and symbols /Gamma1 = { a , b , # } . We shall have the set of states Q = { s 0 , s 1 , h } and the set of rules This program leaves everything alone. It simply reads each letter and th… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{he tape to the left or to the right. Some definitions allow a machine either to print a letter or to move the head, but not both. Thus it requires two separate rules to print a letter and move the position on the tape. Weshall begin with a program that simply moves the position …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=312509 \begin{verbatim} he tape to the left or to the right. Some definitions allow a machine either to print a letter or to move the head, but not both. Thus it requires two separate rules to print a letter and move the position on the tape. Weshall begin with a program that simply moves the position of the machine on the tape from the beginning to the end of a string. The alphabet is /Sigma1 = { a , b } and symbols /Gamma1 = { a , b , # } . We shall have the set of states Q = { s 0 , s 1 , h } and the set of rules This program leaves everything alone. It simply reads each letter and th… \end{verbatim} ``` </details>
303. ph-b888ad4d1df4714420fa — automata/docling_md/AutomataTheory.md ### Plain (markdown context) d . As we demonstrate this program, it would be rather tiresome to continually draw the Turing machine so rather than draw ![Image](./AutomataTheory_artifacts/image_000205_64e9f170c5ccc95c50b0d86fa97c8052043867611bbceb8dbf7c12f355037ae4.png) which shows the position of the machine at the second square of the tape and in state s 1, while the first and third squares of the tape contain an a , the second, fourth and fifth squares contain a b and the other squares are blank; we replace this with where the line below the b denotes the location of the head, and the 1 ab… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{d . As we demonstrate this program, it would be rather tiresome to continually draw the Turing machine so rather than draw ![Image](./AutomataTheory\_artifacts/image\_000205\_64e9f170c5ccc95c50b0d86fa97c8052043867611bbceb8dbf7c12f355037ae4.png) which shows the position of the machi…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=313553 \begin{verbatim} d . As we demonstrate this program, it would be rather tiresome to continually draw the Turing machine so rather than draw ![Image](./AutomataTheory_artifacts/image_000205_64e9f170c5ccc95c50b0d86fa97c8052043867611bbceb8dbf7c12f355037ae4.png) which shows the position of the machine at the second square of the tape and in state s 1, while the first and third squares of the tape contain an a , the second, fourth and fifth squares contain a b and the other squares are blank; we replace this with where the line below the b denotes the location of the head, and the 1 ab… \end{verbatim} ``` </details>
304. ph-a9288b65b010e3f85a45 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) the machine at the second square of the tape and in state s 1, while the first and third squares of the tape contain an a , the second, fourth and fifth squares contain a b and the other squares are blank; we replace this with where the line below the b denotes the location of the head, and the 1 above the b denotes the current state of the machine. We shall call this the configuration of the Turing machine. As we begin our program the machine has configuration moving the head to the right and changing from… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{the machine at the second square of the tape and in state s 1, while the first and third squares of the tape contain an a , the second, fourth and fifth squares contain a b and the other squares are blank; we replace this with where the line below th…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=313824 \begin{verbatim} the machine at the second square of the tape and in state s 1, while the first and third squares of the tape contain an a , the second, fourth and fifth squares contain a b and the other squares are blank; we replace this with where the line below the b denotes the location of the head, and the 1 above the b denotes the current state of the machine. We shall call this the configuration of the Turing machine. As we begin our program the machine has configuration moving the head to the right and changing from… \end{verbatim} ``` </details>
305. ph-4618975b9e439bc831ea — automata/docling_md/AutomataTheory.md ### Plain (markdown context) are of the tape and in state s 1, while the first and third squares of the tape contain an a , the second, fourth and fifth squares contain a b and the other squares are blank; we replace this with where the line below the b denotes the location of the head, and the 1 above the b denotes the current state of the machine. We shall call this the configuration of the Turing machine. As we begin our program the machine has configuration moving the head to the right and changing from state s 0 to state s 1 and o… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{are of the tape and in state s 1, while the first and third squares of the tape contain an a , the second, fourth and fifth squares contain a b and the other squares are blank; we replace this with where the line below the b denotes the location of t…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=313854 \begin{verbatim} are of the tape and in state s 1, while the first and third squares of the tape contain an a , the second, fourth and fifth squares contain a b and the other squares are blank; we replace this with where the line below the b denotes the location of the head, and the 1 above the b denotes the current state of the machine. We shall call this the configuration of the Turing machine. As we begin our program the machine has configuration moving the head to the right and changing from state s 0 to state s 1 and o… \end{verbatim} ``` </details>
306. ph-5cd8c4ea90aea1414cbf — automata/docling_md/AutomataTheory.md ### Plain (markdown context) a b and the other squares are blank; we replace this with where the line below the b denotes the location of the head, and the 1 above the b denotes the current state of the machine. We shall call this the configuration of the Turing machine. As we begin our program the machine has configuration moving the head to the right and changing from state s 0 to state s 1 and our machine then has configuration moving the head to the right again and our machi… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{a b and the other squares are blank; we replace this with where the line below the b denotes the location of the head, and the 1 above the b denotes the current state of the machine. We shall call this the configuration of the Turing machine. As we b…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=313994 \begin{verbatim} a b and the other squares are blank; we replace this with where the line below the b denotes the location of the head, and the 1 above the b denotes the current state of the machine. We shall call this the configuration of the Turing machine. As we begin our program the machine has configuration moving the head to the right and changing from state s 0 to state s 1 and our machine then has configuration moving the head to the right again and our machi… \end{verbatim} ``` </details>
307. ph-5b73083a48985cb8a62a — automata/docling_md/AutomataTheory.md ### Plain (markdown context) blank; we replace this with where the line below the b denotes the location of the head, and the 1 above the b denotes the current state of the machine. We shall call this the configuration of the Turing machine. As we begin our program the machine has configuration moving the head to the right and changing from state s 0 to state s 1 and our machine then has configuration moving the head to the right again and our machine then has configuration where the line below the b denotes the location of the head, and the 1 above the b denotes the current state of the machine. We shall call this the configuration of the Turing machine. As we begin our program the machine h…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=314024 \begin{verbatim} blank; we replace this with where the line below the b denotes the location of the head, and the 1 above the b denotes the current state of the machine. We shall call this the configuration of the Turing machine. As we begin our program the machine has configuration moving the head to the right and changing from state s 0 to state s 1 and our machine then has configuration moving the head to the right again and our machine then has configuration moving the head to the right and changing from state s 0 to state s 1 and our machine then has configuration moving the head to the right again and our machine then has configuration moving the head to the right again and our machin… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{n of the head, and the 1 above the b denotes the current state of the machine. We shall call this the configuration of the Turing machine. As we begin our program the machine has configuration moving the head to the right…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=314129 \begin{verbatim} n of the head, and the 1 above the b denotes the current state of the machine. We shall call this the configuration of the Turing machine. As we begin our program the machine has configuration moving the head to the right and changing from state s 0 to state s 1 and our machine then has configuration moving the head to the right again and our machine then has configuration moving the head to the right again and our machin… \end{verbatim} ``` </details>
309. ph-6820d5499bd90e23df0f — automata/docling_md/AutomataTheory.md ### Plain (markdown context) the b denotes the current state of the machine. We shall call this the configuration of the Turing machine. As we begin our program the machine has configuration moving the head to the right and changing from state s 0 to state s 1 and our machine then has configuration moving the head to the right again and our machine then has configuration moving the head to the right again and our machine then has configuration moving the head to the right and changing from state s 0 to…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=314159 \begin{verbatim} the b denotes the current state of the machine. We shall call this the configuration of the Turing machine. As we begin our program the machine has configuration moving the head to the right and changing from state s 0 to state s 1 and our machine then has configuration moving the head to the right again and our machine then has configuration moving the head to the right again and our machine then has configuration moving the head to the right and changing from state s 0 to state s 1 and our machine then has configuration moving the head to the right again and our machine then has configuration moving the head to the right again and our machine then has configuration moving the head to the right again and our machine t… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ne. As we begin our program the machine has configuration moving the head to the right and changing from state s 0 to state s 1 and our machine then has configuration moving the head to the right and changing from state s 0 to state s 1 and our machine then has configuration moving the head to the right again and our machine then has configuration moving the head to the right again and our machine then has configuration moving the head to the right again and our machine t… \end{verbatim} ``` </details>
311. ph-35d2e50a0f0919784a76 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) he machine has configuration moving the head to the right and changing from state s 0 to state s 1 and our machine then has configuration moving the head to the right again and our machine then has configuration moving the head to the right again and our machine then has configuration moving the head to the right again and our machine then has configuration moving the head to the right and changing from state s 0 to state s 1 and our machine then has configuration moving the head to the ri…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=314294 \begin{verbatim} he machine has configuration moving the head to the right and changing from state s 0 to state s 1 and our machine then has configuration moving the head to the right again and our machine then has configuration moving the head to the right again and our machine then has configuration moving the head to the right again and our machine then has configuration moving the head to the right again and our machine then has configuration moving the head to the right again and our machine then has configuration moving the head to the right again and our machine then has configuration We then apply rule We then apply rule We then apply rule We again apply rule We a… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{to the right and changing from state s 0 to state s 1 and our machine then has configuration moving the head to the right again and our machine then has configuration moving the head to the right again and our machine then has configuration moving the head to the right again and our machine then has configuration moving the head to the right again and our machine then has configuration We then apply rule We then apply rule We then apply rule We again apply rule We a… \end{verbatim} ``` </details>
313. ph-fde30236fd72c96844ba — automata/docling_md/AutomataTheory.md ### Plain (markdown context) --> moving the head to the right again and our machine then has configuration moving the head to the right again and our machine then has configuration moving the head to the right again and our machine then has configuration We then apply rule We then apply rule We then apply rule We again apply rule We apply the same rule again and have We then use rule and the machine shuts down. We mention… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{--> moving the head to the right again and our machine then has configuration moving the head to the right again and our machine then has configuration moving the …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=314549 \begin{verbatim} --> moving the head to the right again and our machine then has configuration moving the head to the right again and our machine then has configuration moving the head to the right again and our machine then has configuration We then apply rule We then apply rule We then apply rule We again apply rule We apply the same rule again and have We then use rule and the machine shuts down. We mention… \end{verbatim} ``` </details>
314. ph-3c63a178bcada74db87b — automata/docling_md/AutomataTheory.md ### Plain (markdown context) machine then has configuration moving the head to the right again and our machine then has configuration moving the head to the right again and our machine then has configuration We then apply rule We then apply rule We then apply rule We again apply rule We apply the same rule again and have We then use rule and the machine shuts down. We mentioned previously that if the position on the tape … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{machine then has configuration moving the head to the right again and our machine then has configuration moving the head to the right again and our machine then ha…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=314597 \begin{verbatim} machine then has configuration moving the head to the right again and our machine then has configuration moving the head to the right again and our machine then has configuration We then apply rule We then apply rule We then apply rule We again apply rule We apply the same rule again and have We then use rule and the machine shuts down. We mentioned previously that if the position on the tape … \end{verbatim} ``` </details>
315. ph-777685ab31e11d283f8d — automata/docling_md/AutomataTheory.md ### Plain (markdown context) d the machine shuts down. We mentioned previously that if the position on the tape is on the leftmost position on the tape and gets an instruction to move left, we say that the machine crashes , and the machine ceases functioning. We shall now construct a rather unusual program. This program causes the machine to crash. We shall again let the input alphabet /Gamma1 be the set { a , b , # } . We shall also assume we have states Q = { s 0 , s 1 , . . . , s j , . . . } . It shall have the rules If we have a larger alphabet, we simply add more rules, so that regardles… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{d the machine shuts down. We mentioned previously that if the position on the tape is on the leftmost position on the tape and gets an instruction to move left, we say that the machine crashes , and the machine ceases functioning. We shall now construct a rather unusual program.…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=315129 \begin{verbatim} d the machine shuts down. We mentioned previously that if the position on the tape is on the leftmost position on the tape and gets an instruction to move left, we say that the machine crashes , and the machine ceases functioning. We shall now construct a rather unusual program. This program causes the machine to crash. We shall again let the input alphabet /Gamma1 be the set { a , b , # } . We shall also assume we have states Q = { s 0 , s 1 , . . . , s j , . . . } . It shall have the rules If we have a larger alphabet, we simply add more rules, so that regardles… \end{verbatim} ``` </details>
316. ph-eeb319d29a9ce5c10cad — automata/docling_md/AutomataTheory.md ### Plain (markdown context) efine a function. If the rules do not define a function then there is a state s and an input letter a for which there is no rule. When this happens, we say that the system hangs , since it cannot go on. We shall again meet this problem with nondeterministic automata. Suppose we would like the set of rules to define a function, but we still want the program to stop when it is in state s and reads a . The system cannot hang since the function is defined for every input. We can however add a rule which puts the system into the suicide state and causes it to crash usi… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{efine a function. If the rules do not define a function then there is a state s and an input letter a for which there is no rule. When this happens, we say that the system hangs , since it cannot go on. We shall again meet this problem with nondeterministic automata. Suppose we …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=316602 \begin{verbatim} efine a function. If the rules do not define a function then there is a state s and an input letter a for which there is no rule. When this happens, we say that the system hangs , since it cannot go on. We shall again meet this problem with nondeterministic automata. Suppose we would like the set of rules to define a function, but we still want the program to stop when it is in state s and reads a . The system cannot hang since the function is defined for every input. We can however add a rule which puts the system into the suicide state and causes it to crash usi… \end{verbatim} ``` </details>
317. ph-303e6456adafbc629c25 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) c automata. Suppose we would like the set of rules to define a function, but we still want the program to stop when it is in state s and reads a . The system cannot hang since the function is defined for every input. We can however add a rule which puts the system into the suicide state and causes it to crash using go-crash. Thus the system crashes instead of hanging and we have expanded our rules so that we have a function. In this discussion, we will state only relevant rules with the understanding that we could produce a function us… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{c automata. Suppose we would like the set of rules to define a function, but we still want the program to stop when it is in state s and reads a . The system cannot hang since the function is defined for every input. We can however add a rule which p…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=316858 \begin{verbatim} c automata. Suppose we would like the set of rules to define a function, but we still want the program to stop when it is in state s and reads a . The system cannot hang since the function is defined for every input. We can however add a rule which puts the system into the suicide state and causes it to crash using go-crash. Thus the system crashes instead of hanging and we have expanded our rules so that we have a function. In this discussion, we will state only relevant rules with the understanding that we could produce a function us… \end{verbatim} ``` </details>
318. ph-22ed12ef3b242bbde419 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) for the Turing machine. We begin by showing some of its properties as a text editor. Our first step is not exactly a giant one. We show how to move the position on the tape to the right n steps. Again we assume both the input and output alphabet are the set { a , b } . If we have a larger alphabet, we simply add appropriate rules for each new letter. The set of states Q = { s 1 , . . . , s j , . . . , sn , sn + 1 } . We shall call this new subroutine go-right ( n ). It has the following rules: It is easily seen that if we begin in state s 1, each application of a … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{for the Turing machine. We begin by showing some of its properties as a text editor. Our first step is not exactly a giant one. We show how to move the position on the tape to the right n steps. Again we assume both the input and output alphabet are the set \{ a , b \} . If we hav…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=317569 \begin{verbatim} for the Turing machine. We begin by showing some of its properties as a text editor. Our first step is not exactly a giant one. We show how to move the position on the tape to the right n steps. Again we assume both the input and output alphabet are the set { a , b } . If we have a larger alphabet, we simply add appropriate rules for each new letter. The set of states Q = { s 1 , . . . , s j , . . . , sn , sn + 1 } . We shall call this new subroutine go-right ( n ). It has the following rules: It is easily seen that if we begin in state s 1, each application of a … \end{verbatim} ``` </details>
319. ph-f209c17d40c1f02eaa18 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) position on the tape one step to the right and increases the state. After n steps the head has been moved to the right by n squares and we are in state sn + 1 . It is hoped that, with little effort, the reader can create a subroutine for moving to the left by n squares. Suppose that after moving left or right by n squares, or without moving at all we want to change the letter in the current square occupied from a to b . Assuming that we are in state si at the time then we simply use the rule Moving along, suppose that /Gamma1 = /Sigma1… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{position on the tape one step to the right and increases the state. After n steps the head has been moved to the right by n squares and we are in state sn + 1 . It is hoped that, with little effort, the reader can create a subroutine for moving to the left by n squares. Suppose …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=318220 \begin{verbatim} position on the tape one step to the right and increases the state. After n steps the head has been moved to the right by n squares and we are in state sn + 1 . It is hoped that, with little effort, the reader can create a subroutine for moving to the left by n squares. Suppose that after moving left or right by n squares, or without moving at all we want to change the letter in the current square occupied from a to b . Assuming that we are in state si at the time then we simply use the rule Moving along, suppose that /Gamma1 = /Sigma1… \end{verbatim} ``` </details>
320. ph-6063f1191f5fefa94d60 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) the state. After n steps the head has been moved to the right by n squares and we are in state sn + 1 . It is hoped that, with little effort, the reader can create a subroutine for moving to the left by n squares. Suppose that after moving left or right by n squares, or without moving at all we want to change the letter in the current square occupied from a to b . Assuming that we are in state si at the time then we simply use the rule Moving along, suppose that /Gamma1 = /Sigma1 ∪ { # } and we want to replace Moving along, suppose that /Gamma1 = /Sigma1 ∪ { # } and we want to replace Moving along, suppose that /Gamma1 = /Sigma1 ∪ { # } and we want to replace We first use go-right ( i ) to move to the prop… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{e are in state sn + 1 . It is hoped that, with little effort, the reader can create a subroutine for moving to the left by n squares. Suppose that after moving left or right by n squares, or without moving at all we want to change the letter in the current square occupied from a…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=318358 \begin{verbatim} e are in state sn + 1 . It is hoped that, with little effort, the reader can create a subroutine for moving to the left by n squares. Suppose that after moving left or right by n squares, or without moving at all we want to change the letter in the current square occupied from a to b . Assuming that we are in state si at the time then we simply use the rule Moving along, suppose that /Gamma1 = /Sigma1 ∪ { # } and we want to replace We first use go-right ( i ) to move to the prop… \end{verbatim} ``` </details>
322. ph-00f226ed85970c5dced1 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) hoped that, with little effort, the reader can create a subroutine for moving to the left by n squares. Suppose that after moving left or right by n squares, or without moving at all we want to change the letter in the current square occupied from a to b . Assuming that we are in state si at the time then we simply use the rule Moving along, suppose that /Gamma1 = /Sigma1 ∪ { # } and we want to replace We first use go-right ( i ) to move to the proper position so the head is on … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{hoped that, with little effort, the reader can create a subroutine for moving to the left by n squares. Suppose that after moving left or right by n squares, or without moving at all we want to change the letter in the current square occupied from a to b . Assuming that we are i…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=318388 \begin{verbatim} hoped that, with little effort, the reader can create a subroutine for moving to the left by n squares. Suppose that after moving left or right by n squares, or without moving at all we want to change the letter in the current square occupied from a to b . Assuming that we are in state si at the time then we simply use the rule Moving along, suppose that /Gamma1 = /Sigma1 ∪ { # } and we want to replace We first use go-right ( i ) to move to the proper position so the head is on … \end{verbatim} ``` </details>
323. ph-642a45920452d9f4c369 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) t moving at all we want to change the letter in the current square occupied from a to b . Assuming that we are in state si at the time then we simply use the rule Moving along, suppose that /Gamma1 = /Sigma1 ∪ { # } and we want to replace We first use go-right ( i ) to move to the proper position so the head is on ai + 1 . Assume we are in state s ′ , We then use the rules to replace the letters and use go-left ( j ) to return to the original spot. w… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{t moving at all we want to change the letter in the current square occupied from a to b . Assuming that we are in state si at the time then we simply use the rule Moving along, suppose that /Gamma1 = /Sigma1 ∪ \{ \# \} and w…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=318556 \begin{verbatim} t moving at all we want to change the letter in the current square occupied from a to b . Assuming that we are in state si at the time then we simply use the rule Moving along, suppose that /Gamma1 = /Sigma1 ∪ { # } and we want to replace We first use go-right ( i ) to move to the proper position so the head is on ai + 1 . Assume we are in state s ′ , We then use the rules to replace the letters and use go-left ( j ) to return to the original spot. w… \end{verbatim} ``` </details>
324. ph-ec5e76a1e220407bd71c — automata/docling_md/AutomataTheory.md ### Plain (markdown context) > We first use go-right ( i ) to move to the proper position so the head is on ai + 1 . Assume we are in state s ′ , We then use the rules to replace the letters and use go-left ( j ) to return to the original spot. with The next text edit feature which we shall illustrate is to insert a letter in a string. We shall find this feature very handy in the near future. We shall call this subroutine insert(c) . Say that we have a string and we want to replace it with so that the strin… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{> We first use go-right ( i ) to move to the proper position so the head is on ai + 1 . Assume we are in state s ′ , We then use the rules to replace the letters and use go-left ( j ) to return to the original spot. with …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=318885 \begin{verbatim} > We first use go-right ( i ) to move to the proper position so the head is on ai + 1 . Assume we are in state s ′ , We then use the rules to replace the letters and use go-left ( j ) to return to the original spot. with The next text edit feature which we shall illustrate is to insert a letter in a string. We shall find this feature very handy in the near future. We shall call this subroutine insert(c) . Say that we have a string and we want to replace it with so that the strin… \end{verbatim} ``` </details>
325. ph-85e5b32ee1e2f1a7d213 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) o move to the proper position so the head is on ai + 1 . Assume we are in state s ′ , We then use the rules to replace the letters and use go-left ( j ) to return to the original spot. with The next text edit feature which we shall illustrate is to insert a letter in a string. We shall find this feature very handy in the near future. We shall call this subroutine insert(c) . Say that we have a string and we want to replace it with so that the string ai + 1 · · · an -1 an must be moved one square to the righ… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{o move to the proper position so the head is on ai + 1 . Assume we are in state s ′ , We then use the rules to replace the letters and use go-left ( j ) to return to the original spot. with The next text edit feature which we shall illustrate is to i…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=318947 \begin{verbatim} o move to the proper position so the head is on ai + 1 . Assume we are in state s ′ , We then use the rules to replace the letters and use go-left ( j ) to return to the original spot. with The next text edit feature which we shall illustrate is to insert a letter in a string. We shall find this feature very handy in the near future. We shall call this subroutine insert(c) . Say that we have a string and we want to replace it with so that the string ai + 1 · · · an -1 an must be moved one square to the righ… \end{verbatim} ``` </details>
326. ph-82eb74a0f5c7ae1089b7 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) ponding to the letter destroyed and in this way 'remember' this letter so it can be printed in the next square. Remember in state sai + j , we print ai + j regardless of the letter read. Finally, when we reach a blank square, we print an and then use go-left ( n ) to return to the beginning of the string. Also it is possible that c occurs elsewhere in the string; however, we shall assume that ai + 1 is not already c . Thus our rules for actually printing c and moving over the other letters are and we end up in state sy . For example assume we have the word abbac a… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ponding to the letter destroyed and in this way 'remember' this letter so it can be printed in the next square. Remember in state sai + j , we print ai + j regardless of the letter read. Finally, when we reach a blank square, we print an and then use go-left ( n ) to return to t…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=320736 \begin{verbatim} ponding to the letter destroyed and in this way 'remember' this letter so it can be printed in the next square. Remember in state sai + j , we print ai + j regardless of the letter read. Finally, when we reach a blank square, we print an and then use go-left ( n ) to return to the beginning of the string. Also it is possible that c occurs elsewhere in the string; however, we shall assume that ai + 1 is not already c . Thus our rules for actually printing c and moving over the other letters are and we end up in state sy . For example assume we have the word abbac a… \end{verbatim} ``` </details>
327. ph-f26e60f7c65c3e8aec6e — automata/docling_md/AutomataTheory.md ### Plain (markdown context) ally, when we reach a blank square, we print an and then use go-left ( n ) to return to the beginning of the string. Also it is possible that c occurs elsewhere in the string; however, we shall assume that ai + 1 is not already c . Thus our rules for actually printing c and moving over the other letters are and we end up in state sy . For example assume we have the word abbac and want to insert c so that we have abcbbc . Using go-right (2), we have configuration ## Applying rule we have configuration and we end up in state sy . For example assume we have the word abbac and want to insert c so that we have abcbbc . Using go-right (2), we have configuration ## Applying rule we have configuration and we end up in state sy . For example assume we have the word abbac and want to insert c so that we have abcbbc . Using go-right (2), we have configuration ## Applying rule we have configuration In the future we will condense this statement to… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{n ) to return to the beginning of the string. Also it is possible that c occurs elsewhere in the string; however, we shall assume that ai + 1 is not already c . Thus our rules for actually printing c and moving over the other letters are and we end u…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=320997 \begin{verbatim} n ) to return to the beginning of the string. Also it is possible that c occurs elsewhere in the string; however, we shall assume that ai + 1 is not already c . Thus our rules for actually printing c and moving over the other letters are and we end up in state sy . For example assume we have the word abbac and want to insert c so that we have abcbbc . Using go-right (2), we have configuration ## Applying rule we have configuration In the future we will condense this statement to… \end{verbatim} ``` </details>
329. ph-86ccee8493b91ff2e26b — automata/docling_md/AutomataTheory.md ### Plain (markdown context) of the string. Also it is possible that c occurs elsewhere in the string; however, we shall assume that ai + 1 is not already c . Thus our rules for actually printing c and moving over the other letters are and we end up in state sy . For example assume we have the word abbac and want to insert c so that we have abcbbc . Using go-right (2), we have configuration ## Applying rule we have configuration In the future we will condense this statement to W… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{of the string. Also it is possible that c occurs elsewhere in the string; however, we shall assume that ai + 1 is not already c . Thus our rules for actually printing c and moving over the other letters are and we end up in state sy . For example ass…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=321027 \begin{verbatim} of the string. Also it is possible that c occurs elsewhere in the string; however, we shall assume that ai + 1 is not already c . Thus our rules for actually printing c and moving over the other letters are and we end up in state sy . For example assume we have the word abbac and want to insert c so that we have abcbbc . Using go-right (2), we have configuration ## Applying rule we have configuration In the future we will condense this statement to W… \end{verbatim} ``` </details>
330. ph-1e8d92d490d2b69444a6 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) er, we shall assume that ai + 1 is not already c . Thus our rules for actually printing c and moving over the other letters are and we end up in state sy . For example assume we have the word abbac and want to insert c so that we have abcbbc . Using go-right (2), we have configuration ## Applying rule we have configuration In the future we will condense this statement to We then have the following rules and configurations … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{er, we shall assume that ai + 1 is not already c . Thus our rules for actually printing c and moving over the other letters are and we end up in state sy . For example assume we have the word abbac and want to insert c so that we have abcbbc . Using …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=321107 \begin{verbatim} er, we shall assume that ai + 1 is not already c . Thus our rules for actually printing c and moving over the other letters are and we end up in state sy . For example assume we have the word abbac and want to insert c so that we have abcbbc . Using go-right (2), we have configuration ## Applying rule we have configuration In the future we will condense this statement to We then have the following rules and configurations … \end{verbatim} ``` </details>
331. ph-672f1f2fcaf8553751ce — automata/docling_md/AutomataTheory.md ### Plain (markdown context) ting c and moving over the other letters are and we end up in state sy . For example assume we have the word abbac and want to insert c so that we have abcbbc . Using go-right (2), we have configuration ## Applying rule we have configuration In the future we will condense this statement to We then have the following rules and configurations and we now use go-left (5) to return to our original … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ting c and moving over the other letters are and we end up in state sy . For example assume we have the word abbac and want to insert c so that we have abcbbc . Using go-right (2), we have configuration \#\# Applying rule w…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=321190 \begin{verbatim} ting c and moving over the other letters are and we end up in state sy . For example assume we have the word abbac and want to insert c so that we have abcbbc . Using go-right (2), we have configuration ## Applying rule we have configuration In the future we will condense this statement to We then have the following rules and configurations and we now use go-left (5) to return to our original … \end{verbatim} ``` </details>
332. ph-d109ec7349d7475d68a9 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) er letters are and we end up in state sy . For example assume we have the word abbac and want to insert c so that we have abcbbc . Using go-right (2), we have configuration ## Applying rule we have configuration In the future we will condense this statement to We then have the following rules and configurations and we now use go-left (5) to return to our original position. Suppose we began at … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{er letters are and we end up in state sy . For example assume we have the word abbac and want to insert c so that we have abcbbc . Using go-right (2), we have configuration \#\# Applying rule we have configuration and we end up in state sy . For example assume we have the word abbac and want to insert c so that we have abcbbc . Using go-right (2), we have configuration ## Applying rule we have configuration In the future we will condense this statement to We then have the following rules and configurations and we now use go-left (5) to return to our original position. Suppose we began at … \end{verbatim} ```
333. ph-f52bb6500c68f784945d — automata/docling_md/AutomataTheory.md ### Plain (markdown context) nd replace the marker with that letter. Then go right again to the letter which has been duplicated and replace it with a marker. Continue this process until reaching the end of the string. Let /Delta1 be the special marker. Assume that we have used go-right ( i ) to reach the letter c to be deleted. Again assume that we begin in state sx and want to end in state sy . We shall also let the /Sigma1 = { a , b , c } and /Gamma1 = { a , b , c } ∪ { # , } . We then have the following set of rules: Note that the marker is not actually needed. It is used to make the rule… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{nd replace the marker with that letter. Then go right again to the letter which has been duplicated and replace it with a marker. Continue this process until reaching the end of the string. Let /Delta1 be the special marker. Assume that we have used go-right ( i ) to reach the l…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=323308 \begin{verbatim} nd replace the marker with that letter. Then go right again to the letter which has been duplicated and replace it with a marker. Continue this process until reaching the end of the string. Let /Delta1 be the special marker. Assume that we have used go-right ( i ) to reach the letter c to be deleted. Again assume that we begin in state sx and want to end in state sy . We shall also let the /Sigma1 = { a , b , c } and /Gamma1 = { a , b , c } ∪ { # , } . We then have the following set of rules: Note that the marker is not actually needed. It is used to make the rule… \end{verbatim} ``` </details>
334. ph-392a3ff4b896118c6479 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) o be deleted. Again assume that we begin in state sx and want to end in state sy . We shall also let the /Sigma1 = { a , b , c } and /Gamma1 = { a , b , c } ∪ { # , } . We then have the following set of rules: Note that the marker is not actually needed. It is used to make the rules easier to read. For example, suppose we have the string abcbac and wish to remove the c in the third space. We use go-right (2) to get to the desired space and have the configuration We then have the following rules and configurations: Note that the marker is not actually nee…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=323597 \begin{verbatim} o be deleted. Again assume that we begin in state sx and want to end in state sy . We shall also let the /Sigma1 = { a , b , c } and /Gamma1 = { a , b , c } ∪ { # , } . We then have the following set of rules: Note that the marker is not actually needed. It is used to make the rules easier to read. For example, suppose we have the string abcbac and wish to remove the c in the third space. We use go-right (2) to get to the desired space and have the configuration We then have the following rules and configurations: Note that the marker is not actually needed. It is used to make the rules easier to read. For example, suppose we have the string abcbac and wish to remove the c in the third space. We use go-right (2) to get to the desired space and have the configuration We then have the following rules and configurations: Finally applying rule we have configuration <!-… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{e shall also let the /Sigma1 = \{ a , b , c \} and /Gamma1 = \{ a , b , c \} ∪ \{ \# , \} . We then have the following set of rules: Note that the marker is not actually needed. It is used to make the rules easier to read. For example, suppose we have the s…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=323681 \begin{verbatim} e shall also let the /Sigma1 = { a , b , c } and /Gamma1 = { a , b , c } ∪ { # , } . We then have the following set of rules: Note that the marker is not actually needed. It is used to make the rules easier to read. For example, suppose we have the string abcbac and wish to remove the c in the third space. We use go-right (2) to get to the desired space and have the configuration We then have the following rules and configurations: Finally applying rule we have configuration <!-… \end{verbatim} ``` </details>
336. ph-eab291a78504faf752c5 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) { # , } . We then have the following set of rules: Note that the marker is not actually needed. It is used to make the rules easier to read. For example, suppose we have the string abcbac and wish to remove the c in the third space. We use go-right (2) to get to the desired space and have the configuration We then have the following rules and configurations: Finally applying rule we have configuration and using go-left (5), we return to the beginning… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{\{ \# , \} . We then have the following set of rules: Note that the marker is not actually needed. It is used to make the rules easier to read. For example, suppose we have the string abcbac and wish to remove the c in the third space. We use go-right (…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=323756 \begin{verbatim} { # , } . We then have the following set of rules: Note that the marker is not actually needed. It is used to make the rules easier to read. For example, suppose we have the string abcbac and wish to remove the c in the third space. We use go-right (2) to get to the desired space and have the configuration We then have the following rules and configurations: Finally applying rule we have configuration and using go-left (5), we return to the beginning… \end{verbatim} ``` </details>
337. ph-23b2ba978b3588db8418 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) lowing set of rules: Note that the marker is not actually needed. It is used to make the rules easier to read. For example, suppose we have the string abcbac and wish to remove the c in the third space. We use go-right (2) to get to the desired space and have the configuration We then have the following rules and configurations: Finally applying rule we have configuration and using go-left (5), we return to the beginning of the string. Finally we sho… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{lowing set of rules: Note that the marker is not actually needed. It is used to make the rules easier to read. For example, suppose we have the string abcbac and wish to remove the c in the third space. We use go-right (2) to get to the desired space…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=323786 \begin{verbatim} lowing set of rules: Note that the marker is not actually needed. It is used to make the rules easier to read. For example, suppose we have the string abcbac and wish to remove the c in the third space. We use go-right (2) to get to the desired space and have the configuration We then have the following rules and configurations: Finally applying rule we have configuration and using go-left (5), we return to the beginning of the string. Finally we sho… \end{verbatim} ``` </details>
338. ph-78a7056921f125760410 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) tring is replaced by λ a if the letter is a and by λ b if the letter is b . We then go to the end of the string and place a corresponding σ a , or σ b . We then return to the first symbol and replace it with the original letter, go to the second letter and repeat the process. We continue until we have a string followed by corresponding σ a s, and σ b s. We then replace each σ a with an a and σ b with a b . Assumethatwestartinstate sx andendinstate sy . We then have the following set of rules: For example, we shall duplicate the word bab . The initial configuration… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{tring is replaced by λ a if the letter is a and by λ b if the letter is b . We then go to the end of the string and place a corresponding σ a , or σ b . We then return to the first symbol and replace it with the original letter, go to the second letter and repeat the process. We…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=324723 \begin{verbatim} tring is replaced by λ a if the letter is a and by λ b if the letter is b . We then go to the end of the string and place a corresponding σ a , or σ b . We then return to the first symbol and replace it with the original letter, go to the second letter and repeat the process. We continue until we have a string followed by corresponding σ a s, and σ b s. We then replace each σ a with an a and σ b with a b . Assumethatwestartinstate sx andendinstate sy . We then have the following set of rules: For example, we shall duplicate the word bab . The initial configuration… \end{verbatim} ``` </details>
339. ph-49a4d9e711c411bd4120 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) ring and place a corresponding σ a , or σ b . We then return to the first symbol and replace it with the original letter, go to the second letter and repeat the process. We continue until we have a string followed by corresponding σ a s, and σ b s. We then replace each σ a with an a and σ b with a b . Assumethatwestartinstate sx andendinstate sy . We then have the following set of rules: For example, we shall duplicate the word bab . The initial configuration is We then have the following rules and configurations: For example, we shall duplicate the word bab . The initial configuration is We then have the following rules and configurations: For example, we shall duplicate the word bab . The initial configuration is We then have the following rules and configurations: ⇒ ( sx ′ , a , sx ′ , a , L ) /turnstileleft x ′ λ b a b σ b ⇒ ( sx ′ , λ b ,… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{replace it with the original letter, go to the second letter and repeat the process. We continue until we have a string followed by corresponding σ a s, and σ b s. We then replace each σ a with an a and σ b with a b . Assumethatwestartinstate sx andendinstate sy . We then have t…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=324914 \begin{verbatim} replace it with the original letter, go to the second letter and repeat the process. We continue until we have a string followed by corresponding σ a s, and σ b s. We then replace each σ a with an a and σ b with a b . Assumethatwestartinstate sx andendinstate sy . We then have the following set of rules: For example, we shall duplicate the word bab . The initial configuration is We then have the following rules and configurations: ⇒ ( sx ′ , a , sx ′ , a , L ) /turnstileleft x ′ λ b a b σ b ⇒ ( sx ′ , λ b ,… \end{verbatim} ``` </details>
341. ph-42a94726f466d438cb83 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) g that a Turing machine can recognize a regular language. We already know that an automaton recognizes a regular language, so what we shall basically do is program it to imitate an automaton. Assume that we have a word in the Turing machine which we want the machine to read so that it can determine whether it wants to accept it. An automaton reads a word beginning with the first letter and reads from left to right until it has reached the last letter. We need our Turing machine to do the same. and we are ready to begin. Wehave another way of representing a Turing … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{g that a Turing machine can recognize a regular language. We already know that an automaton recognizes a regular language, so what we shall basically do is program it to imitate an automaton. Assume that we have a word in the Turing machine which we want the machine to read so t…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=327061 \begin{verbatim} g that a Turing machine can recognize a regular language. We already know that an automaton recognizes a regular language, so what we shall basically do is program it to imitate an automaton. Assume that we have a word in the Turing machine which we want the machine to read so that it can determine whether it wants to accept it. An automaton reads a word beginning with the first letter and reads from left to right until it has reached the last letter. We need our Turing machine to do the same. and we are ready to begin. Wehave another way of representing a Turing … \end{verbatim} ``` </details>
342. ph-0c711b17123215df9fc1 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) tomaton. Assume that we have a word in the Turing machine which we want the machine to read so that it can determine whether it wants to accept it. An automaton reads a word beginning with the first letter and reads from left to right until it has reached the last letter. We need our Turing machine to do the same. and we are ready to begin. Wehave another way of representing a Turing machine which makes it look more like an automaton. We shall represent the rule so that the program go-end which has rules <!… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{tomaton. Assume that we have a word in the Turing machine which we want the machine to read so that it can determine whether it wants to accept it. An automaton reads a word beginning with the first letter and reads from left to right until it has reached the last letter. We nee…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=327244 \begin{verbatim} tomaton. Assume that we have a word in the Turing machine which we want the machine to read so that it can determine whether it wants to accept it. An automaton reads a word beginning with the first letter and reads from left to right until it has reached the last letter. We need our Turing machine to do the same. and we are ready to begin. Wehave another way of representing a Turing machine which makes it look more like an automaton. We shall represent the rule so that the program go-end which has rules <!… \end{verbatim} ``` </details>
343. ph-1829f7ed01982e235d00 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) word in the Turing machine which we want the machine to read so that it can determine whether it wants to accept it. An automaton reads a word beginning with the first letter and reads from left to right until it has reached the last letter. We need our Turing machine to do the same. and we are ready to begin. Wehave another way of representing a Turing machine which makes it look more like an automaton. We shall represent the rule so that the program go-end which has rules by t… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{word in the Turing machine which we want the machine to read so that it can determine whether it wants to accept it. An automaton reads a word beginning with the first letter and reads from left to right until it has reached the last letter. We need our Turing machine to do the …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=327274 \begin{verbatim} word in the Turing machine which we want the machine to read so that it can determine whether it wants to accept it. An automaton reads a word beginning with the first letter and reads from left to right until it has reached the last letter. We need our Turing machine to do the same. and we are ready to begin. Wehave another way of representing a Turing machine which makes it look more like an automaton. We shall represent the rule so that the program go-end which has rules by t… \end{verbatim} ``` </details>
344. ph-6236023764f1a4f361cf — automata/docling_md/AutomataTheory.md ### Plain (markdown context) an determine whether it wants to accept it. An automaton reads a word beginning with the first letter and reads from left to right until it has reached the last letter. We need our Turing machine to do the same. and we are ready to begin. Wehave another way of representing a Turing machine which makes it look more like an automaton. We shall represent the rule so that the program go-end which has rules by the symbol may b… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{an determine whether it wants to accept it. An automaton reads a word beginning with the first letter and reads from left to right until it has reached the last letter. We need our Turing machine to do the same. and we are ready to begin. Wehave anot…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=327348 \begin{verbatim} an determine whether it wants to accept it. An automaton reads a word beginning with the first letter and reads from left to right until it has reached the last letter. We need our Turing machine to do the same. and we are ready to begin. Wehave another way of representing a Turing machine which makes it look more like an automaton. We shall represent the rule so that the program go-end which has rules by the symbol may b… \end{verbatim} ``` </details>
345. ph-56accc550dadc4bd700c — automata/docling_md/AutomataTheory.md ### Plain (markdown context) n automaton reads a word beginning with the first letter and reads from left to right until it has reached the last letter. We need our Turing machine to do the same. and we are ready to begin. Wehave another way of representing a Turing machine which makes it look more like an automaton. We shall represent the rule so that the program go-end which has rules by the symbol may be represented by or the symbol shown as ![Ima… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{n automaton reads a word beginning with the first letter and reads from left to right until it has reached the last letter. We need our Turing machine to do the same. and we are ready to begin. Wehave another way of representing a Turing machine whic…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=327393 \begin{verbatim} n automaton reads a word beginning with the first letter and reads from left to right until it has reached the last letter. We need our Turing machine to do the same. and we are ready to begin. Wehave another way of representing a Turing machine which makes it look more like an automaton. We shall represent the rule so that the program go-end which has rules by the symbol may be represented by or the symbol shown as ![Ima… \end{verbatim} ``` </details>
346. ph-638894e7be8a7b68f546 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) ning with the first letter and reads from left to right until it has reached the last letter. We need our Turing machine to do the same. and we are ready to begin. Wehave another way of representing a Turing machine which makes it look more like an automaton. We shall represent the rule so that the program go-end which has rules by the symbol may be represented by or the symbol shown as ![Image](./AutomataTheory_artifacts… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ning with the first letter and reads from left to right until it has reached the last letter. We need our Turing machine to do the same. and we are ready to begin. Wehave another way of representing a Turing machine which makes it look more like an a…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=327423 \begin{verbatim} ning with the first letter and reads from left to right until it has reached the last letter. We need our Turing machine to do the same. and we are ready to begin. Wehave another way of representing a Turing machine which makes it look more like an automaton. We shall represent the rule so that the program go-end which has rules by the symbol may be represented by or the symbol shown as ![Image](./AutomataTheory_artifacts… \end{verbatim} ``` </details>
347. ph-47b8b2425c1e2f64a815 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) nted out. We could print a # in each square as it is read or we could simply print back the letter that is read. We shall choose to do the latter. Each time a letter is read, we wish the machine to move one square to the left, so that the next letter is read. We are now ready to imitate an automaton. If the symbol a ![Image](./AutomataTheory_artifacts/image_000207_98dcb5ef041f06761b94e94218522149a14a68488147ae18880c0c34c5aef771.png) occurs in an automaton, we shall imitate it with the rule ![Image](./AutomataTheory_artifacts/image_000208_cb4113b9fe290fd03719777786… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{nted out. We could print a \# in each square as it is read or we could simply print back the letter that is read. We shall choose to do the latter. Each time a letter is read, we wish the machine to move one square to the left, so that the next letter is read. We are now ready to…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=328236 \begin{verbatim} nted out. We could print a # in each square as it is read or we could simply print back the letter that is read. We shall choose to do the latter. Each time a letter is read, we wish the machine to move one square to the left, so that the next letter is read. We are now ready to imitate an automaton. If the symbol a ![Image](./AutomataTheory_artifacts/image_000207_98dcb5ef041f06761b94e94218522149a14a68488147ae18880c0c34c5aef771.png) occurs in an automaton, we shall imitate it with the rule ![Image](./AutomataTheory_artifacts/image_000208_cb4113b9fe290fd03719777786… \end{verbatim} ``` </details>
348. ph-90ec36a3102928cd01f8 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) y_artifacts/image_000207_98dcb5ef041f06761b94e94218522149a14a68488147ae18880c0c34c5aef771.png) occurs in an automaton, we shall imitate it with the rule ![Image](./AutomataTheory_artifacts/image_000208_cb4113b9fe290fd03719777786524fe1a737760b2c167b6ca62948e65e1df94a.png) It may be recalled that a word is accepted by an automaton if, after the word is read, the automaton is in an acceptance state. For every acceptance state s of the automaton, we will add a rule ![Image](./AutomataTheory_artifacts/image_000209_65c2a96dd26b6505ef98b2f88b… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{y\_artifacts/image\_000207\_98dcb5ef041f06761b94e94218522149a14a68488147ae18880c0c34c5aef771.png) occurs in an automaton, we shall imitate it with the rule ![Image](./AutomataTheory\_artifacts/image\_000208\_cb4113b9fe290fd03719777786524fe1a737760b2c167b6c…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=328581 \begin{verbatim} y_artifacts/image_000207_98dcb5ef041f06761b94e94218522149a14a68488147ae18880c0c34c5aef771.png) occurs in an automaton, we shall imitate it with the rule ![Image](./AutomataTheory_artifacts/image_000208_cb4113b9fe290fd03719777786524fe1a737760b2c167b6ca62948e65e1df94a.png) It may be recalled that a word is accepted by an automaton if, after the word is read, the automaton is in an acceptance state. For every acceptance state s of the automaton, we will add a rule ![Image](./AutomataTheory_artifacts/image_000209_65c2a96dd26b6505ef98b2f88b… \end{verbatim} ``` </details>
349. ph-4ef531ed7d95107a26b5 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) tive integer } , which is not context-free. We begin by designing a program for a Turing machine that will recognize the language { a n b n : n is a positive integer } . We basically want the Turing machine to read an a , then read a b , return to read an a , and continue until all of the a s and b s have been read, if there is an equal number of them. We begin by reading an a in the first square. We want to know that we have counted this a , so we shall change it to A . We do this with rule We now want to go right until we reach a b , which we shall change to a B… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{tive integer \} , which is not context-free. We begin by designing a program for a Turing machine that will recognize the language \{ a n b n : n is a positive integer \} . We basically want the Turing machine to read an a , then read a b , return to read an a , and continue until …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=330502 \begin{verbatim} tive integer } , which is not context-free. We begin by designing a program for a Turing machine that will recognize the language { a n b n : n is a positive integer } . We basically want the Turing machine to read an a , then read a b , return to read an a , and continue until all of the a s and b s have been read, if there is an equal number of them. We begin by reading an a in the first square. We want to know that we have counted this a , so we shall change it to A . We do this with rule We now want to go right until we reach a b , which we shall change to a B… \end{verbatim} ``` </details>
350. ph-31a2dbf5b114d62cda26 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) a s and b s have been read, if there is an equal number of them. We begin by reading an a in the first square. We want to know that we have counted this a , so we shall change it to A . We do this with rule We now want to go right until we reach a b , which we shall change to a B . To get to b , we need to pass over each a without changing it and also after the first time we may have to pass over B s without changing them to reach a b . We do this with the rules When we reach a b , we want to change it to a B and go back left. We do th… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{a s and b s have been read, if there is an equal number of them. We begin by reading an a in the first square. We want to know that we have counted this a , so we shall change it to A . We do this with rule We now want to go right until we reach a b …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=330793 \begin{verbatim} a s and b s have been read, if there is an equal number of them. We begin by reading an a in the first square. We want to know that we have counted this a , so we shall change it to A . We do this with rule We now want to go right until we reach a b , which we shall change to a B . To get to b , we need to pass over each a without changing it and also after the first time we may have to pass over B s without changing them to reach a b . We do this with the rules When we reach a b , we want to change it to a B and go back left. We do th… \end{verbatim} ``` </details>
351. ph-13480f74b787a4f545b1 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) know that we have counted this a , so we shall change it to A . We do this with rule We now want to go right until we reach a b , which we shall change to a B . To get to b , we need to pass over each a without changing it and also after the first time we may have to pass over B s without changing them to reach a b . We do this with the rules When we reach a b , we want to change it to a B and go back left. We do that with the rule We now need to go back to find the second a . To do this we go left until we… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{know that we have counted this a , so we shall change it to A . We do this with rule We now want to go right until we reach a b , which we shall change to a B . To get to b , we need to pass over each a without changing it and also after the first ti…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=330915 \begin{verbatim} know that we have counted this a , so we shall change it to A . We do this with rule We now want to go right until we reach a b , which we shall change to a B . To get to b , we need to pass over each a without changing it and also after the first time we may have to pass over B s without changing them to reach a b . We do this with the rules When we reach a b , we want to change it to a B and go back left. We do that with the rule We now need to go back to find the second a . To do this we go left until we… \end{verbatim} ``` </details>
352. ph-8f7366122607c3904f09 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) me we may have to pass over B s without changing them to reach a b . We do this with the rules When we reach a b , we want to change it to a B and go back left. We do that with the rule We now need to go back to find the second a . To do this we go left until we reach an A . This will tell us that the next letter to the right should be the next a . To go back, we need to pass over B s, and a s to get to A . We do this with the rules When we reach A , we want to go one square to the right to read another a ,… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{me we may have to pass over B s without changing them to reach a b . We do this with the rules When we reach a b , we want to change it to a B and go back left. We do that with the rule We now need to go back to find the …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=331197 \begin{verbatim} me we may have to pass over B s without changing them to reach a b . We do this with the rules When we reach a b , we want to change it to a B and go back left. We do that with the rule We now need to go back to find the second a . To do this we go left until we reach an A . This will tell us that the next letter to the right should be the next a . To go back, we need to pass over B s, and a s to get to A . We do this with the rules When we reach A , we want to go one square to the right to read another a ,… \end{verbatim} ``` </details>
353. ph-493176248ca1cbcbf3bd — automata/docling_md/AutomataTheory.md ### Plain (markdown context) want to change it to a B and go back left. We do that with the rule We now need to go back to find the second a . To do this we go left until we reach an A . This will tell us that the next letter to the right should be the next a . To go back, we need to pass over B s, and a s to get to A . We do this with the rules When we reach A , we want to go one square to the right to read another a , if there is one. We do this with the rule This puts us back into the cycle of reading another a and another b . If we… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{want to change it to a B and go back left. We do that with the rule We now need to go back to find the second a . To do this we go left until we reach an A . This will tell us that the next letter to the right should be the next a . To go back, we ne…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=331346 \begin{verbatim} want to change it to a B and go back left. We do that with the rule We now need to go back to find the second a . To do this we go left until we reach an A . This will tell us that the next letter to the right should be the next a . To go back, we need to pass over B s, and a s to get to A . We do this with the rules When we reach A , we want to go one square to the right to read another a , if there is one. We do this with the rule This puts us back into the cycle of reading another a and another b . If we… \end{verbatim} ``` </details>
354. ph-d8a0feb57d3de3641179 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) ere is one. We do this with the rule This puts us back into the cycle of reading another a and another b . If we run out of b s before we run out of a s the system will be in state s 1 and eventually try to read a blank so it will hang. If we have read the last a , then when we reach A and go right one square, we will read a B . At this point we need to check to see if there is another b . First we change state if we are in s 0 and read a B . We do this with rule In state s 3, read nothing but B s and a blank. Thus we have the rules <!… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ere is one. We do this with the rule This puts us back into the cycle of reading another a and another b . If we run out of b s before we run out of a s the system will be in state s 1 and eventually try to read a blank so it will hang. If we have re…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=331808 \begin{verbatim} ere is one. We do this with the rule This puts us back into the cycle of reading another a and another b . If we run out of b s before we run out of a s the system will be in state s 1 and eventually try to read a blank so it will hang. If we have read the last a , then when we reach A and go right one square, we will read a B . At this point we need to check to see if there is another b . First we change state if we are in s 0 and read a B . We do this with rule In state s 3, read nothing but B s and a blank. Thus we have the rules <!… \end{verbatim} ``` </details>
355. ph-b15ce05d41959e4590a3 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) f reading another a and another b . If we run out of b s before we run out of a s the system will be in state s 1 and eventually try to read a blank so it will hang. If we have read the last a , then when we reach A and go right one square, we will read a B . At this point we need to check to see if there is another b . First we change state if we are in s 0 and read a B . We do this with rule In state s 3, read nothing but B s and a blank. Thus we have the rules ## This may also be shown as the labeled graph ![Image](./AutomataTheory_… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{f reading another a and another b . If we run out of b s before we run out of a s the system will be in state s 1 and eventually try to read a blank so it will hang. If we have read the last a , then when we reach A and go right one square, we will read a B . At this point we ne…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=331910 \begin{verbatim} f reading another a and another b . If we run out of b s before we run out of a s the system will be in state s 1 and eventually try to read a blank so it will hang. If we have read the last a , then when we reach A and go right one square, we will read a B . At this point we need to check to see if there is another b . First we change state if we are in s 0 and read a B . We do this with rule In state s 3, read nothing but B s and a blank. Thus we have the rules ## This may also be shown as the labeled graph ![Image](./AutomataTheory_… \end{verbatim} ``` </details>
356. ph-0b8394d68eff5c152681 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) point we need to check to see if there is another b . First we change state if we are in s 0 and read a B . We do this with rule In state s 3, read nothing but B s and a blank. Thus we have the rules ## This may also be shown as the labeled graph ![Image](./AutomataTheory_artifacts/image_000212_127ef03fe2afa9feb2186c5df8414e0ff7003f9d4eaa3c13fa91382090c32e81.png) For example consider the string aabb . The initial configuration is We then have the following rules and configurations: In state s 3, read nothing but B s and a blank. Thus we have the rules \#\# This may also be s…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=332177 \begin{verbatim} point we need to check to see if there is another b . First we change state if we are in s 0 and read a B . We do this with rule In state s 3, read nothing but B s and a blank. Thus we have the rules ## This may also be shown as the labeled graph ![Image](./AutomataTheory_artifacts/image_000212_127ef03fe2afa9feb2186c5df8414e0ff7003f9d4eaa3c13fa91382090c32e81.png) For example consider the string aabb . The initial configuration is We then have the following rules and configurations: In state s 3, read nothing but B s and a blank. Thus we have the rules ## This may also be shown as the labeled graph ![Image](./AutomataTheory_artifacts/image_000212_127ef03fe2afa9feb2186c5df8414e0ff7003f9d4eaa3c13fa91382090c32e81.png) For example consider the string aabb . The initial configuration is We then have the following rules and configurations: Next we design a program for a Turing machine that… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{re in s 0 and read a B . We do this with rule In state s 3, read nothing but B s and a blank. Thus we have the rules \#\# This may also be shown as the labeled graph ![Image](./AutomataTheory\_artifacts/image\_000212\_127ef03f…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=332261 \begin{verbatim} re in s 0 and read a B . We do this with rule In state s 3, read nothing but B s and a blank. Thus we have the rules ## This may also be shown as the labeled graph ![Image](./AutomataTheory_artifacts/image_000212_127ef03fe2afa9feb2186c5df8414e0ff7003f9d4eaa3c13fa91382090c32e81.png) For example consider the string aabb . The initial configuration is We then have the following rules and configurations: Next we design a program for a Turing machine that… \end{verbatim} ``` </details>
358. ph-6a1ab000a86f3b1be541 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) this with rule In state s 3, read nothing but B s and a blank. Thus we have the rules ## This may also be shown as the labeled graph ![Image](./AutomataTheory_artifacts/image_000212_127ef03fe2afa9feb2186c5df8414e0ff7003f9d4eaa3c13fa91382090c32e81.png) For example consider the string aabb . The initial configuration is We then have the following rules and configurations: Next we design a program for a Turing machine that will recognize the language { … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{this with rule In state s 3, read nothing but B s and a blank. Thus we have the rules \#\# This may also be shown as the labeled graph ![Image](./AutomataTheory\_artifacts/image\_000212\_127ef03fe2afa9feb2186c5df8414e0ff7003f9…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=332291 \begin{verbatim} this with rule In state s 3, read nothing but B s and a blank. Thus we have the rules ## This may also be shown as the labeled graph ![Image](./AutomataTheory_artifacts/image_000212_127ef03fe2afa9feb2186c5df8414e0ff7003f9d4eaa3c13fa91382090c32e81.png) For example consider the string aabb . The initial configuration is We then have the following rules and configurations: Next we design a program for a Turing machine that will recognize the language { … \end{verbatim} ``` </details>
359. ph-e4132b95bfe9a0845e70 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) 3fa91382090c32e81.png) For example consider the string aabb . The initial configuration is We then have the following rules and configurations: Next we design a program for a Turing machine that will recognize the language { a n b n : n is a positive integer } . We begin by reading an a in the first square. We want to know that we have counted this a , so we shall change it to A . We do this with rule We now want to go right until we reach a b , which we shall change to a B . To… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{3fa91382090c32e81.png) For example consider the string aabb . The initial configuration is We then have the following rules and configurations: Next we design a program for a Turing machine th…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=332584 \begin{verbatim} 3fa91382090c32e81.png) For example consider the string aabb . The initial configuration is We then have the following rules and configurations: Next we design a program for a Turing machine that will recognize the language { a n b n : n is a positive integer } . We begin by reading an a in the first square. We want to know that we have counted this a , so we shall change it to A . We do this with rule We now want to go right until we reach a b , which we shall change to a B . To… \end{verbatim} ``` </details>
360. ph-c41d19d24f4015b8412b — automata/docling_md/AutomataTheory.md ### Plain (markdown context) l recognize the language { a n b n : n is a positive integer } . We begin by reading an a in the first square. We want to know that we have counted this a , so we shall change it to A . We do this with rule We now want to go right until we reach a b , which we shall change to a B . To get to b , we need to pass over each a without changing it and also after the first time we may have to pass over B s without changing them to reach a b . We do this with the rules When we reach a b , we want to change it to a B and start back to look for… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{l recognize the language \{ a n b n : n is a positive integer \} . We begin by reading an a in the first square. We want to know that we have counted this a , so we shall change it to A . We do this with rule We now want to go right until we reach a b …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=332875 \begin{verbatim} l recognize the language { a n b n : n is a positive integer } . We begin by reading an a in the first square. We want to know that we have counted this a , so we shall change it to A . We do this with rule We now want to go right until we reach a b , which we shall change to a B . To get to b , we need to pass over each a without changing it and also after the first time we may have to pass over B s without changing them to reach a b . We do this with the rules When we reach a b , we want to change it to a B and start back to look for… \end{verbatim} ``` </details>
361. ph-6744ecc4da07d3651ce1 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) unted this a , so we shall change it to A . We do this with rule We now want to go right until we reach a b , which we shall change to a B . To get to b , we need to pass over each a without changing it and also after the first time we may have to pass over B s without changing them to reach a b . We do this with the rules When we reach a b , we want to change it to a B and start back to look for another a . We do this with the rule To go back, we need to pass over B s and a s to get to A . We do this with … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{unted this a , so we shall change it to A . We do this with rule We now want to go right until we reach a b , which we shall change to a B . To get to b , we need to pass over each a without changing it and also after the first time we may have to pa…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=333018 \begin{verbatim} unted this a , so we shall change it to A . We do this with rule We now want to go right until we reach a b , which we shall change to a B . To get to b , we need to pass over each a without changing it and also after the first time we may have to pass over B s without changing them to reach a b . We do this with the rules When we reach a b , we want to change it to a B and start back to look for another a . We do this with the rule To go back, we need to pass over B s and a s to get to A . We do this with … \end{verbatim} ``` </details>
362. ph-fba3f4a4975141d3733e — automata/docling_md/AutomataTheory.md ### Plain (markdown context) ght until we reach a b , which we shall change to a B . To get to b , we need to pass over each a without changing it and also after the first time we may have to pass over B s without changing them to reach a b . We do this with the rules When we reach a b , we want to change it to a B and start back to look for another a . We do this with the rule To go back, we need to pass over B s and a s to get to A . We do this with the rules When we reach A , we want to go one square to the right to read another a ,… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ght until we reach a b , which we shall change to a B . To get to b , we need to pass over each a without changing it and also after the first time we may have to pass over B s without changing them to reach a b . We do this with the rules When we re…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=333134 \begin{verbatim} ght until we reach a b , which we shall change to a B . To get to b , we need to pass over each a without changing it and also after the first time we may have to pass over B s without changing them to reach a b . We do this with the rules When we reach a b , we want to change it to a B and start back to look for another a . We do this with the rule To go back, we need to pass over B s and a s to get to A . We do this with the rules When we reach A , we want to go one square to the right to read another a ,… \end{verbatim} ``` </details>
363. ph-2ea927161ab329826861 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) e may have to pass over B s without changing them to reach a b . We do this with the rules When we reach a b , we want to change it to a B and start back to look for another a . We do this with the rule To go back, we need to pass over B s and a s to get to A . We do this with the rules When we reach A , we want to go one square to the right to read another a , if there is one. We do this with the rule This puts us back into the cycle of reading another a and b . If we run out o… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{e may have to pass over B s without changing them to reach a b . We do this with the rules When we reach a b , we want to change it to a B and start back to look for another a . We do this with the rule To go back, we nee…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=333283 \begin{verbatim} e may have to pass over B s without changing them to reach a b . We do this with the rules When we reach a b , we want to change it to a B and start back to look for another a . We do this with the rule To go back, we need to pass over B s and a s to get to A . We do this with the rules When we reach A , we want to go one square to the right to read another a , if there is one. We do this with the rule This puts us back into the cycle of reading another a and b . If we run out o… \end{verbatim} ``` </details>
364. ph-c47f77173f546afbdea7 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) ch A , we want to go one square to the right to read another a , if there is one. We do this with the rule This puts us back into the cycle of reading another a and b . If we run out of b s before we run out of a s the system will hang. If we have read the last a , then when we reach A and go right one square, we will read a B . At this point we need to check to see if there is another b . First we change state if we are in s 0 and read a B . We do this with rule In state s 3, we expect to read nothing but B s, b , and a blank. Thus we… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ch A , we want to go one square to the right to read another a , if there is one. We do this with the rule This puts us back into the cycle of reading another a and b . If we run out of b s before we run out of a s the system will hang. If we have re…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=333675 \begin{verbatim} ch A , we want to go one square to the right to read another a , if there is one. We do this with the rule This puts us back into the cycle of reading another a and b . If we run out of b s before we run out of a s the system will hang. If we have read the last a , then when we reach A and go right one square, we will read a B . At this point we need to check to see if there is another b . First we change state if we are in s 0 and read a B . We do this with rule In state s 3, we expect to read nothing but B s, b , and a blank. Thus we… \end{verbatim} ``` </details>
365. ph-cbc9b6b1de4891101c58 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) -not-decoded --> This puts us back into the cycle of reading another a and b . If we run out of b s before we run out of a s the system will hang. If we have read the last a , then when we reach A and go right one square, we will read a B . At this point we need to check to see if there is another b . First we change state if we are in s 0 and read a B . We do this with rule In state s 3, we expect to read nothing but B s, b , and a blank. Thus we have the rules We now design a program for a Turing machine that will recognize the langu… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{-not-decoded --> This puts us back into the cycle of reading another a and b . If we run out of b s before we run out of a s the system will hang. If we have read the last a , then when we reach A and go right one square, we will read a B . At this point we need to check to see …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=333795 \begin{verbatim} -not-decoded --> This puts us back into the cycle of reading another a and b . If we run out of b s before we run out of a s the system will hang. If we have read the last a , then when we reach A and go right one square, we will read a B . At this point we need to check to see if there is another b . First we change state if we are in s 0 and read a B . We do this with rule In state s 3, we expect to read nothing but B s, b , and a blank. Thus we have the rules We now design a program for a Turing machine that will recognize the langu… \end{verbatim} ``` </details>
366. ph-44397a5f96ec2664241b — automata/docling_md/AutomataTheory.md ### Plain (markdown context) t-decoded --> We now design a program for a Turing machine that will recognize the language { a n b n c n : n is a positive integer } . In a manner similar to the previous example we want the Turing machine to read an a , then read a b , then read a c , and continue until all of the a s, b s, and c s have been read, if there are an equal number of them. We begin by reading an a in the first square. We want to know that we have counted this a , so we shall change it to A . We do this with rule We now want to go right until we reach a b , which we shall change to a … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{t-decoded --> We now design a program for a Turing machine that will recognize the language \{ a n b n c n : n is a positive integer \} . In a manner similar to the previous example we want the Turing machine to read an a , then read a b , then read a c , and continue until all of…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=334310 \begin{verbatim} t-decoded --> We now design a program for a Turing machine that will recognize the language { a n b n c n : n is a positive integer } . In a manner similar to the previous example we want the Turing machine to read an a , then read a b , then read a c , and continue until all of the a s, b s, and c s have been read, if there are an equal number of them. We begin by reading an a in the first square. We want to know that we have counted this a , so we shall change it to A . We do this with rule We now want to go right until we reach a b , which we shall change to a … \end{verbatim} ``` </details>
367. ph-9b3e5dfc2f320668c718 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) s, and c s have been read, if there are an equal number of them. We begin by reading an a in the first square. We want to know that we have counted this a , so we shall change it to A . We do this with rule We now want to go right until we reach a b , which we shall change to a B . To get to b , we need to pass over each a without changing it and also after the first time we may have to pass over B s without changing them to reach a b . We do this with the rules When we reach a b , we want to change it to a B and continue onward. We do… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{s, and c s have been read, if there are an equal number of them. We begin by reading an a in the first square. We want to know that we have counted this a , so we shall change it to A . We do this with rule We now want to go right until we reach a b …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=334601 \begin{verbatim} s, and c s have been read, if there are an equal number of them. We begin by reading an a in the first square. We want to know that we have counted this a , so we shall change it to A . We do this with rule We now want to go right until we reach a b , which we shall change to a B . To get to b , we need to pass over each a without changing it and also after the first time we may have to pass over B s without changing them to reach a b . We do this with the rules When we reach a b , we want to change it to a B and continue onward. We do… \end{verbatim} ``` </details>
368. ph-73e16d0770033965dcc2 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) ow that we have counted this a , so we shall change it to A . We do this with rule We now want to go right until we reach a b , which we shall change to a B . To get to b , we need to pass over each a without changing it and also after the first time we may have to pass over B s without changing them to reach a b . We do this with the rules When we reach a b , we want to change it to a B and continue onward. We do that with the rule We now need to continue until we find a c . We will need to pass over b s a… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ow that we have counted this a , so we shall change it to A . We do this with rule We now want to go right until we reach a b , which we shall change to a B . To get to b , we need to pass over each a without changing it and also after the first time…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=334726 \begin{verbatim} ow that we have counted this a , so we shall change it to A . We do this with rule We now want to go right until we reach a b , which we shall change to a B . To get to b , we need to pass over each a without changing it and also after the first time we may have to pass over B s without changing them to reach a b . We do this with the rules When we reach a b , we want to change it to a B and continue onward. We do that with the rule We now need to continue until we find a c . We will need to pass over b s a… \end{verbatim} ``` </details>
369. ph-843e23d0f864af22eb0b — automata/docling_md/AutomataTheory.md ### Plain (markdown context) til we reach a b , which we shall change to a B . To get to b , we need to pass over each a without changing it and also after the first time we may have to pass over B s without changing them to reach a b . We do this with the rules When we reach a b , we want to change it to a B and continue onward. We do that with the rule We now need to continue until we find a c . We will need to pass over b s and C s. We do this with the rules Wenext want to read c , replace it with a C , and start back to look for an… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{til we reach a b , which we shall change to a B . To get to b , we need to pass over each a without changing it and also after the first time we may have to pass over B s without changing them to reach a b . We do this with the rules When we reach a …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=334866 \begin{verbatim} til we reach a b , which we shall change to a B . To get to b , we need to pass over each a without changing it and also after the first time we may have to pass over B s without changing them to reach a b . We do this with the rules When we reach a b , we want to change it to a B and continue onward. We do that with the rule We now need to continue until we find a c . We will need to pass over b s and C s. We do this with the rules Wenext want to read c , replace it with a C , and start back to look for an… \end{verbatim} ``` </details>
370. ph-07ac59689de0dd6b0696 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) we may have to pass over B s without changing them to reach a b . We do this with the rules When we reach a b , we want to change it to a B and continue onward. We do that with the rule We now need to continue until we find a c . We will need to pass over b s and C s. We do this with the rules Wenext want to read c , replace it with a C , and start back to look for another a . We do this with the rule To go back, we need to pass over C s, b s, B s, and a s to get to A . We do th… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{we may have to pass over B s without changing them to reach a b . We do this with the rules When we reach a b , we want to change it to a B and continue onward. We do that with the rule We now need to continue until we fi…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=335007 \begin{verbatim} we may have to pass over B s without changing them to reach a b . We do this with the rules When we reach a b , we want to change it to a B and continue onward. We do that with the rule We now need to continue until we find a c . We will need to pass over b s and C s. We do this with the rules Wenext want to read c , replace it with a C , and start back to look for another a . We do this with the rule To go back, we need to pass over C s, b s, B s, and a s to get to A . We do th… \end{verbatim} ``` </details>
371. ph-284f3ffb46a6e5b5f3e2 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) n we reach a b , we want to change it to a B and continue onward. We do that with the rule We now need to continue until we find a c . We will need to pass over b s and C s. We do this with the rules Wenext want to read c , replace it with a C , and start back to look for another a . We do this with the rule To go back, we need to pass over C s, b s, B s, and a s to get to A . We do this with the rules When we reach A , we want to go one square to the right to read another a , i… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{n we reach a b , we want to change it to a B and continue onward. We do that with the rule We now need to continue until we find a c . We will need to pass over b s and C s. We do this with the rules Wenext want to read c…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=335134 \begin{verbatim} n we reach a b , we want to change it to a B and continue onward. We do that with the rule We now need to continue until we find a c . We will need to pass over b s and C s. We do this with the rules Wenext want to read c , replace it with a C , and start back to look for another a . We do this with the rule To go back, we need to pass over C s, b s, B s, and a s to get to A . We do this with the rules When we reach A , we want to go one square to the right to read another a , i… \end{verbatim} ``` </details>
372. ph-3e44dd741c2807f8481a — automata/docling_md/AutomataTheory.md ### Plain (markdown context) il we find a c . We will need to pass over b s and C s. We do this with the rules Wenext want to read c , replace it with a C , and start back to look for another a . We do this with the rule To go back, we need to pass over C s, b s, B s, and a s to get to A . We do this with the rules When we reach A , we want to go one square to the right to read another a , if there is one. We do this with the rule This puts us back into the cycle of reading another a , b , and c . If we run… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{il we find a c . We will need to pass over b s and C s. We do this with the rules Wenext want to read c , replace it with a C , and start back to look for another a . We do this with the rule To go back, we need to pass o…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=335283 \begin{verbatim} il we find a c . We will need to pass over b s and C s. We do this with the rules Wenext want to read c , replace it with a C , and start back to look for another a . We do this with the rule To go back, we need to pass over C s, b s, B s, and a s to get to A . We do this with the rules When we reach A , we want to go one square to the right to read another a , if there is one. We do this with the rule This puts us back into the cycle of reading another a , b , and c . If we run… \end{verbatim} ``` </details>
373. ph-6c44ae17aaf32a19c77b — automata/docling_md/AutomataTheory.md ### Plain (markdown context) t to go one square to the right to read another a , if there is one. We do this with the rule This puts us back into the cycle of reading another a , b , and c . If we run out of b s or c s before we run out of a s the system will hang. If we have read the last a , then when we reach A and go right one square, we will read a B . At this point we need to check to see if there is another b . First we change state if we are in s 0 and read a B . We do this with rule In state s 4, we expect to read nothing but B s, C s, and a blank. Thus w… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{t to go one square to the right to read another a , if there is one. We do this with the rule This puts us back into the cycle of reading another a , b , and c . If we run out of b s or c s before we run out of a s the system will hang. If we have re…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=335688 \begin{verbatim} t to go one square to the right to read another a , if there is one. We do this with the rule This puts us back into the cycle of reading another a , b , and c . If we run out of b s or c s before we run out of a s the system will hang. If we have read the last a , then when we reach A and go right one square, we will read a B . At this point we need to check to see if there is another b . First we change state if we are in s 0 and read a B . We do this with rule In state s 4, we expect to read nothing but B s, C s, and a blank. Thus w… \end{verbatim} ``` </details>
374. ph-e3a9661ec4031e33f350 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) -> This puts us back into the cycle of reading another a , b , and c . If we run out of b s or c s before we run out of a s the system will hang. If we have read the last a , then when we reach A and go right one square, we will read a B . At this point we need to check to see if there is another b . First we change state if we are in s 0 and read a B . We do this with rule In state s 4, we expect to read nothing but B s, C s, and a blank. Thus we have the rules This may also be shown as the labeled directed graph ![Image](./AutomataTh… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{-> This puts us back into the cycle of reading another a , b , and c . If we run out of b s or c s before we run out of a s the system will hang. If we have read the last a , then when we reach A and go right one square, we will read a B . At this point we need to check to see i…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=335809 \begin{verbatim} -> This puts us back into the cycle of reading another a , b , and c . If we run out of b s or c s before we run out of a s the system will hang. If we have read the last a , then when we reach A and go right one square, we will read a B . At this point we need to check to see if there is another b . First we change state if we are in s 0 and read a B . We do this with rule In state s 4, we expect to read nothing but B s, C s, and a blank. Thus we have the rules This may also be shown as the labeled directed graph ![Image](./AutomataTh… \end{verbatim} ``` </details>
375. ph-3553abf2358f961f63d5 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) see if there is another b . First we change state if we are in s 0 and read a B . We do this with rule In state s 4, we expect to read nothing but B s, C s, and a blank. Thus we have the rules This may also be shown as the labeled directed graph ![Image](./AutomataTheory_artifacts/image_000213_67e323c0d0c3806682cd955a3528a306a11f5ffd40534ed4d616f6a74ef4c6c6.png) For example consider the string aabbcc . The initial configuration is We then have the following rules and configurati… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{see if there is another b . First we change state if we are in s 0 and read a B . We do this with rule In state s 4, we expect to read nothing but B s, C s, and a blank. Thus we have the rules This may also be shown as th…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=336084 \begin{verbatim} see if there is another b . First we change state if we are in s 0 and read a B . We do this with rule In state s 4, we expect to read nothing but B s, C s, and a blank. Thus we have the rules This may also be shown as the labeled directed graph ![Image](./AutomataTheory_artifacts/image_000213_67e323c0d0c3806682cd955a3528a306a11f5ffd40534ed4d616f6a74ef4c6c6.png) For example consider the string aabbcc . The initial configuration is We then have the following rules and configurati… \end{verbatim} ``` </details>
376. ph-23ddd3eeb97bc2b12b6b — automata/docling_md/AutomataTheory.md ### Plain (markdown context) rst we change state if we are in s 0 and read a B . We do this with rule In state s 4, we expect to read nothing but B s, C s, and a blank. Thus we have the rules This may also be shown as the labeled directed graph ![Image](./AutomataTheory_artifacts/image_000213_67e323c0d0c3806682cd955a3528a306a11f5ffd40534ed4d616f6a74ef4c6c6.png) For example consider the string aabbcc . The initial configuration is We then have the following rules and configurations: In state s 4, we expect to read nothing but B s, C s, and a blank. Thus we have the rules This may also be shown as the labeled directed graph ![Ima…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=336114 \begin{verbatim} rst we change state if we are in s 0 and read a B . We do this with rule In state s 4, we expect to read nothing but B s, C s, and a blank. Thus we have the rules This may also be shown as the labeled directed graph ![Image](./AutomataTheory_artifacts/image_000213_67e323c0d0c3806682cd955a3528a306a11f5ffd40534ed4d616f6a74ef4c6c6.png) For example consider the string aabbcc . The initial configuration is We then have the following rules and configurations: In state s 4, we expect to read nothing but B s, C s, and a blank. Thus we have the rules This may also be shown as the labeled directed graph ![Image](./AutomataTheory_artifacts/image_000213_67e323c0d0c3806682cd955a3528a306a11f5ffd40534ed4d616f6a74ef4c6c6.png) For example consider the string aabbcc . The initial configuration is We then have the following rules and configurations: WenowshowhowtoperformtwoarithmeticoperationsonaTur… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{la-not-decoded --> In state s 4, we expect to read nothing but B s, C s, and a blank. Thus we have the rules This may also be shown as the labeled directed graph ![Image](./AutomataTheory\_artifacts/image\_000213\_67e323c0d0c3806682cd955a3528a306a11f5ff…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=336198 \begin{verbatim} la-not-decoded --> In state s 4, we expect to read nothing but B s, C s, and a blank. Thus we have the rules This may also be shown as the labeled directed graph ![Image](./AutomataTheory_artifacts/image_000213_67e323c0d0c3806682cd955a3528a306a11f5ffd40534ed4d616f6a74ef4c6c6.png) For example consider the string aabbcc . The initial configuration is We then have the following rules and configurations: WenowshowhowtoperformtwoarithmeticoperationsonaTur… \end{verbatim} ``` </details>
378. ph-a220f552f9edccd5593f — automata/docling_md/AutomataTheory.md ### Plain (markdown context) 4, we expect to read nothing but B s, C s, and a blank. Thus we have the rules This may also be shown as the labeled directed graph ![Image](./AutomataTheory_artifacts/image_000213_67e323c0d0c3806682cd955a3528a306a11f5ffd40534ed4d616f6a74ef4c6c6.png) For example consider the string aabbcc . The initial configuration is We then have the following rules and configurations: WenowshowhowtoperformtwoarithmeticoperationsonaTuringmachine. The first of these… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{4, we expect to read nothing but B s, C s, and a blank. Thus we have the rules This may also be shown as the labeled directed graph ![Image](./AutomataTheory\_artifacts/image\_000213\_67e323c0d0c3806682cd955a3528a306a11f5ffd40534ed4d616f6a74ef4c6c6.png)…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=336228 \begin{verbatim} 4, we expect to read nothing but B s, C s, and a blank. Thus we have the rules This may also be shown as the labeled directed graph ![Image](./AutomataTheory_artifacts/image_000213_67e323c0d0c3806682cd955a3528a306a11f5ffd40534ed4d616f6a74ef4c6c6.png) For example consider the string aabbcc . The initial configuration is We then have the following rules and configurations: WenowshowhowtoperformtwoarithmeticoperationsonaTuringmachine. The first of these… \end{verbatim} ``` </details>
379. ph-4e52783b14ebe53b7ee7 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) θ m 2 , . . . , θ mp , which together with the state and input would give us the rules to use. If we apply all possible relevant sequences, we can produce all possible computations . Hence if a word is accepted by the Turing machine T , it will be accepted in one of these computations. The next problem is the production of the sequences of integers. We shall label these sequences N 0 , N 1 , N 2 , . . . , Ni , . . . We begin with N 0 = 0 and simply count in base n + 1. Thus the sequences are The sequence following is and the sequence following is The sequence following is and the sequence following is The sequence following is and the sequence following is (1 , 3 , 4 , n , n , n ) The sequence following is and the sequence following is (1 , 3 , 4 , n , n , n ) The sequence following is and the sequence following is (1 , 3 , 4 , n , n , n ) The subroutine in whic… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{le relevant sequences, we can produce all possible computations . Hence if a word is accepted by the Turing machine T , it will be accepted in one of these computations. The next problem is the production of the sequences of integers. We shall label these sequences N 0 , N 1 , N…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=345220 \begin{verbatim} le relevant sequences, we can produce all possible computations . Hence if a word is accepted by the Turing machine T , it will be accepted in one of these computations. The next problem is the production of the sequences of integers. We shall label these sequences N 0 , N 1 , N 2 , . . . , Ni , . . . We begin with N 0 = 0 and simply count in base n + 1. Thus the sequences are The sequence following is and the sequence following is (1 , 3 , 4 , n , n , n ) The subroutine in whic… \end{verbatim} ``` </details>
382. ph-9aa1a1617ebd8e182e4a — automata/docling_md/AutomataTheory.md ### Plain (markdown context) tations . Hence if a word is accepted by the Turing machine T , it will be accepted in one of these computations. The next problem is the production of the sequences of integers. We shall label these sequences N 0 , N 1 , N 2 , . . . , Ni , . . . We begin with N 0 = 0 and simply count in base n + 1. Thus the sequences are The sequence following is and the sequence following is (1 , 3 , 4 , n , n , n ) The subroutine in which a Turing machine changes the number Nk to Nk + 1 is st… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{tations . Hence if a word is accepted by the Turing machine T , it will be accepted in one of these computations. The next problem is the production of the sequences of integers. We shall label these sequences N 0 , N 1 , N 2 , . . . , Ni , . . . We begin with N 0 = 0 and simply…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=345276 \begin{verbatim} tations . Hence if a word is accepted by the Turing machine T , it will be accepted in one of these computations. The next problem is the production of the sequences of integers. We shall label these sequences N 0 , N 1 , N 2 , . . . , Ni , . . . We begin with N 0 = 0 and simply count in base n + 1. Thus the sequences are The sequence following is and the sequence following is (1 , 3 , 4 , n , n , n ) The subroutine in which a Turing machine changes the number Nk to Nk + 1 is st… \end{verbatim} ``` </details>
383. ph-db9bf261d50b846ecf5d — automata/docling_md/AutomataTheory.md ### Plain (markdown context) nce we mark it with a ′ , so that we proceed each time to the first unmarked number in the sequence, select it, mark it, and then return to the marked letter with the information needed to select the proper path for the Turing machine to take, given the state and the letter being read. For example suppose the Turing machine is in state s , reads letter a , and finds that j is the number selected, it then proceeds with θ j ( s , a ) to supply the rule to use. As an illustration suppose we have We change b to bs 4 so we have We then move… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{nce we mark it with a ′ , so that we proceed each time to the first unmarked number in the sequence, select it, mark it, and then return to the marked letter with the information needed to select the proper path for the Turing machine to take, given the state and the letter bein…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=346572 \begin{verbatim} nce we mark it with a ′ , so that we proceed each time to the first unmarked number in the sequence, select it, mark it, and then return to the marked letter with the information needed to select the proper path for the Turing machine to take, given the state and the letter being read. For example suppose the Turing machine is in state s , reads letter a , and finds that j is the number selected, it then proceeds with θ j ( s , a ) to supply the rule to use. As an illustration suppose we have We change b to bs 4 so we have We then move… \end{verbatim} ``` </details>
384. ph-8e8582b40697d7d972b2 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) first unmarked number in the sequence, select it, mark it, and then return to the marked letter with the information needed to select the proper path for the Turing machine to take, given the state and the letter being read. For example suppose the Turing machine is in state s , reads letter a , and finds that j is the number selected, it then proceeds with θ j ( s , a ) to supply the rule to use. As an illustration suppose we have We change b to bs 4 so we have We then move to 1, the first unmarked integer and mark it, so we have We change b to bs 4 so we have We then move to 1, the first unmarked integer and mark it, so we have We change b to bs 4 so we have We then move to 1, the first unmarked integer and mark it, so we have where the subscript of the state is the number selected. We then return to b… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{the information needed to select the proper path for the Turing machine to take, given the state and the letter being read. For example suppose the Turing machine is in state s , reads letter a , and finds that j is the number selected, it then proceeds with θ j ( s , a ) to sup…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=346735 \begin{verbatim} the information needed to select the proper path for the Turing machine to take, given the state and the letter being read. For example suppose the Turing machine is in state s , reads letter a , and finds that j is the number selected, it then proceeds with θ j ( s , a ) to supply the rule to use. As an illustration suppose we have We change b to bs 4 so we have We then move to 1, the first unmarked integer and mark it, so we have where the subscript of the state is the number selected. We then return to b… \end{verbatim} ``` </details>
386. ph-5b9958d888de04493fe1 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) inds that j is the number selected, it then proceeds with θ j ( s , a ) to supply the rule to use. As an illustration suppose we have We change b to bs 4 so we have We then move to 1, the first unmarked integer and mark it, so we have where the subscript of the state is the number selected. We then return to bs 4 where we have θ ∗ ( q 1 , bs 4 ) = θ 1 ( s 4 , b ). The instructions could be as follows Informally we state the procedure for testing a word for acceptance by a Turing… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{inds that j is the number selected, it then proceeds with θ j ( s , a ) to supply the rule to use. As an illustration suppose we have We change b to bs 4 so we have We then move to 1, the first unmarked integer and mark i…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=346936 \begin{verbatim} inds that j is the number selected, it then proceeds with θ j ( s , a ) to supply the rule to use. As an illustration suppose we have We change b to bs 4 so we have We then move to 1, the first unmarked integer and mark it, so we have where the subscript of the state is the number selected. We then return to bs 4 where we have θ ∗ ( q 1 , bs 4 ) = θ 1 ( s 4 , b ). The instructions could be as follows Informally we state the procedure for testing a word for acceptance by a Turing… \end{verbatim} ``` </details>
387. ph-994637b8b35c93b462e4 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) > We then move to 1, the first unmarked integer and mark it, so we have where the subscript of the state is the number selected. We then return to bs 4 where we have θ ∗ ( q 1 , bs 4 ) = θ 1 ( s 4 , b ). The instructions could be as follows Informally we state the procedure for testing a word for acceptance by a Turing machine as follows: First, given the word, duplicate the word and follow it by the first sequence so that we have Perform the process above for testing the second copy of the word w following… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{> We then move to 1, the first unmarked integer and mark it, so we have where the subscript of the state is the number selected. We then return to bs 4 where we have θ ∗ ( q 1 , bs 4 ) = θ 1 ( s 4 , b ). The instructions could be as follows where the subscript of the state is the number selected. We then return to bs 4 where we have θ ∗ ( q 1 , bs 4 ) = θ 1 ( s 4 , b ). The instructions could be as follows Informally we state the procedure for testing a word for acceptance by a Turing machine as follows: First, given the word, duplicate the word and follow it by the first sequence so that we have Perform the process above for testing the second copy of the word w following… \end{verbatim} ```
388. ph-4a135ff94815e7733faf — automata/docling_md/AutomataTheory.md ### Plain (markdown context) y a Turing machine M ′′ . Hence L is Turing decidable and, by Theorem 5.4, its complement L ′ is also Turing decidable. /square Before proceeding further we need to show that every Turing machine with alphabet A = { a , b } can be uniquely described by a string of a s and b s. It is obvious that a Turing machine is uniquely determined by the set of rules for the machine. We shall show this for the set of states S = { s 1 , s 2 , s 3 , . . . , sn } . It may be recalled that a rule has the form where the first and third components are states, the second and fourth c… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{y a Turing machine M ′′ . Hence L is Turing decidable and, by Theorem 5.4, its complement L ′ is also Turing decidable. /square Before proceeding further we need to show that every Turing machine with alphabet A = \{ a , b \} can be uniquely described by a string of a s and b s. I…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=353166 \begin{verbatim} y a Turing machine M ′′ . Hence L is Turing decidable and, by Theorem 5.4, its complement L ′ is also Turing decidable. /square Before proceeding further we need to show that every Turing machine with alphabet A = { a , b } can be uniquely described by a string of a s and b s. It is obvious that a Turing machine is uniquely determined by the set of rules for the machine. We shall show this for the set of states S = { s 1 , s 2 , s 3 , . . . , sn } . It may be recalled that a rule has the form where the first and third components are states, the second and fourth c… \end{verbatim} ``` </details>
389. ph-5fc4816863d862a6a1f6 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) hm that would satisfy the halting problem is the algorithm which describes MM 2 and it does not exist. Theorem 5.8 Given a Turing machine T and an input string w , there is no algorithm which will determine whether the Turing machine T , given the input string w , will reach the halt state. ## Exercises - (1) Show that a finite set is Turing decidable. - (2) Find the string representing the rule ( s 5 , /Delta1 , s 2 , a , R ) . - (3) Find c ( M ) where M is the machine defined by the rules - (4) Find c ( M ) where M is the machine defined by the rules - (4) Find c ( M ) where M is the machine defined by the rules - (4) Find c ( M ) where M is the machine defined by the rules - (5) Find the rule that corresponds to the string aaabababbaa . - (6) Find … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{t exist. Theorem 5.8 Given a Turing machine T and an input string w , there is no algorithm which will determine whether the Turing machine T , given the input string w , will reach the halt state. \#\# Exercises - (1) Show that a finite set is Turing decidable. - (2) Find the str…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=359703 \begin{verbatim} t exist. Theorem 5.8 Given a Turing machine T and an input string w , there is no algorithm which will determine whether the Turing machine T , given the input string w , will reach the halt state. ## Exercises - (1) Show that a finite set is Turing decidable. - (2) Find the string representing the rule ( s 5 , /Delta1 , s 2 , a , R ) . - (3) Find c ( M ) where M is the machine defined by the rules - (4) Find c ( M ) where M is the machine defined by the rules - (5) Find the rule that corresponds to the string aaabababbaa . - (6) Find … \end{verbatim} ``` </details>
391. ph-1412a08ddadee6e48b06 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) abaaaabbbbabbbaabbbbaabbbabbbaababaaababbbaaabbb . - (11) Find the Turing machine and input that correspond to the string abaaaabbbbabbbaabaabaabbbabbbaaabaaabbbaaababaa ababababaabb . - (12) Devise a method of coding that allows the use of A and B as well as a and b by allowing strings of length 3 to represent input and output symbols. - (13) Use the coding in the previous problem to find the string corresponding to ( s 1 , a , s 3 , A , R ). - (14) Find the string that represents the machine together with input ababaab . - (15) Find the string that represents th… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{abaaaabbbbabbbaabbbbaabbbabbbaababaaababbbaaabbb . - (11) Find the Turing machine and input that correspond to the string abaaaabbbbabbbaabaabaabbbabbbaaabaaabbbaaababaa ababababaabb . - (12) Devise a method of coding that allows the use of A and B as well as a and b by allowing…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=360788 \begin{verbatim} abaaaabbbbabbbaabbbbaabbbabbbaababaaababbbaaabbb . - (11) Find the Turing machine and input that correspond to the string abaaaabbbbabbbaabaabaabbbabbbaaabaaabbbaaababaa ababababaabb . - (12) Devise a method of coding that allows the use of A and B as well as a and b by allowing strings of length 3 to represent input and output symbols. - (13) Use the coding in the previous problem to find the string corresponding to ( s 1 , a , s 3 , A , R ). - (14) Find the string that represents the machine together with input ababaab . - (15) Find the string that represents th… \end{verbatim} ``` </details>
392. ph-56e346ebc5562382b414 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) e string abaaaabbbbabbbaabaabaabbbabbbaaabaaabbbaaababaa ababababaabb . - (12) Devise a method of coding that allows the use of A and B as well as a and b by allowing strings of length 3 to represent input and output symbols. - (13) Use the coding in the previous problem to find the string corresponding to ( s 1 , a , s 3 , A , R ). - (14) Find the string that represents the machine together with input ababaab . - (15) Find the string that represents the machine together with input babbab . - (16) Let L be a language. Prove that one an… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{e string abaaaabbbbabbbaabaabaabbbabbbaaabaaabbbaaababaa ababababaabb . - (12) Devise a method of coding that allows the use of A and B as well as a and b by allowing strings of length 3 to represent input and output symbols. - (13) Use the coding in the previous problem to find…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=360901 \begin{verbatim} e string abaaaabbbbabbbaabaabaabbbabbbaaabaaabbbaaababaa ababababaabb . - (12) Devise a method of coding that allows the use of A and B as well as a and b by allowing strings of length 3 to represent input and output symbols. - (13) Use the coding in the previous problem to find the string corresponding to ( s 1 , a , s 3 , A , R ). - (14) Find the string that represents the machine together with input ababaab . - (15) Find the string that represents the machine together with input babbab . - (16) Let L be a language. Prove that one an… \end{verbatim} ``` </details>
393. ph-1e754ba979696e0e3987 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) de, ( /star $ , $) in P 2 . It is obvious that only ( L ( u 0 ) , L ( u 0 ) /star ) can begin a match in P 2 , since it is the only pair where we do not have one word in the pair beginning with a star while the other does not. It is also obvious that the only pair that can end a pair in P 2 , is ( /star $ , $), since it is the only word where the last symbols match, that is we do not have one ending in a star while the other does not. It is also obvious that if there exist a sequence of pairs in P 1, such that w = u 0 ui 1 ui 2 · · · uim = v 0 v i 1 v i 2 · · · v … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{de, ( /star $ , $) in P 2 . It is obvious that only ( L ( u 0 ) , L ( u 0 ) /star ) can begin a match in P 2 , since it is the only pair where we do not have one word in the pair beginning with a star while the other does not. It is also obvious that the only pair that can end a…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=364898 \begin{verbatim} de, ( /star $ , $) in P 2 . It is obvious that only ( L ( u 0 ) , L ( u 0 ) /star ) can begin a match in P 2 , since it is the only pair where we do not have one word in the pair beginning with a star while the other does not. It is also obvious that the only pair that can end a pair in P 2 , is ( /star $ , $), since it is the only word where the last symbols match, that is we do not have one ending in a star while the other does not. It is also obvious that if there exist a sequence of pairs in P 1, such that w = u 0 ui 1 ui 2 · · · uim = v 0 v i 1 v i 2 · · · v … \end{verbatim} ``` </details>
394. ph-adcf2b0cfdd0862612f1 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) only pair where we do not have one word in the pair beginning with a star while the other does not. It is also obvious that the only pair that can end a pair in P 2 , is ( /star $ , $), since it is the only word where the last symbols match, that is we do not have one ending in a star while the other does not. It is also obvious that if there exist a sequence of pairs in P 1, such that w = u 0 ui 1 ui 2 · · · uim = v 0 v i 1 v i 2 · · · v im . Then the sequence produces a match in P 2. The words and in P 1, such that w = u 0 ui 1 ui 2 · · · uim = v 0 v i 1 v i 2 · · · v im . Then the sequence produces a match in P 2. The words and in P 1, such that w = u 0 ui 1 ui 2 · · · uim = v 0 v i 1 v i 2 · · · v im . Then the sequence produces a match in P 2. The words and in P 1, such that w = u 0 ui 1 ui 2 · · · uim = v 0 v i 1 v i 2 · · · v im . Then the sequence produces a match in P 2. The words and in P 1, such that w = u 0 ui 1 ui 2 · · · uim = v 0 v i 1 v i 2 · · · v im . Then the sequence produces a match in P 2. The words and in P 2 differ from the words u 0 ui 1 ui 2 · · · … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{It is also obvious that the only pair that can end a pair in P 2 , is ( /star $ , $), since it is the only word where the last symbols match, that is we do not have one ending in a star while the other does not. It is also obvious that if there exist a sequence of pairs in P 1, such that w = u 0 ui 1 ui 2 · · · uim = v 0 v i 1 v i 2 · · · v im . Then the sequence produces a match in P 2. The words and in P 2 differ from the words u 0 ui 1 ui 2 · · · … \end{verbatim} ```
397. ph-9b1eb9b60da74c2fa4a3 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) ly pair that can end a pair in P 2 , is ( /star $ , $), since it is the only word where the last symbols match, that is we do not have one ending in a star while the other does not. It is also obvious that if there exist a sequence of pairs in P 1, such that w = u 0 ui 1 ui 2 · · · uim = v 0 v i 1 v i 2 · · · v im . Then the sequence produces a match in P 2. The words and in P 2 differ from the words u 0 ui 1 ui 2 · · · uim and v 0 v i 1 v i 2 · · · … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ly pair that can end a pair in P 2 , is ( /star $ , $), since it is the only word where the last symbols match, that is we do not have one ending in a star while the other does not. It is also obvious that if there exist a sequence of pairs in P 1, s…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=365155 \begin{verbatim} ly pair that can end a pair in P 2 , is ( /star $ , $), since it is the only word where the last symbols match, that is we do not have one ending in a star while the other does not. It is also obvious that if there exist a sequence of pairs in P 1, such that w = u 0 ui 1 ui 2 · · · uim = v 0 v i 1 v i 2 · · · v im . Then the sequence produces a match in P 2. The words and in P 2 differ from the words u 0 ui 1 ui 2 · · · uim and v 0 v i 1 v i 2 · · · … \end{verbatim} ``` </details>
398. ph-6f6b70b92886afd19ed6 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) -decoded --> in P 2 differ from the words u 0 ui 1 ui 2 · · · uim and v 0 v i 1 v i 2 · · · v im respectively in P 1 in the fact that that they have stars between the letters and end in $. Hence, since a match in the modified Post's correspondence system has a corresponding match in Post's correspondence system, if Post's Correspondence Problem is decidable, then the modified Post's Correspondence Problem is decidable. /square ## Example 5.2 Using the previous modified Post's correspondence with match abcded f , we have with match ∗ a … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{-decoded --> in P 2 differ from the words u 0 ui 1 ui 2 · · · uim and v 0 v i 1 v i 2 · · · v im respectively in P 1 in the fact that that they have stars between the letters and end in $. Hence, since a match in the modified Post's correspondence system has a corresponding matc…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=365671 \begin{verbatim} -decoded --> in P 2 differ from the words u 0 ui 1 ui 2 · · · uim and v 0 v i 1 v i 2 · · · v im respectively in P 1 in the fact that that they have stars between the letters and end in $. Hence, since a match in the modified Post's correspondence system has a corresponding match in Post's correspondence system, if Post's Correspondence Problem is decidable, then the modified Post's Correspondence Problem is decidable. /square ## Example 5.2 Using the previous modified Post's correspondence with match abcded f , we have with match ∗ a … \end{verbatim} ``` </details>
399. ph-e5257cb33f97cdfee2c7 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) · uim and v 0 v i 1 v i 2 · · · v im respectively in P 1 in the fact that that they have stars between the letters and end in $. Hence, since a match in the modified Post's correspondence system has a corresponding match in Post's correspondence system, if Post's Correspondence Problem is decidable, then the modified Post's Correspondence Problem is decidable. /square ## Example 5.2 Using the previous modified Post's correspondence with match abcded f , we have with match ∗ a ∗ b ∗ c ∗ d ∗ e ∗ d ∗ f /star $. Theorem 5.9 Post's Correspo… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{· uim and v 0 v i 1 v i 2 · · · v im respectively in P 1 in the fact that that they have stars between the letters and end in $. Hence, since a match in the modified Post's correspondence system has a corresponding match in Post's correspondence system, if Post's Correspondence …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=365732 \begin{verbatim} · uim and v 0 v i 1 v i 2 · · · v im respectively in P 1 in the fact that that they have stars between the letters and end in $. Hence, since a match in the modified Post's correspondence system has a corresponding match in Post's correspondence system, if Post's Correspondence Problem is decidable, then the modified Post's Correspondence Problem is decidable. /square ## Example 5.2 Using the previous modified Post's correspondence with match abcded f , we have with match ∗ a ∗ b ∗ c ∗ d ∗ e ∗ d ∗ f /star $. Theorem 5.9 Post's Correspo… \end{verbatim} ``` </details>
400. ph-30513587a5c8d0d42526 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) Correspondence Problem is undecidable by showing that the modified Post's Correspondence Problem is undecidable. We do this by showing that if the modified Post's Correspondence Problem is decidable, then L 0 (see previous section) is acceptable, which means that it is decidable if a Turing machine accepts a given word. Assuming the sequence for a given Turing machine and word, we construct a modified Post's correspondence system that has a match if and only if M accepts w . Intuitively assume describes the process used by the Turing machine to read w , where each… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{Correspondence Problem is undecidable by showing that the modified Post's Correspondence Problem is undecidable. We do this by showing that if the modified Post's Correspondence Problem is decidable, then L 0 (see previous section) is acceptable, which means that it is decidable…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=366395 \begin{verbatim} Correspondence Problem is undecidable by showing that the modified Post's Correspondence Problem is undecidable. We do this by showing that if the modified Post's Correspondence Problem is decidable, then L 0 (see previous section) is acceptable, which means that it is decidable if a Turing machine accepts a given word. Assuming the sequence for a given Turing machine and word, we construct a modified Post's correspondence system that has a match if and only if M accepts w . Intuitively assume describes the process used by the Turing machine to read w , where each… \end{verbatim} ``` </details>
401. ph-c84ec32a9c0d23085cb2 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) 523d12327bad90b61b04fb37d87f08efbf2c12024a7dd442.png) Since this is a modified Post's correspondence system, we can require that we begin with this pair. For each X in /Gamma1 we have ![Image](./AutomataTheory_artifacts/image_000217_03250beda82ca304c7e4564bdee577c2eec62f0611e205b5a01177a95696b5d5.png) Wenext use the following pairs to guide us in selecting the next string in our match: For each state s , which is not a final state and each state s ′ , and symbols X , Y , and Z in /Gamma1 , ![Image](./AutomataTheory_artifacts/image_000218_541c7f305d6e24ef2385b7ddb0… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{523d12327bad90b61b04fb37d87f08efbf2c12024a7dd442.png) Since this is a modified Post's correspondence system, we can require that we begin with this pair. For each X in /Gamma1 we have ![Image](./AutomataTheory\_artifacts/image\_000217\_03250beda82ca304c7e4564bdee577c2eec62f0611e205…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=368132 \begin{verbatim} 523d12327bad90b61b04fb37d87f08efbf2c12024a7dd442.png) Since this is a modified Post's correspondence system, we can require that we begin with this pair. For each X in /Gamma1 we have ![Image](./AutomataTheory_artifacts/image_000217_03250beda82ca304c7e4564bdee577c2eec62f0611e205b5a01177a95696b5d5.png) Wenext use the following pairs to guide us in selecting the next string in our match: For each state s , which is not a final state and each state s ′ , and symbols X , Y , and Z in /Gamma1 , ![Image](./AutomataTheory_artifacts/image_000218_541c7f305d6e24ef2385b7ddb0… \end{verbatim} ``` </details>
402. ph-506cb7279b6fbfa47377 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) 77a95696b5d5.png) Wenext use the following pairs to guide us in selecting the next string in our match: For each state s , which is not a final state and each state s ′ , and symbols X , Y , and Z in /Gamma1 , ![Image](./AutomataTheory_artifacts/image_000218_541c7f305d6e24ef2385b7ddb0c359c22b5789564aafc4752fd95dd99ad7d81c.png) We shall call these the pairs generated by δ. In trying to get our match this set guides us to the next string. For example if we have in our match and one of the pairs above is ( si 0 , 1 s j ), we will want the… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{77a95696b5d5.png) Wenext use the following pairs to guide us in selecting the next string in our match: For each state s , which is not a final state and each state s ′ , and symbols X , Y , and Z in /Gamma1 , ![Image](./AutomataTheory\_artifacts/imag…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=368419 \begin{verbatim} 77a95696b5d5.png) Wenext use the following pairs to guide us in selecting the next string in our match: For each state s , which is not a final state and each state s ′ , and symbols X , Y , and Z in /Gamma1 , ![Image](./AutomataTheory_artifacts/image_000218_541c7f305d6e24ef2385b7ddb0c359c22b5789564aafc4752fd95dd99ad7d81c.png) We shall call these the pairs generated by δ. In trying to get our match this set guides us to the next string. For example if we have in our match and one of the pairs above is ( si 0 , 1 s j ), we will want the… \end{verbatim} ``` </details>
403. ph-1b2c30b35ea7eeac0a17 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) 541c7f305d6e24ef2385b7ddb0c359c22b5789564aafc4752fd95dd99ad7d81c.png) We shall call these the pairs generated by δ. In trying to get our match this set guides us to the next string. For example if we have in our match and one of the pairs above is ( si 0 , 1 s j ), we will want the next string to be #111 s j 11#. Note however that the two 1s at the beginning and end of the string are not affected by the pair above. Hence we need pairs (# , #) and (1 , 1) to get More precisely we would use 1 1 , 1 1 , si 0 1 , 1 1 , 1 1 , # # . Hence we… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{541c7f305d6e24ef2385b7ddb0c359c22b5789564aafc4752fd95dd99ad7d81c.png) We shall call these the pairs generated by δ. In trying to get our match this set guides us to the next string. For example if we have in our match and one of the pairs above is ( …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=368711 \begin{verbatim} 541c7f305d6e24ef2385b7ddb0c359c22b5789564aafc4752fd95dd99ad7d81c.png) We shall call these the pairs generated by δ. In trying to get our match this set guides us to the next string. For example if we have in our match and one of the pairs above is ( si 0 , 1 s j ), we will want the next string to be #111 s j 11#. Note however that the two 1s at the beginning and end of the string are not affected by the pair above. Hence we need pairs (# , #) and (1 , 1) to get More precisely we would use 1 1 , 1 1 , si 0 1 , 1 1 , 1 1 , # # . Hence we… \end{verbatim} ``` </details>
404. ph-391f6f112e748976c38c — automata/docling_md/AutomataTheory.md ### Plain (markdown context) In trying to get our match this set guides us to the next string. For example if we have in our match and one of the pairs above is ( si 0 , 1 s j ), we will want the next string to be #111 s j 11#. Note however that the two 1s at the beginning and end of the string are not affected by the pair above. Hence we need pairs (# , #) and (1 , 1) to get More precisely we would use 1 1 , 1 1 , si 0 1 , 1 1 , 1 1 , # # . Hence we need pairs Obviously if we never get to an acceptance sta… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{In trying to get our match this set guides us to the next string. For example if we have in our match and one of the pairs above is ( si 0 , 1 s j ), we will want the next string to be \#111 s j 11\#. Note however that the two 1s at the beginning and e…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=368829 \begin{verbatim} In trying to get our match this set guides us to the next string. For example if we have in our match and one of the pairs above is ( si 0 , 1 s j ), we will want the next string to be #111 s j 11#. Note however that the two 1s at the beginning and end of the string are not affected by the pair above. Hence we need pairs (# , #) and (1 , 1) to get More precisely we would use 1 1 , 1 1 , si 0 1 , 1 1 , 1 1 , # # . Hence we need pairs Obviously if we never get to an acceptance sta… \end{verbatim} ``` </details>
405. ph-2053c4e97cc38da334f0 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) s set guides us to the next string. For example if we have in our match and one of the pairs above is ( si 0 , 1 s j ), we will want the next string to be #111 s j 11#. Note however that the two 1s at the beginning and end of the string are not affected by the pair above. Hence we need pairs (# , #) and (1 , 1) to get More precisely we would use 1 1 , 1 1 , si 0 1 , 1 1 , 1 1 , # # . Hence we need pairs Obviously if we never get to an acceptance state (and hence a final state) w… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{s set guides us to the next string. For example if we have in our match and one of the pairs above is ( si 0 , 1 s j ), we will want the next string to be \#111 s j 11\#. Note however that the two 1s at the beginning and end of the string are not affec…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=368859 \begin{verbatim} s set guides us to the next string. For example if we have in our match and one of the pairs above is ( si 0 , 1 s j ), we will want the next string to be #111 s j 11#. Note however that the two 1s at the beginning and end of the string are not affected by the pair above. Hence we need pairs (# , #) and (1 , 1) to get More precisely we would use 1 1 , 1 1 , si 0 1 , 1 1 , 1 1 , # # . Hence we need pairs Obviously if we never get to an acceptance state (and hence a final state) w… \end{verbatim} ``` </details>
406. ph-620bad14355f6213b5f5 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) Hence we need pairs (# , #) and (1 , 1) to get More precisely we would use 1 1 , 1 1 , si 0 1 , 1 1 , 1 1 , # # . Hence we need pairs Obviously if we never get to an acceptance state (and hence a final state) we will never have a match since there will always be an overlap at the bottom. We thus need rules to get a match if we reach a halt state h . We use the following pairs to get rid of the overlap. The last term gets rid of the overlap sm when all of the other symbols have b… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{Hence we need pairs (\# , \#) and (1 , 1) to get More precisely we would use 1 1 , 1 1 , si 0 1 , 1 1 , 1 1 , \# \# . Hence we need pairs Obviously if we never get to an acceptance state (and henc…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=369163 \begin{verbatim} Hence we need pairs (# , #) and (1 , 1) to get More precisely we would use 1 1 , 1 1 , si 0 1 , 1 1 , 1 1 , # # . Hence we need pairs Obviously if we never get to an acceptance state (and hence a final state) we will never have a match since there will always be an overlap at the bottom. We thus need rules to get a match if we reach a halt state h . We use the following pairs to get rid of the overlap. The last term gets rid of the overlap sm when all of the other symbols have b… \end{verbatim} ``` </details>
407. ph-359869712f616eb546d3 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) . Hence we need pairs Obviously if we never get to an acceptance state (and hence a final state) we will never have a match since there will always be an overlap at the bottom. We thus need rules to get a match if we reach a halt state h . We use the following pairs to get rid of the overlap. The last term gets rid of the overlap sm when all of the other symbols have been eliminated. Thus if we reached as follows Obviously if we never get to an acceptance state (and hence a final state) we will never have a match since there will always be an overlap at the bottom. We thus need rules to get a match if we reac…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=369306 \begin{verbatim} . Hence we need pairs Obviously if we never get to an acceptance state (and hence a final state) we will never have a match since there will always be an overlap at the bottom. We thus need rules to get a match if we reach a halt state h . We use the following pairs to get rid of the overlap. The last term gets rid of the overlap sm when all of the other symbols have been eliminated. Thus if we reached as follows Obviously if we never get to an acceptance state (and hence a final state) we will never have a match since there will always be an overlap at the bottom. We thus need rules to get a match if we reach a halt state h . We use the following pairs to get rid of the overlap. The last term gets rid of the overlap sm when all of the other symbols have been eliminated. Thus if we reached as follows Formally… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{rmula-not-decoded --> Obviously if we never get to an acceptance state (and hence a final state) we will never have a match since there will always be an overlap at the bottom. We thus need rules to get a match if we reach a halt state h . We use the…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=369336 \begin{verbatim} rmula-not-decoded --> Obviously if we never get to an acceptance state (and hence a final state) we will never have a match since there will always be an overlap at the bottom. We thus need rules to get a match if we reach a halt state h . We use the following pairs to get rid of the overlap. The last term gets rid of the overlap sm when all of the other symbols have been eliminated. Thus if we reached as follows Formally… \end{verbatim} ``` </details>
409. ph-831cd15077e08135a4dd — automata/docling_md/AutomataTheory.md ### Plain (markdown context) coded --> Obviously if we never get to an acceptance state (and hence a final state) we will never have a match since there will always be an overlap at the bottom. We thus need rules to get a match if we reach a halt state h . We use the following pairs to get rid of the overlap. The last term gets rid of the overlap sm when all of the other symbols have been eliminated. Thus if we reached as follows Formally we give a proof of the theorem. If we ha… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{coded --> Obviously if we never get to an acceptance state (and hence a final state) we will never have a match since there will always be an overlap at the bottom. We thus need rules to get a match if we reach a halt state h . We use the following pairs to get rid of the overla…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=369378 \begin{verbatim} coded --> Obviously if we never get to an acceptance state (and hence a final state) we will never have a match since there will always be an overlap at the bottom. We thus need rules to get a match if we reach a halt state h . We use the following pairs to get rid of the overlap. The last term gets rid of the overlap sm when all of the other symbols have been eliminated. Thus if we reached as follows Formally we give a proof of the theorem. If we ha… \end{verbatim} ``` </details>
410. ph-083846f57f5e15f30dcc — automata/docling_md/AutomataTheory.md ### Plain (markdown context) er get to an acceptance state (and hence a final state) we will never have a match since there will always be an overlap at the bottom. We thus need rules to get a match if we reach a halt state h . We use the following pairs to get rid of the overlap. The last term gets rid of the overlap sm when all of the other symbols have been eliminated. Thus if we reached as follows Formally we give a proof of the theorem. If we have a valid set of sequences d… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{er get to an acceptance state (and hence a final state) we will never have a match since there will always be an overlap at the bottom. We thus need rules to get a match if we reach a halt state h . We use the following pairs to get rid of the overlap. The last term gets rid of the overlap sm when all of the other symbols have been eliminated. Thus if we reached as follows Formally we give a proof of the theorem. If we have a valid set of sequences d… \end{verbatim} ```
411. ph-2beb50dede7d3336cd24 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) rid of the overlap. The last term gets rid of the overlap sm when all of the other symbols have been eliminated. Thus if we reached as follows Formally we give a proof of the theorem. If we have a valid set of sequences describing the acceptance of w by M , using induction on the number of computations we show that there is a partial solution For n = 0, we have ![Image](./AutomataTheory_artifacts/image_000219_5b727be33d51… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{rid of the overlap. The last term gets rid of the overlap sm when all of the other symbols have been eliminated. Thus if we reached as follows The last term gets rid of the overlap sm when all of the other symbols have been eliminated. Thus if we reached as follows Formally we give a proof of the theorem. If we have a valid set of sequences describing the acceptance of w by M , using induction on the number of computations we show that there is a partial solution For n = 0, we have ![Image](./AutomataTheory_artifacts/image_000219_5b727be33d51… \end{verbatim} ```
412. ph-2aa5e58372e4a963e7f1 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) la-not-decoded --> Formally we give a proof of the theorem. If we have a valid set of sequences describing the acceptance of w by M , using induction on the number of computations we show that there is a partial solution For n = 0, we have ![Image](./AutomataTheory_artifacts/image_000219_5b727be33d51dd2f49e9af007eaafec411a761064191798997f44cbbd4f68413.png) Assuming the statement is true for k , and sk is not the halt state we have The next pairs are chosen so the string at the top forms # α k sk β k # using… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{la-not-decoded --> Formally we give a proof of the theorem. If we have a valid set of sequences describing the acceptance of w by M , using induction on the number of computations we show that there is a partial solution …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=369888 \begin{verbatim} la-not-decoded --> Formally we give a proof of the theorem. If we have a valid set of sequences describing the acceptance of w by M , using induction on the number of computations we show that there is a partial solution For n = 0, we have ![Image](./AutomataTheory_artifacts/image_000219_5b727be33d51dd2f49e9af007eaafec411a761064191798997f44cbbd4f68413.png) Assuming the statement is true for k , and sk is not the halt state we have The next pairs are chosen so the string at the top forms # α k sk β k # using… \end{verbatim} ``` </details>
413. ph-a960235298100d4a08a6 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) we show that there is a partial solution For n = 0, we have ![Image](./AutomataTheory_artifacts/image_000219_5b727be33d51dd2f49e9af007eaafec411a761064191798997f44cbbd4f68413.png) Assuming the statement is true for k , and sk is not the halt state we have The next pairs are chosen so the string at the top forms # α k sk β k # using the rules above. There is at most one pair in the pairs generated by δ that works. We can thus form and we have extended a new partial solution. Since rules generated by δ apply t… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{we show that there is a partial solution For n = 0, we have ![Image](./AutomataTheory\_artifacts/image\_000219\_5b727be33d51dd2f49e9af007eaafec411a761064191798997f44cbbd4f68413.png) Assuming the statement is true for k , and sk is not the halt state we …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=370098 \begin{verbatim} we show that there is a partial solution For n = 0, we have ![Image](./AutomataTheory_artifacts/image_000219_5b727be33d51dd2f49e9af007eaafec411a761064191798997f44cbbd4f68413.png) Assuming the statement is true for k , and sk is not the halt state we have The next pairs are chosen so the string at the top forms # α k sk β k # using the rules above. There is at most one pair in the pairs generated by δ that works. We can thus form and we have extended a new partial solution. Since rules generated by δ apply t… \end{verbatim} ``` </details>
414. ph-9ad40e4c894d2712a677 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) is not a rule and there can be no match. If, for some k , β k is a halt state, then as mentioned above, there are rules to make the upper and lower lists agree. As already mentioned, if we do not reach the halt state, we cannot have a match. If we do reach the halt state, we can produce a match. Hence if the modified Post's Correspondence Problem is decidable, L 0 is decidable. Therefore the modified Post's Correspondence Problem is undecidable. /square ## Example 5.3 Let the Turing Machine and word 0110 where with corresponding pairs In addition we have pairs <!-… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{is not a rule and there can be no match. If, for some k , β k is a halt state, then as mentioned above, there are rules to make the upper and lower lists agree. As already mentioned, if we do not reach the halt state, we cannot have a match. If we do reach the halt state, we can…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=370934 \begin{verbatim} is not a rule and there can be no match. If, for some k , β k is a halt state, then as mentioned above, there are rules to make the upper and lower lists agree. As already mentioned, if we do not reach the halt state, we cannot have a match. If we do reach the halt state, we can produce a match. Hence if the modified Post's Correspondence Problem is decidable, L 0 is decidable. Therefore the modified Post's Correspondence Problem is undecidable. /square ## Example 5.3 Let the Turing Machine and word 0110 where with corresponding pairs In addition we have pairs <!-… \end{verbatim} ``` </details>
415. ph-38da4de0e7330e24fc04 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) , there are rules to make the upper and lower lists agree. As already mentioned, if we do not reach the halt state, we cannot have a match. If we do reach the halt state, we can produce a match. Hence if the modified Post's Correspondence Problem is decidable, L 0 is decidable. Therefore the modified Post's Correspondence Problem is undecidable. /square ## Example 5.3 Let the Turing Machine and word 0110 where with corresponding pairs In addition we have pairs Our first pair is … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{, there are rules to make the upper and lower lists agree. As already mentioned, if we do not reach the halt state, we cannot have a match. If we do reach the halt state, we can produce a match. Hence if the modified Post's Correspondence Problem is decidable, L 0 is decidable. …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=371037 \begin{verbatim} , there are rules to make the upper and lower lists agree. As already mentioned, if we do not reach the halt state, we cannot have a match. If we do reach the halt state, we can produce a match. Hence if the modified Post's Correspondence Problem is decidable, L 0 is decidable. Therefore the modified Post's Correspondence Problem is undecidable. /square ## Example 5.3 Let the Turing Machine and word 0110 where with corresponding pairs In addition we have pairs Our first pair is … \end{verbatim} ``` </details>
416. ph-fff03bb58cb347e8424b — automata/docling_md/AutomataTheory.md ### Plain (markdown context) upper and lower lists agree. As already mentioned, if we do not reach the halt state, we cannot have a match. If we do reach the halt state, we can produce a match. Hence if the modified Post's Correspondence Problem is decidable, L 0 is decidable. Therefore the modified Post's Correspondence Problem is undecidable. /square ## Example 5.3 Let the Turing Machine and word 0110 where with corresponding pairs In addition we have pairs Our first pair is (# , # s 0 w ) which produces … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{upper and lower lists agree. As already mentioned, if we do not reach the halt state, we cannot have a match. If we do reach the halt state, we can produce a match. Hence if the modified Post's Correspondence Problem is decidable, L 0 is decidable. Therefore the modified Post's …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=371067 \begin{verbatim} upper and lower lists agree. As already mentioned, if we do not reach the halt state, we cannot have a match. If we do reach the halt state, we can produce a match. Hence if the modified Post's Correspondence Problem is decidable, L 0 is decidable. Therefore the modified Post's Correspondence Problem is undecidable. /square ## Example 5.3 Let the Turing Machine and word 0110 where with corresponding pairs In addition we have pairs Our first pair is (# , # s 0 w ) which produces … \end{verbatim} ``` </details>
417. ph-7df2c0bab36601ced4de — automata/docling_md/AutomataTheory.md ### Plain (markdown context) As already mentioned, if we do not reach the halt state, we cannot have a match. If we do reach the halt state, we can produce a match. Hence if the modified Post's Correspondence Problem is decidable, L 0 is decidable. Therefore the modified Post's Correspondence Problem is undecidable. /square ## Example 5.3 Let the Turing Machine and word 0110 where with corresponding pairs In addition we have pairs Our first pair is (# , # s 0 w ) which produces # # s 0010#. and word 0110 where with corresponding pairs In addition we have pairs Our first pair is (# , # s 0 w ) which produces # # s 0010#. and word 0110 where with corresponding pairs In addition we have pairs Our first pair is (# , # s 0 w ) which produces # # s 0010#. We then use (1 , 1) twice, (0 , 0), and (# , #) to get and word …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=371191 \begin{verbatim} h the halt state, we can produce a match. Hence if the modified Post's Correspondence Problem is decidable, L 0 is decidable. Therefore the modified Post's Correspondence Problem is undecidable. /square ## Example 5.3 Let the Turing Machine and word 0110 where with corresponding pairs In addition we have pairs Our first pair is (# , # s 0 w ) which produces # # s 0010#. We then use (1 , 1) twice, (0 , 0), and (# , #) to get and word 0110 where with corresponding pairs In addition we have pairs Our first pair is (# , # s 0 w ) which produces # # s 0010#. We then use (1 , 1) twice, (0 , 0), and (# , #) to get We next use ( /star, /star ) , ( s 11 , 1 s 1 ) , (1 , 1),(0 , 0), and (# , #) , to… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{Problem is decidable, L 0 is decidable. Therefore the modified Post's Correspondence Problem is undecidable. /square \#\# Example 5.3 Let the Turing Machine and word 0110 where with corresponding pairs In addition we have pairs and word 0110 where with corresponding pairs In addition we have pairs Our first pair is (# , # s 0 w ) which produces # # s 0010#. We then use (1 , 1) twice, (0 , 0), and (# , #) to get We next use ( /star, /star ) , ( s 11 , 1 s 1 ) , (1 , 1),(0 , 0), and (# , #) , to… \end{verbatim} ``` </details>
420. ph-9bf57e7253a4b959d5df — automata/docling_md/AutomataTheory.md ### Plain (markdown context) Our first pair is (# , # s 0 w ) which produces # # s 0010#. We then use (1 , 1) twice, (0 , 0), and (# , #) to get We next use ( /star, /star ) , ( s 11 , 1 s 1 ) , (1 , 1),(0 , 0), and (# , #) , to get # s # 00110# s 00110# /star /star s s 1110# 1110# /star 1 s 110#, again using ( /star, /star ) , (1 , 1) , ( s 11 , 1 s 1 ) , (0 , 0), and (# , #) we get Now using ( /star, /star ) , (1 , 1) , (1 s 10 , s 210), and (# , #) we get Our first pair is (\# , \# s 0 w ) which produces \# \# s 0010\#. We then use (1 , 1) twice, (0 , 0), and (\# , \#) to get We next use ( /star, /star ) , ( s 11 , 1 s 1 )…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=371566 \begin{verbatim} Our first pair is (# , # s 0 w ) which produces # # s 0010#. We then use (1 , 1) twice, (0 , 0), and (# , #) to get We next use ( /star, /star ) , ( s 11 , 1 s 1 ) , (1 , 1),(0 , 0), and (# , #) , to get # s # 00110# s 00110# /star /star s s 1110# 1110# /star 1 s 110#, again using ( /star, /star ) , (1 , 1) , ( s 11 , 1 s 1 ) , (0 , 0), and (# , #) we get Now using ( /star, /star ) , (1 , 1) , (1 s 10 , s 210), and (# , #) we get We then use (1 , 1) twice, (0 , 0), and (# , #) to get We next use ( /star, /star ) , ( s 11 , 1 s 1 ) , (1 , 1),(0 , 0), and (# , #) , to get # s # 00110# s 00110# /star /star s s 1110# 1110# /star 1 s 110#, again using ( /star, /star ) , (1 , 1) , ( s 11 , 1 s 1 ) , (0 , 0), and (# , #) we get Now using ( /star, /star ) , (1 , 1) , (1 s 10 , s 210), and (# , #) we get Using ( /star, /star ) , ( /star s 11 , s 2 /star 1) , (1 , 1) , (0 , 0), and (# , #) we get <!… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{s \# \# s 0010\#. We then use (1 , 1) twice, (0 , 0), and (\# , \#) to get We next use ( /star, /star ) , ( s 11 , 1 s 1 ) , (1 , 1),(0 , 0), and (\# , \#) , to get \# s \# 00110\# s 00110\# /star /star s s 1110\# 1110\# /star 1 s 110…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=371673 \begin{verbatim} s # # s 0010#. We then use (1 , 1) twice, (0 , 0), and (# , #) to get We next use ( /star, /star ) , ( s 11 , 1 s 1 ) , (1 , 1),(0 , 0), and (# , #) , to get # s # 00110# s 00110# /star /star s s 1110# 1110# /star 1 s 110#, again using ( /star, /star ) , (1 , 1) , ( s 11 , 1 s 1 ) , (0 , 0), and (# , #) we get Now using ( /star, /star ) , (1 , 1) , (1 s 10 , s 210), and (# , #) we get Using ( /star, /star ) , ( /star s 11 , s 2 /star 1) , (1 , 1) , (0 , 0), and (# , #) we get <!… \end{verbatim} ``` </details>
422. ph-32129c3da0474cbb83c2 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) oded --> We next use ( /star, /star ) , ( s 11 , 1 s 1 ) , (1 , 1),(0 , 0), and (# , #) , to get # s # 00110# s 00110# /star /star s s 1110# 1110# /star 1 s 110#, again using ( /star, /star ) , (1 , 1) , ( s 11 , 1 s 1 ) , (0 , 0), and (# , #) we get Now using ( /star, /star ) , (1 , 1) , (1 s 10 , s 210), and (# , #) we get Using ( /star, /star ) , ( /star s 11 , s 2 /star 1) , (1 , 1) , (0 , 0), and (# , #) we get Now using ( s 2 /star, h 0), (1 , 1) twice, (0 , 0), and (# , #) , we get We next use ( /star, /star ) , ( s 11 , 1 s 1 ) , (1 , 1),(0 , 0), and (\# , \#) , to get \# s \# 00110\# s 00110\# /star /star s s 1110\# 1110\# /star 1 s 110\#, again using ( /star, /star ) , (1 , 1) , ( s 11 , 1 s 1 ) , (0 , 0), and (\# , \#) we get …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=371797 \begin{verbatim} oded --> We next use ( /star, /star ) , ( s 11 , 1 s 1 ) , (1 , 1),(0 , 0), and (# , #) , to get # s # 00110# s 00110# /star /star s s 1110# 1110# /star 1 s 110#, again using ( /star, /star ) , (1 , 1) , ( s 11 , 1 s 1 ) , (0 , 0), and (# , #) we get Now using ( /star, /star ) , (1 , 1) , (1 s 10 , s 210), and (# , #) we get Using ( /star, /star ) , ( /star s 11 , s 2 /star 1) , (1 , 1) , (0 , 0), and (# , #) we get Now using ( s 2 /star, h 0), (1 , 1) twice, (0 , 0), and (# , #) , we get Now using ( /star, /star ) , (1 , 1) , (1 s 10 , s 210), and (# , #) we get Using ( /star, /star ) , ( /star s 11 , s 2 /star 1) , (1 , 1) , (0 , 0), and (# , #) we get Now using ( s 2 /star, h 0), (1 , 1) twice, (0 , 0), and (# , #) , we get Finally, using the pairs containing h , together with (1 , 1) , (0 , 0), and (# , #) , we … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{\# 00110\# s 00110\# /star /star s s 1110\# 1110\# /star 1 s 110\#, again using ( /star, /star ) , (1 , 1) , ( s 11 , 1 s 1 ) , (0 , 0), and (\# , \#) we get Now using ( /star, /star ) , (1 , 1) , (1 s 10 , s 210), and (\# , \#) we get Now using ( /star, /star ) , (1 , 1) , (1 s 10 , s 210), and (# , #) we get Using ( /star, /star ) , ( /star s 11 , s 2 /star 1) , (1 , 1) , (0 , 0), and (# , #) we get Now using ( s 2 /star, h 0), (1 , 1) twice, (0 , 0), and (# , #) , we get Finally, using the pairs containing h , together with (1 , 1) , (0 , 0), and (# , #) , we … \end{verbatim} ``` </details>
424. ph-b6a14a0412ab33f2e5d2 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) 1 s 1 ) , (0 , 0), and (# , #) we get Now using ( /star, /star ) , (1 , 1) , (1 s 10 , s 210), and (# , #) we get Using ( /star, /star ) , ( /star s 11 , s 2 /star 1) , (1 , 1) , (0 , 0), and (# , #) we get Now using ( s 2 /star, h 0), (1 , 1) twice, (0 , 0), and (# , #) , we get Finally, using the pairs containing h , together with (1 , 1) , (0 , 0), and (# , #) , we get Wecan now use the fact that Post's Correspondence Problem is undecidable to sol… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{1 s 1 ) , (0 , 0), and (\# , \#) we get Now using ( /star, /star ) , (1 , 1) , (1 s 10 , s 210), and (\# , \#) we get Using ( /star, /star ) , ( /star s 11 , s 2 /star 1) , (1 , 1) , (0 , 0), and (\# , \#) we get Now using ( /star, /star ) , (1 , 1) , (1 s 10 , s 210), and (# , #) we get Using ( /star, /star ) , ( /star s 11 , s 2 /star 1) , (1 , 1) , (0 , 0), and (# , #) we get Now using ( s 2 /star, h 0), (1 , 1) twice, (0 , 0), and (# , #) , we get Finally, using the pairs containing h , together with (1 , 1) , (0 , 0), and (# , #) , we get Wecan now use the fact that Post's Correspondence Problem is undecidable to sol… \end{verbatim} ```
425. ph-df6b6a42d478d7c5bb5f — automata/docling_md/AutomataTheory.md ### Plain (markdown context) ions about solvability with regard to context-free languages. We now use ( s 00 , /star s 1 ) to get Theorem 5.10 It is undecidable for arbitrary context-free grammars G 1 and G 2 whether L ( G 1 ) ∩ L ( G 2 ) = ∅ . Proof Let P ⊂ /Sigma1 ∗ × /Sigma1 ∗ be an arbitrary correspondence system with pairs ( u 0 , v 0 ) , ( u 1 , v 1 ) , ( u 2 , v 2 ) , . . . , ( un , v n ). In the following, w -1 will be w with the letters reversed. For example 1101 -1 is 1011. Let G 1 be generated by productions Thus every word in L ( G 1) has the form ui 0… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ions about solvability with regard to context-free languages. We now use ( s 00 , /star s 1 ) to get Theorem 5.10 It is undecidable for arbitrary context-free grammars G 1 and G 2 whether L ( G 1 ) ∩ L ( G 2 ) = ∅ . Proof Let P ⊂ /Sigma1 ∗ × /Sigma1 ∗ be an arbitrary corresponde…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=372658 \begin{verbatim} ions about solvability with regard to context-free languages. We now use ( s 00 , /star s 1 ) to get Theorem 5.10 It is undecidable for arbitrary context-free grammars G 1 and G 2 whether L ( G 1 ) ∩ L ( G 2 ) = ∅ . Proof Let P ⊂ /Sigma1 ∗ × /Sigma1 ∗ be an arbitrary correspondence system with pairs ( u 0 , v 0 ) , ( u 1 , v 1 ) , ( u 2 , v 2 ) , . . . , ( un , v n ). In the following, w -1 will be w with the letters reversed. For example 1101 -1 is 1011. Let G 1 be generated by productions Thus every word in L ( G 1) has the form ui 0… \end{verbatim} ``` </details>
426. ph-a94a1d9a8fe50be32de5 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) gard to context-free languages. We now use ( s 00 , /star s 1 ) to get Theorem 5.10 It is undecidable for arbitrary context-free grammars G 1 and G 2 whether L ( G 1 ) ∩ L ( G 2 ) = ∅ . Proof Let P ⊂ /Sigma1 ∗ × /Sigma1 ∗ be an arbitrary correspondence system with pairs ( u 0 , v 0 ) , ( u 1 , v 1 ) , ( u 2 , v 2 ) , . . . , ( un , v n ). In the following, w -1 will be w with the letters reversed. For example 1101 -1 is 1011. Let G 1 be generated by productions Thus every word in L ( G 1) has the form ui 0 ui 1 ui 2 . . . uim c v -1 im… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{gard to context-free languages. We now use ( s 00 , /star s 1 ) to get Theorem 5.10 It is undecidable for arbitrary context-free grammars G 1 and G 2 whether L ( G 1 ) ∩ L ( G 2 ) = ∅ . Proof Let P ⊂ /Sigma1 ∗ × /Sigma1 ∗ be an arbitrary correspondence system with pairs ( u 0 , …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=372688 \begin{verbatim} gard to context-free languages. We now use ( s 00 , /star s 1 ) to get Theorem 5.10 It is undecidable for arbitrary context-free grammars G 1 and G 2 whether L ( G 1 ) ∩ L ( G 2 ) = ∅ . Proof Let P ⊂ /Sigma1 ∗ × /Sigma1 ∗ be an arbitrary correspondence system with pairs ( u 0 , v 0 ) , ( u 1 , v 1 ) , ( u 2 , v 2 ) , . . . , ( un , v n ). In the following, w -1 will be w with the letters reversed. For example 1101 -1 is 1011. Let G 1 be generated by productions Thus every word in L ( G 1) has the form ui 0 ui 1 ui 2 . . . uim c v -1 im… \end{verbatim} ``` </details>
427. ph-2c883c79956c3964fcf5 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) ly if w = ui 0 ui 1 ui 2 . . . uim = v i 0 v i 1 v i 2 . . . v im which is a solution to the Post's correspondence system. Hence it is undecidable for arbitrary context-free grammars G G whether L ( G ) L ( G ) . 1 and 2 1 ∩ 2 = ∅ /square Definition 5.8 A context-free grammar is ambiguous if there are two leftmost generations of the same word. Example 5.4 Let /Gamma1 = ( N , /Sigma1 , S , P ) be the grammar defined by N = { S , A , B } , /Sigma1 = { a , b } , and P be the set of productions Obviously a n b n can be generated in two different ways. Theorem 5.11 It … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ly if w = ui 0 ui 1 ui 2 . . . uim = v i 0 v i 1 v i 2 . . . v im which is a solution to the Post's correspondence system. Hence it is undecidable for arbitrary context-free grammars G G whether L ( G ) L ( G ) . 1 and 2 1 ∩ 2 = ∅ /square Definition 5.8 A context-free grammar is…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=373416 \begin{verbatim} ly if w = ui 0 ui 1 ui 2 . . . uim = v i 0 v i 1 v i 2 . . . v im which is a solution to the Post's correspondence system. Hence it is undecidable for arbitrary context-free grammars G G whether L ( G ) L ( G ) . 1 and 2 1 ∩ 2 = ∅ /square Definition 5.8 A context-free grammar is ambiguous if there are two leftmost generations of the same word. Example 5.4 Let /Gamma1 = ( N , /Sigma1 , S , P ) be the grammar defined by N = { S , A , B } , /Sigma1 = { a , b } , and P be the set of productions Obviously a n b n can be generated in two different ways. Theorem 5.11 It … \end{verbatim} ``` </details>
428. ph-013d7aef9810363bca35 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) /Sigma1 = { a , b } , and P be the set of productions Obviously a n b n can be generated in two different ways. Theorem 5.11 It is undecidable whether an arbitrary context-free grammar is ambiguous. Proof Let P ⊂ /Sigma1 + × /Sigma1 + be an arbitrary correspondence system with pairs ( u 0 , v 0 )( u 1 , v 1 ) , ( u 2 , v 2 ) , . . . , ( un , v n ). Let α 0 , α 1 , α 2 , . . . , α n be symbols not in /Sigma1 ∗ . We construct two grammars G 1 and G 2 as follows: where N 1 = { S 1 } , /Sigma1 a = /Sigma1 ∪ { α 0 , α 1 , α 2 , . . . , α n … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{/Sigma1 = \{ a , b \} , and P be the set of productions Obviously a n b n can be generated in two different ways. Theorem 5.11 It is undecidable whether an arbitrary context-free grammar is ambiguous. Proof Let P ⊂ /Sigma1 + × /Sigma1 + be an arbitrary…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=373860 \begin{verbatim} /Sigma1 = { a , b } , and P be the set of productions Obviously a n b n can be generated in two different ways. Theorem 5.11 It is undecidable whether an arbitrary context-free grammar is ambiguous. Proof Let P ⊂ /Sigma1 + × /Sigma1 + be an arbitrary correspondence system with pairs ( u 0 , v 0 )( u 1 , v 1 ) , ( u 2 , v 2 ) , . . . , ( un , v n ). Let α 0 , α 1 , α 2 , . . . , α n be symbols not in /Sigma1 ∗ . We construct two grammars G 1 and G 2 as follows: where N 1 = { S 1 } , /Sigma1 a = /Sigma1 ∪ { α 0 , α 1 , α 2 , . . . , α n … \end{verbatim} ``` </details>
429. ph-ce0200b6b8890dbe41e2 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) an arbitrary context-free grammar is ambiguous. Proof Let P ⊂ /Sigma1 + × /Sigma1 + be an arbitrary correspondence system with pairs ( u 0 , v 0 )( u 1 , v 1 ) , ( u 2 , v 2 ) , . . . , ( un , v n ). Let α 0 , α 1 , α 2 , . . . , α n be symbols not in /Sigma1 ∗ . We construct two grammars G 1 and G 2 as follows: where N 1 = { S 1 } , /Sigma1 a = /Sigma1 ∪ { α 0 , α 1 , α 2 , . . . , α n } , and P 1 = { S 1 → α i S 1 ui for i = 0 , 1 , . . . , n , and S 1 → λ } . where N 2 = { S 2 } , /Sigma1 a = /Sigma1 ∪ { α 0 , α 1 , α 2 , . . . , α … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{an arbitrary context-free grammar is ambiguous. Proof Let P ⊂ /Sigma1 + × /Sigma1 + be an arbitrary correspondence system with pairs ( u 0 , v 0 )( u 1 , v 1 ) , ( u 2 , v 2 ) , . . . , ( un , v n ). Let α 0 , α 1 , α 2 , . . . , α n be symbols not in /Sigma1 ∗ . We construct tw…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=374044 \begin{verbatim} an arbitrary context-free grammar is ambiguous. Proof Let P ⊂ /Sigma1 + × /Sigma1 + be an arbitrary correspondence system with pairs ( u 0 , v 0 )( u 1 , v 1 ) , ( u 2 , v 2 ) , . . . , ( un , v n ). Let α 0 , α 1 , α 2 , . . . , α n be symbols not in /Sigma1 ∗ . We construct two grammars G 1 and G 2 as follows: where N 1 = { S 1 } , /Sigma1 a = /Sigma1 ∪ { α 0 , α 1 , α 2 , . . . , α n } , and P 1 = { S 1 → α i S 1 ui for i = 0 , 1 , . . . , n , and S 1 → λ } . where N 2 = { S 2 } , /Sigma1 a = /Sigma1 ∪ { α 0 , α 1 , α 2 , . . . , α … \end{verbatim} ``` </details>
430. ph-a3ce84e8efc7ba138bac — automata/docling_md/AutomataTheory.md ### Plain (markdown context) λ . Proposition 6.2 For each word u in /Sigma1 + there is a unique pair ( q , n ) , with q in Q and n in N, for which u = q n . Proof For each i with 1 ≤ i ≤ | u | , let u i be the prefix of u of length i . Compute successively the u / ui until a j occurs at which u / u j = ( m , λ ). Such a j will certainly occur since u / u | u | = (1 , λ ). For the pair ( u j , m ) we have u j in Q. Consequently u = u m j has the required form. Suppose now that u = p m = q n , where both p and q are in Q. and Thus pq = qp and by Proposition 6.1, p a… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{λ . Proposition 6.2 For each word u in /Sigma1 + there is a unique pair ( q , n ) , with q in Q and n in N, for which u = q n . Proof For each i with 1 ≤ i ≤ | u | , let u i be the prefix of u of length i . Compute successively the u / ui until a j occurs at which u / u j = ( m …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=383957 \begin{verbatim} λ . Proposition 6.2 For each word u in /Sigma1 + there is a unique pair ( q , n ) , with q in Q and n in N, for which u = q n . Proof For each i with 1 ≤ i ≤ | u | , let u i be the prefix of u of length i . Compute successively the u / ui until a j occurs at which u / u j = ( m , λ ). Such a j will certainly occur since u / u | u | = (1 , λ ). For the pair ( u j , m ) we have u j in Q. Consequently u = u m j has the required form. Suppose now that u = p m = q n , where both p and q are in Q. and Thus pq = qp and by Proposition 6.1, p a… \end{verbatim} ``` </details>
431. ph-f7cfeb93c82c61998512 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) u in /Sigma1 + there is a unique pair ( q , n ) , with q in Q and n in N, for which u = q n . Proof For each i with 1 ≤ i ≤ | u | , let u i be the prefix of u of length i . Compute successively the u / ui until a j occurs at which u / u j = ( m , λ ). Such a j will certainly occur since u / u | u | = (1 , λ ). For the pair ( u j , m ) we have u j in Q. Consequently u = u m j has the required form. Suppose now that u = p m = q n , where both p and q are in Q. and Thus pq = qp and by Proposition 6.1, p and q must be powers of a common wo… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{u in /Sigma1 + there is a unique pair ( q , n ) , with q in Q and n in N, for which u = q n . Proof For each i with 1 ≤ i ≤ | u | , let u i be the prefix of u of length i . Compute successively the u / ui until a j occurs at which u / u j = ( m , λ ). Such a j will certainly occ…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=383992 \begin{verbatim} u in /Sigma1 + there is a unique pair ( q , n ) , with q in Q and n in N, for which u = q n . Proof For each i with 1 ≤ i ≤ | u | , let u i be the prefix of u of length i . Compute successively the u / ui until a j occurs at which u / u j = ( m , λ ). Such a j will certainly occur since u / u | u | = (1 , λ ). For the pair ( u j , m ) we have u j in Q. Consequently u = u m j has the required form. Suppose now that u = p m = q n , where both p and q are in Q. and Thus pq = qp and by Proposition 6.1, p and q must be powers of a common wo… \end{verbatim} ``` </details>
432. ph-9269896a867024a65901 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) /Sigma1 + is either ∅ or { 1 } . 8. (c) For the language LL , determine the spectra of ab , abab , and ababab . 9. (d) Describe P ( LL ) and Su( LL ). ## 6.5 Visualizing languages In order to spell out the visualization of a language L within Q × N , we begin with the usual x -y plane with each point having associated real number coordinates ( x , y ). We use only the upper half plane , { ( x , y ) : y > 0 } . With each integer i and each positive integer n we associate the unit rectangle In this way the upper half plane is partitioned into nonoverlapping unit … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{/Sigma1 + is either ∅ or \{ 1 \} . 8. (c) For the language LL , determine the spectra of ab , abab , and ababab . 9. (d) Describe P ( LL ) and Su( LL ). \#\# 6.5 Visualizing languages In order to spell out the visualization of a language L within Q × N , we begin with the usual x -y…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=395623 \begin{verbatim} /Sigma1 + is either ∅ or { 1 } . 8. (c) For the language LL , determine the spectra of ab , abab , and ababab . 9. (d) Describe P ( LL ) and Su( LL ). ## 6.5 Visualizing languages In order to spell out the visualization of a language L within Q × N , we begin with the usual x -y plane with each point having associated real number coordinates ( x , y ). We use only the upper half plane , { ( x , y ) : y > 0 } . With each integer i and each positive integer n we associate the unit rectangle In this way the upper half plane is partitioned into nonoverlapping unit … \end{verbatim} ``` </details>
433. ph-af6c21ec15eaded4b97e — automata/docling_md/AutomataTheory.md ### Plain (markdown context) f any word is m . The collection of distinct flags { F ( w ) : w ∈ /Sigma1 + } associated with a regular language L is necessarily finite. By a fl ag F of the language L we mean a sequence of states that constitutes the flag, relative to L , of some word in w in /Sigma1 + . With each flag F of L we associate the language I ( F ) = { w ∈ /Sigma1 + : F ( w ) = F } . We call I ( F ) the language of the flag F . For each flag F = { s j : 0 ≤ j ≤ k } , where the s i denote the states in F , we have where each L ( s j , s j + 1) is the language that consists of all word… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{f any word is m . The collection of distinct flags \{ F ( w ) : w ∈ /Sigma1 + \} associated with a regular language L is necessarily finite. By a fl ag F of the language L we mean a sequence of states that constitutes the flag, relative to L , of some word in w in /Sigma1 + . With…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=411124 \begin{verbatim} f any word is m . The collection of distinct flags { F ( w ) : w ∈ /Sigma1 + } associated with a regular language L is necessarily finite. By a fl ag F of the language L we mean a sequence of states that constitutes the flag, relative to L , of some word in w in /Sigma1 + . With each flag F of L we associate the language I ( F ) = { w ∈ /Sigma1 + : F ( w ) = F } . We call I ( F ) the language of the flag F . For each flag F = { s j : 0 ≤ j ≤ k } , where the s i denote the states in F , we have where each L ( s j , s j + 1) is the language that consists of all word… \end{verbatim} ``` </details>
434. ph-dd10b6741327355c7962 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) next proposition it follows that conjugates have the same exponent , which includes the information that the conjugates of primitive words are primitive . This last fact, that conjugates of primitives are primitive, is applied many times in Section 6.9. Proposition 6.3 If u v = p n then v u = q n with q a conjugate of p. Proof Since u v = p n , we may assume that p = u ′′ v ′ where u = p i u ′′ and v = v ′ p j with i and j nonnegative integers for which i + j = n -1. For q = v ′ u ′′ we have Lemma 6.1 Let v be a word for which vv = x v y with x and y nonnull, then… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{next proposition it follows that conjugates have the same exponent , which includes the information that the conjugates of primitive words are primitive . This last fact, that conjugates of primitives are primitive, is applied many times in Section 6.9. Proposition 6.3 If u v = …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=413098 \begin{verbatim} next proposition it follows that conjugates have the same exponent , which includes the information that the conjugates of primitive words are primitive . This last fact, that conjugates of primitives are primitive, is applied many times in Section 6.9. Proposition 6.3 If u v = p n then v u = q n with q a conjugate of p. Proof Since u v = p n , we may assume that p = u ′′ v ′ where u = p i u ′′ and v = v ′ p j with i and j nonnegative integers for which i + j = n -1. For q = v ′ u ′′ we have Lemma 6.1 Let v be a word for which vv = x v y with x and y nonnull, then… \end{verbatim} ``` </details>
435. ph-3184f42f34ff9a0b371b — automata/docling_md/AutomataTheory.md ### Plain (markdown context) e ligase can now paste together the fragments ttttgga ′ and ′ acctttt to yield the dsDNA molecule x ttttggaacctttt. The ligase can also paste together the fragments = ## TTTTGGAACCTTT we know its companion row is ## AAAACCTTGGAAA . Consequently, we need to give only one of the two strands. For efficiency and convenience we will list only one row of each dsDNA molecule. To be certain not to confuse dsDNA and ssDNA, we will use lowercase a, c, g, t to denote the paired deoxyribonucleotides: tttgga ′ and ′ accttt to yield the dsDNA molecule y = tttggaaccttt. The mole… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{e ligase can now paste together the fragments ttttgga ′ and ′ acctttt to yield the dsDNA molecule x ttttggaacctttt. The ligase can also paste together the fragments = \#\# TTTTGGAACCTTT we know its companion row is \#\# AAAACCTTGGAAA . Consequently, we need to give only one of the t…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=439174 \begin{verbatim} e ligase can now paste together the fragments ttttgga ′ and ′ acctttt to yield the dsDNA molecule x ttttggaacctttt. The ligase can also paste together the fragments = ## TTTTGGAACCTTT we know its companion row is ## AAAACCTTGGAAA . Consequently, we need to give only one of the two strands. For efficiency and convenience we will list only one row of each dsDNA molecule. To be certain not to confuse dsDNA and ssDNA, we will use lowercase a, c, g, t to denote the paired deoxyribonucleotides: tttgga ′ and ′ accttt to yield the dsDNA molecule y = tttggaaccttt. The mole… \end{verbatim} ``` </details>
436. ph-d0b985ee6ce067216b69 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) nner, possibly allowing ambiguity between molecules and the words used to represent them. The remainder of this chapter deals specifically with words in a free monoid. (However, all results in the chapter have meaningful interpretations for enzymes acting on dsDNA.) Let /Sigma1 be a finite set to be used as an alphabet. Let /Sigma1 ∗ be the set of all strings over /Sigma1 . By a language we mean a subset of /Sigma1 ∗ . A splicing rule is an element r = ( u , u ′ , v ′ , v ) of the product set The action of the rule r on a language L defines the language r ( L ) = … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{nner, possibly allowing ambiguity between molecules and the words used to represent them. The remainder of this chapter deals specifically with words in a free monoid. (However, all results in the chapter have meaningful interpretations for enzymes acting on dsDNA.) Let /Sigma1 …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=442461 \begin{verbatim} nner, possibly allowing ambiguity between molecules and the words used to represent them. The remainder of this chapter deals specifically with words in a free monoid. (However, all results in the chapter have meaningful interpretations for enzymes acting on dsDNA.) Let /Sigma1 be a finite set to be used as an alphabet. Let /Sigma1 ∗ be the set of all strings over /Sigma1 . By a language we mean a subset of /Sigma1 ∗ . A splicing rule is an element r = ( u , u ′ , v ′ , v ) of the product set The action of the rule r on a language L defines the language r ( L ) = … \end{verbatim} ``` </details>
437. ph-656e480cb2e13fe38a35 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) a pair ( σ, I ) , where σ is a splicing scheme and I is a finite initial language contained in /Sigma1 ∗ . The language generated by ( σ, I ) is L ( σ, I ) = σ ∗ ( I ) . A language L is a splicing language if L = L ( σ, I ) for some splicing system ( σ, I ) . Example 7.1 Let /Sigma1 = { a , c , g , t } . Let r = ( u , u ′ , v ′ , v ) where the four words u , u ′ , v ′ , v in /Sigma1 ∗ appearing in the rule r are u = v ′ = gga and u ′ = v = acc . Let R = { r } . This gives the splicing scheme Let Observe that r applied to the ordered pa… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{a pair ( σ, I ) , where σ is a splicing scheme and I is a finite initial language contained in /Sigma1 ∗ . The language generated by ( σ, I ) is L ( σ, I ) = σ ∗ ( I ) . A language L is a splicing language if L = L ( σ, I ) for some splicing system ( σ, I ) . Example 7.1 Let /Si…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=443934 \begin{verbatim} a pair ( σ, I ) , where σ is a splicing scheme and I is a finite initial language contained in /Sigma1 ∗ . The language generated by ( σ, I ) is L ( σ, I ) = σ ∗ ( I ) . A language L is a splicing language if L = L ( σ, I ) for some splicing system ( σ, I ) . Example 7.1 Let /Sigma1 = { a , c , g , t } . Let r = ( u , u ′ , v ′ , v ) where the four words u , u ′ , v ′ , v in /Sigma1 ∗ appearing in the rule r are u = v ′ = gga and u ′ = v = acc . Let R = { r } . This gives the splicing scheme Let Observe that r applied to the ordered pa… \end{verbatim} ``` </details>
438. ph-1fe8b738b5f97eb54049 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) icing scheme and I is a finite initial language contained in /Sigma1 ∗ . The language generated by ( σ, I ) is L ( σ, I ) = σ ∗ ( I ) . A language L is a splicing language if L = L ( σ, I ) for some splicing system ( σ, I ) . Example 7.1 Let /Sigma1 = { a , c , g , t } . Let r = ( u , u ′ , v ′ , v ) where the four words u , u ′ , v ′ , v in /Sigma1 ∗ appearing in the rule r are u = v ′ = gga and u ′ = v = acc . Let R = { r } . This gives the splicing scheme Let Observe that r applied to the ordered pair ( ttttggaaccttt , tttggaacctttt… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{icing scheme and I is a finite initial language contained in /Sigma1 ∗ . The language generated by ( σ, I ) is L ( σ, I ) = σ ∗ ( I ) . A language L is a splicing language if L = L ( σ, I ) for some splicing system ( σ, I ) . Example 7.1 Let /Sigma1 = \{ a , c , g , t \} . Let r =…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=443969 \begin{verbatim} icing scheme and I is a finite initial language contained in /Sigma1 ∗ . The language generated by ( σ, I ) is L ( σ, I ) = σ ∗ ( I ) . A language L is a splicing language if L = L ( σ, I ) for some splicing system ( σ, I ) . Example 7.1 Let /Sigma1 = { a , c , g , t } . Let r = ( u , u ′ , v ′ , v ) where the four words u , u ′ , v ′ , v in /Sigma1 ∗ appearing in the rule r are u = v ′ = gga and u ′ = v = acc . Let R = { r } . This gives the splicing scheme Let Observe that r applied to the ordered pair ( ttttggaaccttt , tttggaacctttt… \end{verbatim} ``` </details>
439. ph-6c4822b34cfe642cc110 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) for some splicing system ( σ, I ) . Example 7.1 Let /Sigma1 = { a , c , g , t } . Let r = ( u , u ′ , v ′ , v ) where the four words u , u ′ , v ′ , v in /Sigma1 ∗ appearing in the rule r are u = v ′ = gga and u ′ = v = acc . Let R = { r } . This gives the splicing scheme Let Observe that r applied to the ordered pair ( ttttggaaccttt , tttggaacctttt ) of words in I gives the word ttttggaacctttt , and r applied to the ordered pair of words in I gives tttggaaccttt . The less interesting actions of r on I must… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{for some splicing system ( σ, I ) . Example 7.1 Let /Sigma1 = \{ a , c , g , t \} . Let r = ( u , u ′ , v ′ , v ) where the four words u , u ′ , v ′ , v in /Sigma1 ∗ appearing in the rule r are u = v ′ = gga and u ′ = v = acc . Let R = \{ r \} . This gives the splicing scheme Let Observe that r applied to the ordered pair ( ttttggaaccttt , tttggaacctttt ) of words in I gives the word ttttggaacctttt , and r applied to the ordered pair of words in I gives tttggaaccttt . The less interesting actions of r on I must… \end{verbatim} ```
440. ph-89900664ec99798dc300 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) ′ = gga and u ′ = v = acc . Let R = { r } . This gives the splicing scheme Let Observe that r applied to the ordered pair ( ttttggaaccttt , tttggaacctttt ) of words in I gives the word ttttggaacctttt , and r applied to the ordered pair of words in I gives tttggaaccttt . The less interesting actions of r on I must be recognized: When r acts on ordered pairs in the 'diagonal' of I × I , for example on the result is merely ttttggaaccttt which appeared as each coordinate of the pair… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{′ = gga and u ′ = v = acc . Let R = \{ r \} . This gives the splicing scheme Let Observe that r applied to the ordered pair ( ttttggaaccttt , tttggaacctttt ) of words in I gives the word ttttggaacctttt , and r applied to th…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=444357 \begin{verbatim} ′ = gga and u ′ = v = acc . Let R = { r } . This gives the splicing scheme Let Observe that r applied to the ordered pair ( ttttggaaccttt , tttggaacctttt ) of words in I gives the word ttttggaacctttt , and r applied to the ordered pair of words in I gives tttggaaccttt . The less interesting actions of r on I must be recognized: When r acts on ordered pairs in the 'diagonal' of I × I , for example on the result is merely ttttggaaccttt which appeared as each coordinate of the pair… \end{verbatim} ``` </details>
441. ph-f40c60795bea16ad0521 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) ot-decoded --> Observe that r applied to the ordered pair ( ttttggaaccttt , tttggaacctttt ) of words in I gives the word ttttggaacctttt , and r applied to the ordered pair of words in I gives tttggaaccttt . The less interesting actions of r on I must be recognized: When r acts on ordered pairs in the 'diagonal' of I × I , for example on the result is merely ttttggaaccttt which appeared as each coordinate of the pair. Here we have - = I ∪ { ttttggaacctttt , tttggaaccttt , ttttggaaccttt , tttggaacctttt } and … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ot-decoded --> Observe that r applied to the ordered pair ( ttttggaaccttt , tttggaacctttt ) of words in I gives the word ttttggaacctttt , and r applied to the ordered pair of words in I gives tttggaaccttt . The less interesting actions of r on I must…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=444483 \begin{verbatim} ot-decoded --> Observe that r applied to the ordered pair ( ttttggaaccttt , tttggaacctttt ) of words in I gives the word ttttggaacctttt , and r applied to the ordered pair of words in I gives tttggaaccttt . The less interesting actions of r on I must be recognized: When r acts on ordered pairs in the 'diagonal' of I × I , for example on the result is merely ttttggaaccttt which appeared as each coordinate of the pair. Here we have - = I ∪ { ttttggaacctttt , tttggaaccttt , ttttggaaccttt , tttggaacctttt } and … \end{verbatim} ``` </details>
442. ph-0182e294afca8d348fd6 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) ). Then also σ 3 ( I )) = σ 2 ( I ) = σ 1 ( I ) and in fact σ ∗ ( I ) = σ 1 ( I ). Thus L ( σ, I ) is the finite language ``` σ ∗ ( I ) = { ttttggaacctttt , tttggaaccttt , ttttggaaccttt , tttggaacctttt } . ``` This example connects the formal definitions of splicing systems and languages with the less formal introductory remarks of Sections 7.1 and 7.2. Example 7.2 Let /Sigma1 = { a , c , g , t } . Let r = ( c , cccgg , c , cccgg ), R = { r } , and let I contain only one word of length 30, The rule can be applied to the ordered pair wi… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{). Then also σ 3 ( I )) = σ 2 ( I ) = σ 1 ( I ) and in fact σ ∗ ( I ) = σ 1 ( I ). Thus L ( σ, I ) is the finite language ``` σ ∗ ( I ) = \{ ttttggaacctttt , tttggaaccttt , ttttggaaccttt , tttggaacctttt \} . ``` This example connects the formal definitions of splicing systems and …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=445287 \begin{verbatim} ). Then also σ 3 ( I )) = σ 2 ( I ) = σ 1 ( I ) and in fact σ ∗ ( I ) = σ 1 ( I ). Thus L ( σ, I ) is the finite language ``` σ ∗ ( I ) = { ttttggaacctttt , tttggaaccttt , ttttggaaccttt , tttggaacctttt } . ``` This example connects the formal definitions of splicing systems and languages with the less formal introductory remarks of Sections 7.1 and 7.2. Example 7.2 Let /Sigma1 = { a , c , g , t } . Let r = ( c , cccgg , c , cccgg ), R = { r } , and let I contain only one word of length 30, The rule can be applied to the ordered pair wi… \end{verbatim} ``` </details>
443. ph-d7de3e36867b7c71117d — automata/docling_md/AutomataTheory.md ### Plain (markdown context) 1 ( I ). Thus L ( σ, I ) is the finite language ``` σ ∗ ( I ) = { ttttggaacctttt , tttggaaccttt , ttttggaaccttt , tttggaacctttt } . ``` This example connects the formal definitions of splicing systems and languages with the less formal introductory remarks of Sections 7.1 and 7.2. Example 7.2 Let /Sigma1 = { a , c , g , t } . Let r = ( c , cccgg , c , cccgg ), R = { r } , and let I contain only one word of length 30, The rule can be applied to the ordered pair with cuts made using the right occurrence of ccccgg in the first coordinate … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{1 ( I ). Thus L ( σ, I ) is the finite language ``` σ ∗ ( I ) = \{ ttttggaacctttt , tttggaaccttt , ttttggaaccttt , tttggaacctttt \} . ``` This example connects the formal definitions of splicing systems and languages with the less formal introductory remarks of Sections 7.1 and 7.…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=445362 \begin{verbatim} 1 ( I ). Thus L ( σ, I ) is the finite language ``` σ ∗ ( I ) = { ttttggaacctttt , tttggaaccttt , ttttggaaccttt , tttggaacctttt } . ``` This example connects the formal definitions of splicing systems and languages with the less formal introductory remarks of Sections 7.1 and 7.2. Example 7.2 Let /Sigma1 = { a , c , g , t } . Let r = ( c , cccgg , c , cccgg ), R = { r } , and let I contain only one word of length 30, The rule can be applied to the ordered pair with cuts made using the right occurrence of ccccgg in the first coordinate … \end{verbatim} ``` </details>
444. ph-3507a04e5bd66c662d5a — automata/docling_md/AutomataTheory.md ### Plain (markdown context) and languages with the less formal introductory remarks of Sections 7.1 and 7.2. Example 7.2 Let /Sigma1 = { a , c , g , t } . Let r = ( c , cccgg , c , cccgg ), R = { r } , and let I contain only one word of length 30, The rule can be applied to the ordered pair with cuts made using the right occurrence of ccccgg in the first coordinate and the left occurrence of ccccgg in the second coordinate. This gives the word of length 42: The rule can be also applied to the ordered pair using the left occurrence of … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{and languages with the less formal introductory remarks of Sections 7.1 and 7.2. Example 7.2 Let /Sigma1 = \{ a , c , g , t \} . Let r = ( c , cccgg , c , cccgg ), R = \{ r \} , and let I contain only one word of length 30, The rule can be applied to the…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=445564 \begin{verbatim} and languages with the less formal introductory remarks of Sections 7.1 and 7.2. Example 7.2 Let /Sigma1 = { a , c , g , t } . Let r = ( c , cccgg , c , cccgg ), R = { r } , and let I contain only one word of length 30, The rule can be applied to the ordered pair with cuts made using the right occurrence of ccccgg in the first coordinate and the left occurrence of ccccgg in the second coordinate. This gives the word of length 42: The rule can be also applied to the ordered pair using the left occurrence of … \end{verbatim} ``` </details>
445. ph-b32fc25a25be1875e35a — automata/docling_md/AutomataTheory.md ### Plain (markdown context) rule can be applied to the ordered pair with cuts made using the right occurrence of ccccgg in the first coordinate and the left occurrence of ccccgg in the second coordinate. This gives the word of length 42: The rule can be also applied to the ordered pair using the left occurrence of ccccgg in the first coordinate and the right occurrence of ccccgg in the second coordinate. This gives the word of length 18, a 6 ccccgga 6 . Thus Continuing with similar considerations one finds that L ( σ, I ) = σ ∗ ( I ) … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{rule can be applied to the ordered pair with cuts made using the right occurrence of ccccgg in the first coordinate and the left occurrence of ccccgg in the second coordinate. This gives the word of length 42: The rule ca…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=445820 \begin{verbatim} rule can be applied to the ordered pair with cuts made using the right occurrence of ccccgg in the first coordinate and the left occurrence of ccccgg in the second coordinate. This gives the word of length 42: The rule can be also applied to the ordered pair using the left occurrence of ccccgg in the first coordinate and the right occurrence of ccccgg in the second coordinate. This gives the word of length 18, a 6 ccccgga 6 . Thus Continuing with similar considerations one finds that L ( σ, I ) = σ ∗ ( I ) … \end{verbatim} ``` </details>
446. ph-9d19ae9b0d708c7cef47 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) dinate and the left occurrence of ccccgg in the second coordinate. This gives the word of length 42: The rule can be also applied to the ordered pair using the left occurrence of ccccgg in the first coordinate and the right occurrence of ccccgg in the second coordinate. This gives the word of length 18, a 6 ccccgga 6 . Thus Continuing with similar considerations one finds that L ( σ, I ) = σ ∗ ( I ) is the infinite regular language Example 7.3 We may interpret the 30 symbol word given in Example 7.2 as a mo… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{dinate and the left occurrence of ccccgg in the second coordinate. This gives the word of length 42: The rule can be also applied to the ordered pair using the left occurrence of ccccgg in the first coordinate and the right occurrence of ccccgg in th…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=445961 \begin{verbatim} dinate and the left occurrence of ccccgg in the second coordinate. This gives the word of length 42: The rule can be also applied to the ordered pair using the left occurrence of ccccgg in the first coordinate and the right occurrence of ccccgg in the second coordinate. This gives the word of length 18, a 6 ccccgga 6 . Thus Continuing with similar considerations one finds that L ( σ, I ) = σ ∗ ( I ) is the infinite regular language Example 7.3 We may interpret the 30 symbol word given in Example 7.2 as a mo… \end{verbatim} ``` </details>
447. ph-65346a175edc021070c1 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) d of length 18, a 6 ccccgga 6 . Thus Continuing with similar considerations one finds that L ( σ, I ) = σ ∗ ( I ) is the infinite regular language Example 7.3 We may interpret the 30 symbol word given in Example 7.2 as a model of a dsDNA molecule as indicated in Section 7.3. The rule r of Example 7.2 represents the cut and paste activity of the restriction enzyme BsaJ I accompanied by a ligase. With these understandings the language obtained in Example 7.2 is a model of the set of all dsDNA molecules (havin… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{d of length 18, a 6 ccccgga 6 . Thus Continuing with similar considerations one finds that L ( σ, I ) = σ ∗ ( I ) is the infinite regular language Example 7.3 We may interpret the 30 symbol word given in Example 7.2 as a …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=446282 \begin{verbatim} d of length 18, a 6 ccccgga 6 . Thus Continuing with similar considerations one finds that L ( σ, I ) = σ ∗ ( I ) is the infinite regular language Example 7.3 We may interpret the 30 symbol word given in Example 7.2 as a model of a dsDNA molecule as indicated in Section 7.3. The rule r of Example 7.2 represents the cut and paste activity of the restriction enzyme BsaJ I accompanied by a ligase. With these understandings the language obtained in Example 7.2 is a model of the set of all dsDNA molecules (havin… \end{verbatim} ``` </details>
448. ph-b866e55ec96c8316d3fb — automata/docling_md/AutomataTheory.md ### Plain (markdown context) ( cabaab , cabaaab ), and the analyses caba / ab and cab / aaab , the rule r gives the word cabaaaab . (Note that r provides a form of pumping.) Continuing in this way all words caba n b with n ≥ 3 can be obtained. Since caba n b with 0 ≤ n ≤ 2 were given in I , we have, for R = { r } and σ = ( /Sigma1 , R ), L ( σ, I ) = caba ∗ b as desired. In fact it has been shown [ 12 ] that for any regular language L ′ over any alphabet /Sigma1 , by choosing a symbol not in /Sigma1 , say c , the language is generated by a splicing system that can be specified very much as we… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{( cabaab , cabaaab ), and the analyses caba / ab and cab / aaab , the rule r gives the word cabaaaab . (Note that r provides a form of pumping.) Continuing in this way all words caba n b with n ≥ 3 can be obtained. Since caba n b with 0 ≤ n ≤ 2 were given in I , we have, for R =…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=447810 \begin{verbatim} ( cabaab , cabaaab ), and the analyses caba / ab and cab / aaab , the rule r gives the word cabaaaab . (Note that r provides a form of pumping.) Continuing in this way all words caba n b with n ≥ 3 can be obtained. Since caba n b with 0 ≤ n ≤ 2 were given in I , we have, for R = { r } and σ = ( /Sigma1 , R ), L ( σ, I ) = caba ∗ b as desired. In fact it has been shown [ 12 ] that for any regular language L ′ over any alphabet /Sigma1 , by choosing a symbol not in /Sigma1 , say c , the language is generated by a splicing system that can be specified very much as we… \end{verbatim} ``` </details>
449. ph-5818475dbf18bbd1161b — automata/docling_md/AutomataTheory.md ### Plain (markdown context) ormally speaking, each regular language is almost a splicing language. Example 7.5 Let /Sigma1 = { a , b } . The regular language L = ( aa ) ∗ cannot be generated by a splicing system. As the reader may verify, any finite set of rules that allows every word in L to be generated will also generate strings of odd length as well as the strings of even length. Example 7.6 The regular language L ′ = a ∗ ba ∗ ba ∗ cannot be generated by a finite set of rules either: For any nonnegative integer n , generate a ∗ ba n ba ∗ . Consequently, for a… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ormally speaking, each regular language is almost a splicing language. Example 7.5 Let /Sigma1 = \{ a , b \} . The regular language L = ( aa ) ∗ cannot be generated by a splicing system. As the reader may verify, any finite set of rules that allows every word in L to be generated …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=448473 \begin{verbatim} ormally speaking, each regular language is almost a splicing language. Example 7.5 Let /Sigma1 = { a , b } . The regular language L = ( aa ) ∗ cannot be generated by a splicing system. As the reader may verify, any finite set of rules that allows every word in L to be generated will also generate strings of odd length as well as the strings of even length. Example 7.6 The regular language L ′ = a ∗ ba ∗ ba ∗ cannot be generated by a finite set of rules either: For any nonnegative integer n , generate a ∗ ba n ba ∗ . Consequently, for a… \end{verbatim} ``` </details>
450. ph-154f6956a97c8170e343 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) language is almost a splicing language. Example 7.5 Let /Sigma1 = { a , b } . The regular language L = ( aa ) ∗ cannot be generated by a splicing system. As the reader may verify, any finite set of rules that allows every word in L to be generated will also generate strings of odd length as well as the strings of even length. Example 7.6 The regular language L ′ = a ∗ ba ∗ ba ∗ cannot be generated by a finite set of rules either: For any nonnegative integer n , generate a ∗ ba n ba ∗ . Consequently, for any finite subset F of nonnegati… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{language is almost a splicing language. Example 7.5 Let /Sigma1 = \{ a , b \} . The regular language L = ( aa ) ∗ cannot be generated by a splicing system. As the reader may verify, any finite set of rules that allows every word in L to be generated will also generate strings of o…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=448503 \begin{verbatim} language is almost a splicing language. Example 7.5 Let /Sigma1 = { a , b } . The regular language L = ( aa ) ∗ cannot be generated by a splicing system. As the reader may verify, any finite set of rules that allows every word in L to be generated will also generate strings of odd length as well as the strings of even length. Example 7.6 The regular language L ′ = a ∗ ba ∗ ba ∗ cannot be generated by a finite set of rules either: For any nonnegative integer n , generate a ∗ ba n ba ∗ . Consequently, for any finite subset F of nonnegati… \end{verbatim} ``` </details>
451. ph-6a5d42d05b37c6ca3609 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) the [ n ( L )] 4 quadruples of syntactic classes determined in /Sigma1 ∗ by L and, from each such quadruple ( W , X , Y , Z ), we choose one word from each class to obtain one rule ( w, x , y , z ) and then decide whether it respects L . If it does then every rule in W × X × Y × Z respects L . If it does not respect L then no rule in W × X × Y × Z respects L . This discussion has justified the following: Proposition 7.1 Let L be a regular language. The set of rules that respect L has the form where m is a nonnegative integer and each of the sets where m is a nonnegative integer and each of the sets where m is a nonnegative integer and each of the sets is an element of the syntactic monoid of L. Since each syntactic class of … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{om each such quadruple ( W , X , Y , Z ), we choose one word from each class to obtain one rule ( w, x , y , z ) and then decide whether it respects L . If it does then every rule in W × X × Y × Z respects L . If it does not respect L then no rule in W × X × Y × Z respects L . T…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=455822 \begin{verbatim} om each such quadruple ( W , X , Y , Z ), we choose one word from each class to obtain one rule ( w, x , y , z ) and then decide whether it respects L . If it does then every rule in W × X × Y × Z respects L . If it does not respect L then no rule in W × X × Y × Z respects L . This discussion has justified the following: Proposition 7.1 Let L be a regular language. The set of rules that respect L has the form where m is a nonnegative integer and each of the sets is an element of the syntactic monoid of L. Since each syntactic class of … \end{verbatim} ``` </details>
453. ph-8b5b90c5c27d58e7157f — automata/docling_md/AutomataTheory.md ### Plain (markdown context) - formula-not-decoded --> where m is a nonnegative integer and each of the sets is an element of the syntactic monoid of L. Since each syntactic class of a regular language L is itself a regular language, one can list all the strings of length at most k in the class. Consequently when the representation in the proposition has been constructed, the set of all rules of radius at most k that preserve L can be listed with no additional testing: For each of the sets in the representation, list all of the rules ( w, x , y , z ) in where m is a nonnegative integer and each of the sets is an element of the syntactic monoid of L. Since each syntactic class of a regular language L is itself a regular language, one can list all the strings of length at mos…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=456240 \begin{verbatim} - formula-not-decoded --> where m is a nonnegative integer and each of the sets is an element of the syntactic monoid of L. Since each syntactic class of a regular language L is itself a regular language, one can list all the strings of length at most k in the class. Consequently when the representation in the proposition has been constructed, the set of all rules of radius at most k that preserve L can be listed with no additional testing: For each of the sets in the representation, list all of the rules ( w, x , y , z ) in is an element of the syntactic monoid of L. Since each syntactic class of a regular language L is itself a regular language, one can list all the strings of length at most k in the class. Consequently when the representation in the proposition has been constructed, the set of all rules of radius at most k that preserve L can be listed with no additional testing: For each of the sets in the representation, list all of the rules ( w, x , y , z ) in of radius at most k . In order to create such a list without using the synt… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ot-decoded --> is an element of the syntactic monoid of L. Since each syntactic class of a regular language L is itself a regular language, one can list all the strings of length at most k in the class. Consequently when the representation in the proposition has been constructed…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=456336 \begin{verbatim} ot-decoded --> is an element of the syntactic monoid of L. Since each syntactic class of a regular language L is itself a regular language, one can list all the strings of length at most k in the class. Consequently when the representation in the proposition has been constructed, the set of all rules of radius at most k that preserve L can be listed with no additional testing: For each of the sets in the representation, list all of the rules ( w, x , y , z ) in of radius at most k . In order to create such a list without using the synt… \end{verbatim} ``` </details>
455. ph-633ad96a30e634d1bf5d — automata/docling_md/AutomataTheory.md ### Plain (markdown context) ) are also in R . When R is reflexive we say the same of any scheme or system having R as its rule set. In fact, splicing systems that model the cut and paste action of restriction enzymes and a ligase are necessarily reflexive. Consequently, from a modeling perspective, it is the reflexive splicing systems that are of prime interest. Section 7.6 provides the tools to construct, for each regular language L and each positive integer k , the following finite reflexive set Tk of splicing rules: Recall that Tk ( L ) = ∪{ r ( L ) : r ∈ Tk } , which is regular since Tk … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{) are also in R . When R is reflexive we say the same of any scheme or system having R as its rule set. In fact, splicing systems that model the cut and paste action of restriction enzymes and a ligase are necessarily reflexive. Consequently, from a modeling perspective, it is t…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=457615 \begin{verbatim} ) are also in R . When R is reflexive we say the same of any scheme or system having R as its rule set. In fact, splicing systems that model the cut and paste action of restriction enzymes and a ligase are necessarily reflexive. Consequently, from a modeling perspective, it is the reflexive splicing systems that are of prime interest. Section 7.6 provides the tools to construct, for each regular language L and each positive integer k , the following finite reflexive set Tk of splicing rules: Recall that Tk ( L ) = ∪{ r ( L ) : r ∈ Tk } , which is regular since Tk … \end{verbatim} ``` </details>
456. ph-01d20c887b991d3d3778 — automata/docling_md/AutomataTheory.md ### Plain (markdown context) ss, there is no f -1 ( t i ). Let S 1 be the elements of S for which the first result occurs. Let S 2 be the elements of S for which the second result occurs. Let S 3 be the elements of S for which the third result occurs. Obviously these sets are disjoint. Similarly form T 1 , T 2, and T 3 as subsets of T . f is a one-to-one correspondence from S 1 to T 1 . f is also a one-to-one correspondence from S 2 to T 2 . g -1 is a one-to-one correspondence from S 3 to T 3 . Let θ : S → T be defined by θ is a one to one correspondence from S to T . Theorem A.2 For any set … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ss, there is no f -1 ( t i ). Let S 1 be the elements of S for which the first result occurs. Let S 2 be the elements of S for which the second result occurs. Let S 3 be the elements of S for which the third result occurs. Obviously these sets are disjoint. Similarly form T 1 , …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory.md:offset=462088 \begin{verbatim} ss, there is no f -1 ( t i ). Let S 1 be the elements of S for which the first result occurs. Let S 2 be the elements of S for which the second result occurs. Let S 3 be the elements of S for which the third result occurs. Obviously these sets are disjoint. Similarly form T 1 , T 2, and T 3 as subsets of T . f is a one-to-one correspondence from S 1 to T 1 . f is also a one-to-one correspondence from S 2 to T 2 . g -1 is a one-to-one correspondence from S 3 to T 3 . Let θ : S → T be defined by θ is a one to one correspondence from S to T . Theorem A.2 For any set … \end{verbatim} ``` </details>
457. ph-8942098b1d21a32a371a — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md ### Plain (markdown context) s is the set of all strings of symbols in a given alphabet. Definition 1.8 Let A be a set. A ′ = U -A is the set of all elements not in A. Example 1.3 Let A be the set of even integers and U be the set of integers. Then A ′ is the set of odd integers. Example 1.4 Let A = { x : x collects coins } , then A ′ = { x : x does not collect coins } . The proof of the following theorem is left to the reader. Theorem 1.1 Let A, B, and C be subsets of the universal set U (a) Distributive properties - (b) Idempotent properties - (d) De Morgan's la… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{s is the set of all strings of symbols in a given alphabet. Definition 1.8 Let A be a set. A ′ = U -A is the set of all elements not in A. Example 1.3 Let A be the set of even integers and U be the set of integers. Then A ′ is the set of odd integers. Example 1.4 Let A = \{ x : x…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md:offset=19087 \begin{verbatim} s is the set of all strings of symbols in a given alphabet. Definition 1.8 Let A be a set. A ′ = U -A is the set of all elements not in A. Example 1.3 Let A be the set of even integers and U be the set of integers. Then A ′ is the set of odd integers. Example 1.4 Let A = { x : x collects coins } , then A ′ = { x : x does not collect coins } . The proof of the following theorem is left to the reader. Theorem 1.1 Let A, B, and C be subsets of the universal set U (a) Distributive properties - (b) Idempotent properties - (d) De Morgan's la… \end{verbatim} ``` </details>
458. ph-dc307b989c6ec27d67d4 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md ### Plain (markdown context) symbols in a given alphabet. Definition 1.8 Let A be a set. A ′ = U -A is the set of all elements not in A. Example 1.3 Let A be the set of even integers and U be the set of integers. Then A ′ is the set of odd integers. Example 1.4 Let A = { x : x collects coins } , then A ′ = { x : x does not collect coins } . The proof of the following theorem is left to the reader. Theorem 1.1 Let A, B, and C be subsets of the universal set U (a) Distributive properties - (b) Idempotent properties - (d) De Morgan's laws - (e) Commutative properties… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{symbols in a given alphabet. Definition 1.8 Let A be a set. A ′ = U -A is the set of all elements not in A. Example 1.3 Let A be the set of even integers and U be the set of integers. Then A ′ is the set of odd integers. Example 1.4 Let A = \{ x : x collects coins \} , then A ′ = …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md:offset=19117 \begin{verbatim} symbols in a given alphabet. Definition 1.8 Let A be a set. A ′ = U -A is the set of all elements not in A. Example 1.3 Let A be the set of even integers and U be the set of integers. Then A ′ is the set of odd integers. Example 1.4 Let A = { x : x collects coins } , then A ′ = { x : x does not collect coins } . The proof of the following theorem is left to the reader. Theorem 1.1 Let A, B, and C be subsets of the universal set U (a) Distributive properties - (b) Idempotent properties - (d) De Morgan's laws - (e) Commutative properties… \end{verbatim} ``` </details>
459. ph-9dfaa425cccd95c38839 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md ### Plain (markdown context) Example 1.3 Let A be the set of even integers and U be the set of integers. Then A ′ is the set of odd integers. Example 1.4 Let A = { x : x collects coins } , then A ′ = { x : x does not collect coins } . The proof of the following theorem is left to the reader. Theorem 1.1 Let A, B, and C be subsets of the universal set U (a) Distributive properties - (b) Idempotent properties - (d) De Morgan's laws - (e) Commutative properties - (f) Associative laws - (g) Identity properties … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{Example 1.3 Let A be the set of even integers and U be the set of integers. Then A ′ is the set of odd integers. Example 1.4 Let A = \{ x : x collects coins \} , then A ′ = \{ x : x does not collect coins \} . The proof of the following theorem is left to the reader. Theorem 1.1 Let…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md:offset=19228 \begin{verbatim} Example 1.3 Let A be the set of even integers and U be the set of integers. Then A ′ is the set of odd integers. Example 1.4 Let A = { x : x collects coins } , then A ′ = { x : x does not collect coins } . The proof of the following theorem is left to the reader. Theorem 1.1 Let A, B, and C be subsets of the universal set U (a) Distributive properties - (b) Idempotent properties - (d) De Morgan's laws - (e) Commutative properties - (f) Associative laws - (g) Identity properties … \end{verbatim} ``` </details>
460. ph-d097b82a0b499e2b60d1 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md ### Plain (markdown context) the set of integers. Then A ′ is the set of odd integers. Example 1.4 Let A = { x : x collects coins } , then A ′ = { x : x does not collect coins } . The proof of the following theorem is left to the reader. Theorem 1.1 Let A, B, and C be subsets of the universal set U (a) Distributive properties - (b) Idempotent properties - (d) De Morgan's laws - (e) Commutative properties - (f) Associative laws - (g) Identity properties - (h) Complement propertie… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{the set of integers. Then A ′ is the set of odd integers. Example 1.4 Let A = \{ x : x collects coins \} , then A ′ = \{ x : x does not collect coins \} . The proof of the following theorem is left to the reader. Theorem 1.1 Let A, B, and C be subsets of the universal set U (a) Dist…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md:offset=19282 \begin{verbatim} the set of integers. Then A ′ is the set of odd integers. Example 1.4 Let A = { x : x collects coins } , then A ′ = { x : x does not collect coins } . The proof of the following theorem is left to the reader. Theorem 1.1 Let A, B, and C be subsets of the universal set U (a) Distributive properties - (b) Idempotent properties - (d) De Morgan's laws - (e) Commutative properties - (f) Associative laws - (g) Identity properties - (h) Complement propertie… \end{verbatim} ``` </details>
461. ph-43e3a2154ab3d9eb063b — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md ### Plain (markdown context) . Example 1.4 Let A = { x : x collects coins } , then A ′ = { x : x does not collect coins } . The proof of the following theorem is left to the reader. Theorem 1.1 Let A, B, and C be subsets of the universal set U (a) Distributive properties - (b) Idempotent properties - (d) De Morgan's laws - (e) Commutative properties - (f) Associative laws - (g) Identity properties - (h) Complement properties - (c) Double Complement p… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{. Example 1.4 Let A = \{ x : x collects coins \} , then A ′ = \{ x : x does not collect coins \} . The proof of the following theorem is left to the reader. Theorem 1.1 Let A, B, and C be subsets of the universal set U (a) Distributive properties - (b) Idempotent properties - (d) De Morgan's laws - (e) Commutative properties - (f) Associative laws - (g) Identity properties - (h) Complement properties - (c) Double Complement p… \end{verbatim} ```
462. ph-ddd28a86b0003a265a86 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md ### Plain (markdown context) = { x : x does not collect coins } . The proof of the following theorem is left to the reader. Theorem 1.1 Let A, B, and C be subsets of the universal set U (a) Distributive properties - (b) Idempotent properties - (d) De Morgan's laws - (e) Commutative properties - (f) Associative laws - (g) Identity properties - (h) Complement properties - (c) Double Complement property - (b) Idempotent properties - (d) De…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md:offset=19398 \begin{verbatim} = { x : x does not collect coins } . The proof of the following theorem is left to the reader. Theorem 1.1 Let A, B, and C be subsets of the universal set U (a) Distributive properties - (b) Idempotent properties - (d) De Morgan's laws - (e) Commutative properties - (f) Associative laws - (g) Identity properties - (h) Complement properties - (c) Double Complement property - (b) Idempotent properties - (d) De Morgan's laws - (e) Commutative properties - (f) Associative laws - (g) Identity properties - (h) Complement properties - (c) Double Complement property Definition 1.9 The size or … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{theorem is left to the reader. Theorem 1.1 Let A, B, and C be subsets of the universal set U (a) Distributive properties - (b) Idempotent properties - (d) De Morgan's laws - (e) Commutative properties - (b) Idempotent properties - (d) De Morgan's laws - (e) Commutative properties - (f) Associative laws - (g) Identity properties - (h) Complement properties - (c) Double Complement property Definition 1.9 The size or … \end{verbatim} ``` </details>
464. ph-637f14d792d2fd5b77c6 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md ### Plain (markdown context) . Theorem 1.1 Let A, B, and C be subsets of the universal set U (a) Distributive properties - (b) Idempotent properties - (d) De Morgan's laws - (e) Commutative properties - (f) Associative laws - (g) Identity properties - (h) Complement properties - (c) Double Complement property Definition 1.9 The size or cardinality of a finite set A… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{. Theorem 1.1 Let A, B, and C be subsets of the universal set U (a) Distributive properties - (b) Idempotent properties - (d) De Morgan's laws - (e) Commutative properties - (f) Associative la…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md:offset=19492 \begin{verbatim} . Theorem 1.1 Let A, B, and C be subsets of the universal set U (a) Distributive properties - (b) Idempotent properties - (d) De Morgan's laws - (e) Commutative properties - (f) Associative laws - (g) Identity properties - (h) Complement properties - (c) Double Complement property Definition 1.9 The size or cardinality of a finite set A… \end{verbatim} ``` </details>
465. ph-1f8f281c718615da04d4 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md ### Plain (markdown context) be subsets of the universal set U (a) Distributive properties - (b) Idempotent properties - (d) De Morgan's laws - (e) Commutative properties - (f) Associative laws - (g) Identity properties - (h) Complement properties - (c) Double Complement property Definition 1.9 The size or cardinality of a finite set A, denoted by | A | , is the nu… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{be subsets of the universal set U (a) Distributive properties - (b) Idempotent properties - (d) De Morgan's laws - (e) Commutative properties - (f) Associative laws - (b) Idempotent properties - (d) De Morgan's laws - (e) Commutative properties - (f) Associative laws - (g) Identity properties - (h) Complement properties - (c) Double Complement property Definition 1.9 The size or cardinality of a finite set A, denoted by | A | , is the nu… \end{verbatim} ```
466. ph-fd1886c6fdcba11e4939 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md ### Plain (markdown context) countable set. We see that there are two infinite sets, the countable sets and the uncountable sets with different cardinality; however, we shall soon see that there are an infinite number of infinite sets of different cardinality. Further discussion of cardinality will be continued in the appendices. Definition 1.10 Let A and B be sets. The Cartesian product of A and B, denoted by A × B is the set { ( a , b ) : a ∈ A and b ∈ B } . For example, let A = { a , b } and B = { 1 , 2 , 3 } , then ThefamiliarCartesianplane R × R is the set of all ordered pairs of real nu… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{countable set. We see that there are two infinite sets, the countable sets and the uncountable sets with different cardinality; however, we shall soon see that there are an infinite number of infinite sets of different cardinality. Further discussion of cardinality will be conti…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md:offset=21179 \begin{verbatim} countable set. We see that there are two infinite sets, the countable sets and the uncountable sets with different cardinality; however, we shall soon see that there are an infinite number of infinite sets of different cardinality. Further discussion of cardinality will be continued in the appendices. Definition 1.10 Let A and B be sets. The Cartesian product of A and B, denoted by A × B is the set { ( a , b ) : a ∈ A and b ∈ B } . For example, let A = { a , b } and B = { 1 , 2 , 3 } , then ThefamiliarCartesianplane R × R is the set of all ordered pairs of real nu… \end{verbatim} ``` </details>
467. ph-d38b22a4bc0e37ea8286 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md ### Plain (markdown context) nition 1.10 Let A and B be sets. The Cartesian product of A and B, denoted by A × B is the set { ( a , b ) : a ∈ A and b ∈ B } . For example, let A = { a , b } and B = { 1 , 2 , 3 } , then ThefamiliarCartesianplane R × R is the set of all ordered pairs of real numbers. Note that for finite sets | A × B | = | A | × | B | . Definition 1.11 The power set of a set A, denoted by P ( A ) , is the set of all subsets of A. For example the power set of { a , b , c } is In the finite case, it can be easily shown that | P ( A ) | = 2 | A | . ## E… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{nition 1.10 Let A and B be sets. The Cartesian product of A and B, denoted by A × B is the set \{ ( a , b ) : a ∈ A and b ∈ B \} . For example, let A = \{ a , b \} and B = \{ 1 , 2 , 3 \} , then ThefamiliarCartesianplane R × R is the set of all ordered pai…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md:offset=21488 \begin{verbatim} nition 1.10 Let A and B be sets. The Cartesian product of A and B, denoted by A × B is the set { ( a , b ) : a ∈ A and b ∈ B } . For example, let A = { a , b } and B = { 1 , 2 , 3 } , then ThefamiliarCartesianplane R × R is the set of all ordered pairs of real numbers. Note that for finite sets | A × B | = | A | × | B | . Definition 1.11 The power set of a set A, denoted by P ( A ) , is the set of all subsets of A. For example the power set of { a , b , c } is In the finite case, it can be easily shown that | P ( A ) | = 2 | A | . ## E… \end{verbatim} ``` </details>
468. ph-a6013116be47a7702beb — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md ### Plain (markdown context) , b , c } is In the finite case, it can be easily shown that | P ( A ) | = 2 | A | . ## Exercises - (1) State which of the following are true and which are false: 2. (a) {∅} ⊆ A for an arbitrary set A 3. (c) { a , b , c } ⊆ { a , b , { a , b , c }} . 4. (b) ∅ ⊆ A for an arbitrary set A . 5. (d) { a , b , c } ∈ { a , b , { a , b , c }} . - (2) Prove Theorem 1.1. Let A , B , and C be subsets of the universal set U . 7. (e) A ∈ P ( A ). 8. (a) Idempotent property - (b) Double Complement property In the finite case, it can be easily shown that | P ( A ) | = 2 | A | . \#\# Exercises - (1) State which of the following are true and which are false: 2. (a) \{∅\} ⊆ A for an arbitrary set A 3. (c) \{ a , b , c \} ⊆ \{ a , b , \{ a , b , c \}\} .…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md:offset=21973 \begin{verbatim} , b , c } is In the finite case, it can be easily shown that | P ( A ) | = 2 | A | . ## Exercises - (1) State which of the following are true and which are false: 2. (a) {∅} ⊆ A for an arbitrary set A 3. (c) { a , b , c } ⊆ { a , b , { a , b , c }} . 4. (b) ∅ ⊆ A for an arbitrary set A . 5. (d) { a , b , c } ∈ { a , b , { a , b , c }} . - (2) Prove Theorem 1.1. Let A , B , and C be subsets of the universal set U . 7. (e) A ∈ P ( A ). 8. (a) Idempotent property - (b) Double Complement property - (b) Double Complement property - . - (c) De Morgan's laws ## (d) Commutative p… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{it can be easily shown that | P ( A ) | = 2 | A | . \#\# Exercises - (1) State which of the following are true and which are false: 2. (a) \{∅\} ⊆ A for an arbitrary set A 3. (c) \{ a , b , c \} ⊆ \{ a , b , \{ a , b , c \}\} . 4. (b) ∅ ⊆ A for an arbitrary set A . 5. (d) \{ a , b , c \} ∈ …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md:offset=22037 \begin{verbatim} it can be easily shown that | P ( A ) | = 2 | A | . ## Exercises - (1) State which of the following are true and which are false: 2. (a) {∅} ⊆ A for an arbitrary set A 3. (c) { a , b , c } ⊆ { a , b , { a , b , c }} . 4. (b) ∅ ⊆ A for an arbitrary set A . 5. (d) { a , b , c } ∈ { a , b , { a , b , c }} . - (2) Prove Theorem 1.1. Let A , B , and C be subsets of the universal set U . 7. (e) A ∈ P ( A ). 8. (a) Idempotent property - (b) Double Complement property - . - (c) De Morgan's laws ## (d) Commutative p… \end{verbatim} ``` </details>
470. ph-f49d5c76b1cf4ef776c9 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md ### Plain (markdown context) P ( A ) | = 2 | A | . ## Exercises - (1) State which of the following are true and which are false: 2. (a) {∅} ⊆ A for an arbitrary set A 3. (c) { a , b , c } ⊆ { a , b , { a , b , c }} . 4. (b) ∅ ⊆ A for an arbitrary set A . 5. (d) { a , b , c } ∈ { a , b , { a , b , c }} . - (2) Prove Theorem 1.1. Let A , B , and C be subsets of the universal set U . 7. (e) A ∈ P ( A ). 8. (a) Idempotent property - (b) Double Complement property - . - (c) De Morgan's laws ## (d) Commutative properties - (b) Double Complement property - . - (c) De Morgan's laws ## (d) Commutative properties - (b) Double Complement property - . - (c) De Morgan's laws ## (d) Commutative properties - (e) Associative properties - (f) Distributive … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ch are false: 2. (a) \{∅\} ⊆ A for an arbitrary set A 3. (c) \{ a , b , c \} ⊆ \{ a , b , \{ a , b , c \}\} . 4. (b) ∅ ⊆ A for an arbitrary set A . 5. (d) \{ a , b , c \} ∈ \{ a , b , \{ a , b , c \}\} . - (2) Prove Theorem 1.1. Let A , B , and C be subsets of the universal set U . 7. (e) A ∈…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md:offset=22156 \begin{verbatim} ch are false: 2. (a) {∅} ⊆ A for an arbitrary set A 3. (c) { a , b , c } ⊆ { a , b , { a , b , c }} . 4. (b) ∅ ⊆ A for an arbitrary set A . 5. (d) { a , b , c } ∈ { a , b , { a , b , c }} . - (2) Prove Theorem 1.1. Let A , B , and C be subsets of the universal set U . 7. (e) A ∈ P ( A ). 8. (a) Idempotent property - (b) Double Complement property - . - (c) De Morgan's laws ## (d) Commutative properties - (e) Associative properties - (f) Distributive … \end{verbatim} ``` </details>
472. ph-e3aa764cb8781362e61d — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md ### Plain (markdown context) a , b , c } ⊆ { a , b , { a , b , c }} . 4. (b) ∅ ⊆ A for an arbitrary set A . 5. (d) { a , b , c } ∈ { a , b , { a , b , c }} . - (2) Prove Theorem 1.1. Let A , B , and C be subsets of the universal set U . 7. (e) A ∈ P ( A ). 8. (a) Idempotent property - (b) Double Complement property - . - (c) De Morgan's laws ## (d) Commutative properties - (e) Associative properties - (f) Distributive properties - (b) Double Complement property - . - (c) De Morgan's laws ## (d) Commutative properties - (e) Associative properties - (f) Distributive properties - (b) Double Complement property - . - (c) De Morgan's laws ## (d) Commutative properties - (e) Associative properties - (f) Distributive properties - (g) Identity properties ## (h) Complement propertie… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{arbitrary set A . 5. (d) \{ a , b , c \} ∈ \{ a , b , \{ a , b , c \}\} . - (2) Prove Theorem 1.1. Let A , B , and C be subsets of the universal set U . 7. (e) A ∈ P ( A ). 8. (a) Idempotent property - (b) Double Complement property - (b) Double Complement property - . - (c) De Morgan's laws ## (d) Commutative properties - (e) Associative properties - (f) Distributive properties - (g) Identity properties ## (h) Complement propertie… \end{verbatim} ```
474. ph-6d2ec2b3383bab3617d9 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md ### Plain (markdown context) , b , c } ∈ { a , b , { a , b , c }} . - (2) Prove Theorem 1.1. Let A , B , and C be subsets of the universal set U . 7. (e) A ∈ P ( A ). 8. (a) Idempotent property - (b) Double Complement property - . - (c) De Morgan's laws ## (d) Commutative properties - (e) Associative properties - (f) Distributive properties - (g) Identity properties ## (h) Complement properties - (4) If A B , what is A /D… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{, b , c \} ∈ \{ a , b , \{ a , b , c \}\} . - (2) Prove Theorem 1.1. Let A , B , and C be subsets of the universal set U . 7. (e) A ∈ P ( A ). 8. (a) Idempotent property - (b) Double Complement property - (b) Double Complement property - . - (c) De Morgan's laws ## (d) Commutative properties - (e) Associative properties - (f) Distributive properties - (g) Identity properties ## (h) Complement properties - (4) If A B , what is A /D… \end{verbatim} ```
475. ph-97f1a1f7508fbaf6ff7c — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md ### Plain (markdown context) A ∈ P ( A ). 8. (a) Idempotent property - (b) Double Complement property - . - (c) De Morgan's laws ## (d) Commutative properties - (e) Associative properties - (f) Distributive properties - (g) Identity properties ## (h) Complement properties - (4) If A B , what is A /Delta1 B ? - (3) Given a set A ∈ P ( C ), find a set B such that A /… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{A ∈ P ( A ). 8. (a) Idempotent property - (b) Double Complement property - . - (c) De Morgan's laws \#\# (d) Commutative properties - (e) Associative properties - (b) Double Complement property - . - (c) De Morgan's laws ## (d) Commutative properties - (e) Associative properties - (f) Distributive properties - (g) Identity properties ## (h) Complement properties - (4) If A B , what is A /Delta1 B ? - (3) Given a set A ∈ P ( C ), find a set B such that A /… \end{verbatim} ```
476. ph-15de7b712ee1cdf38738 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md ### Plain (markdown context) property - (b) Double Complement property - . - (c) De Morgan's laws ## (d) Commutative properties - (e) Associative properties - (f) Distributive properties - (g) Identity properties ## (h) Complement properties - (4) If A B , what is A /Delta1 B ? - (3) Given a set A ∈ P ( C ), find a set B such that A /Delta1 B = ∅ . - (5) Using the … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{property - (b) Double Complement property - . - (c) De Morgan's laws \#\# (d) Commutative properties - (e) Associative properties - (f) …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md:offset=22462 \begin{verbatim} property - (b) Double Complement property - . - (c) De Morgan's laws ## (d) Commutative properties - (e) Associative properties - (f) Distributive properties - (g) Identity properties ## (h) Complement properties - (4) If A B , what is A /Delta1 B ? - (3) Given a set A ∈ P ( C ), find a set B such that A /Delta1 B = ∅ . - (5) Using the … \end{verbatim} ``` </details>
477. ph-8cc6577b5ff8116061b0 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md ### Plain (markdown context) -decoded --> - (3) Given a set A ∈ P ( C ), find a set B such that A /Delta1 B = ∅ . - (5) Using the properties in Theorem 1.1 prove that A ∩ ( B /Delta1 C ) = ( A ∩ B ) /Delta1 ( A ∩ C ) . 3. ⊆ - (6) Use induction to prove that for any finite set A , | A | < | P ( A ) | . 5. (8) 6. Prove using the properties in Theorem 1.1 - (7) (Russell's Paradox) Let S be the set of all sets. Then S ∈ S . Obviously ∅ / ∈ ∅ . Let W = { A : A / ∈ A } . Discuss whether W ∈ W . - (9) Use the fact that A ∩ ( A ∪ B ) = A to prove that A ∪ ( A ∩ B ) = A… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{-decoded --> - (3) Given a set A ∈ P ( C ), find a set B such that A /Delta1 B = ∅ . - (5) Using the properties in Theorem 1.1 prove that A ∩ ( B /Delta1 C ) = ( A ∩ B ) /Delta1 ( A ∩ C ) . 3. ⊆ - (6) Use induction to prove that for any finite set A …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md:offset=22948 \begin{verbatim} -decoded --> - (3) Given a set A ∈ P ( C ), find a set B such that A /Delta1 B = ∅ . - (5) Using the properties in Theorem 1.1 prove that A ∩ ( B /Delta1 C ) = ( A ∩ B ) /Delta1 ( A ∩ C ) . 3. ⊆ - (6) Use induction to prove that for any finite set A , | A | < | P ( A ) | . 5. (8) 6. Prove using the properties in Theorem 1.1 - (7) (Russell's Paradox) Let S be the set of all sets. Then S ∈ S . Obviously ∅ / ∈ ∅ . Let W = { A : A / ∈ A } . Discuss whether W ∈ W . - (9) Use the fact that A ∩ ( A ∪ B ) = A to prove that A ∪ ( A ∩ B ) = A… \end{verbatim} ``` </details>
478. ph-ddbe59e9637f5f159b34 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md ### Plain (markdown context) B = ∅ . - (5) Using the properties in Theorem 1.1 prove that A ∩ ( B /Delta1 C ) = ( A ∩ B ) /Delta1 ( A ∩ C ) . 3. ⊆ - (6) Use induction to prove that for any finite set A , | A | < | P ( A ) | . 5. (8) 6. Prove using the properties in Theorem 1.1 - (7) (Russell's Paradox) Let S be the set of all sets. Then S ∈ S . Obviously ∅ / ∈ ∅ . Let W = { A : A / ∈ A } . Discuss whether W ∈ W . - (9) Use the fact that A ∩ ( A ∪ B ) = A to prove that A ∪ ( A ∩ B ) = A . - (10) Prove that if two disjoint sets are countable, then their union is … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{B = ∅ . - (5) Using the properties in Theorem 1.1 prove that A ∩ ( B /Delta1 C ) = ( A ∩ B ) /Delta1 ( A ∩ C ) . 3. ⊆ - (6) Use induction to prove that for any finite set A , | A | \< | P ( A ) | . 5. (8) 6. Prove using the properties in Theorem 1.1 - (7) (Russell's Paradox) L…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md:offset=23055 \begin{verbatim} B = ∅ . - (5) Using the properties in Theorem 1.1 prove that A ∩ ( B /Delta1 C ) = ( A ∩ B ) /Delta1 ( A ∩ C ) . 3. ⊆ - (6) Use induction to prove that for any finite set A , | A | < | P ( A ) | . 5. (8) 6. Prove using the properties in Theorem 1.1 - (7) (Russell's Paradox) Let S be the set of all sets. Then S ∈ S . Obviously ∅ / ∈ ∅ . Let W = { A : A / ∈ A } . Discuss whether W ∈ W . - (9) Use the fact that A ∩ ( A ∪ B ) = A to prove that A ∪ ( A ∩ B ) = A . - (10) Prove that if two disjoint sets are countable, then their union is … \end{verbatim} ``` </details>
479. ph-0e7439779d919f98c03b — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md ### Plain (markdown context) S . Obviously ∅ / ∈ ∅ . Let W = { A : A / ∈ A } . Discuss whether W ∈ W . - (9) Use the fact that A ∩ ( A ∪ B ) = A to prove that A ∪ ( A ∩ B ) = A . - (10) Prove that if two disjoint sets are countable, then their union is countable. ## 1.2 Relations Definition 1.12 Given sets A and B, any subset R of A × B is a relation between A and B. If ( a , b ) ∈ R , this is often denoted by a R b. If A = B, R is said to be a relation on A. Note that relations need not have any particular property nor even be describ… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{S . Obviously ∅ / ∈ ∅ . Let W = \{ A : A / ∈ A \} . Discuss whether W ∈ W . - (9) Use the fact that A ∩ ( A ∪ B ) = A to prove that A ∪ ( A ∩ B ) = A . - (10) Prove that if two disjoint sets are countable, then their union …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md:offset=23373 \begin{verbatim} S . Obviously ∅ / ∈ ∅ . Let W = { A : A / ∈ A } . Discuss whether W ∈ W . - (9) Use the fact that A ∩ ( A ∪ B ) = A to prove that A ∪ ( A ∩ B ) = A . - (10) Prove that if two disjoint sets are countable, then their union is countable. ## 1.2 Relations Definition 1.12 Given sets A and B, any subset R of A × B is a relation between A and B. If ( a , b ) ∈ R , this is often denoted by a R b. If A = B, R is said to be a relation on A. Note that relations need not have any particular property nor even be describ… \end{verbatim} ``` </details>
480. ph-4ef6e3e7989d8292f71c — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md ### Plain (markdown context) sets are countable, then their union is countable. ## 1.2 Relations Definition 1.12 Given sets A and B, any subset R of A × B is a relation between A and B. If ( a , b ) ∈ R , this is often denoted by a R b. If A = B, R is said to be a relation on A. Note that relations need not have any particular property nor even be describable. Obviously we will be interested in those relations which are describable and have particular properties which will be shown later. is a relation between A and B . Example 1.7 If … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{sets are countable, then their union is countable. \#\# 1.2 Relations Definition 1.12 Given sets A and B, any subset R of A × B is a relation between A and B. If ( a , b ) ∈ R , this is often denoted by a R b. If A = B, R is said to be a relation on A. …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md:offset=23618 \begin{verbatim} sets are countable, then their union is countable. ## 1.2 Relations Definition 1.12 Given sets A and B, any subset R of A × B is a relation between A and B. If ( a , b ) ∈ R , this is often denoted by a R b. If A = B, R is said to be a relation on A. Note that relations need not have any particular property nor even be describable. Obviously we will be interested in those relations which are describable and have particular properties which will be shown later. is a relation between A and B . Example 1.7 If … \end{verbatim} ``` </details>
481. ph-a2d9d320341fd46fd1be — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md ### Plain (markdown context) ations Definition 1.12 Given sets A and B, any subset R of A × B is a relation between A and B. If ( a , b ) ∈ R , this is often denoted by a R b. If A = B, R is said to be a relation on A. Note that relations need not have any particular property nor even be describable. Obviously we will be interested in those relations which are describable and have particular properties which will be shown later. is a relation between A and B . Example 1.7 If A is the set of people, then a R b if a and b are cousins is … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ations Definition 1.12 Given sets A and B, any subset R of A × B is a relation between A and B. If ( a , b ) ∈ R , this is often denoted by a R b. If A = B, R is said to be a relation on A. Note that relations need not have any particular property no…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md:offset=23681 \begin{verbatim} ations Definition 1.12 Given sets A and B, any subset R of A × B is a relation between A and B. If ( a , b ) ∈ R , this is often denoted by a R b. If A = B, R is said to be a relation on A. Note that relations need not have any particular property nor even be describable. Obviously we will be interested in those relations which are describable and have particular properties which will be shown later. is a relation between A and B . Example 1.7 If A is the set of people, then a R b if a and b are cousins is … \end{verbatim} ``` </details>
482. ph-9ecb321726752e00cb65 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md ### Plain (markdown context) Example 1.8 The domain and range of the relation { ( x , y ) : x 2 + y 2 = 4 } are -2 ≤ x ≤ 2 and -2 ≤ y ≤ 2 respectively. Example 1.9 The relation R is on the set of people. The domain and range of R is the set of people who have cousins. Definition 1.14 Let R be a relation between A and B. The inverse of the relation R denoted by R -1 is a relation been B and A, defined by R -1 = { ( b , a ) : ( a , b ) ∈ R } . Example 1.10 If A = { a , b , c , d , e } and B = { 1 , 2 , 3 , 4 , 5 } , and is a relation between A and B then is a relati… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{Example 1.8 The domain and range of the relation \{ ( x , y ) : x 2 + y 2 = 4 \} are -2 ≤ x ≤ 2 and -2 ≤ y ≤ 2 respectively. Example 1.9 The relation R is on the set of people. The domain and range of R is the set of people who have cousins. Definition 1.14 Let R be a relation bet…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md:offset=24535 \begin{verbatim} Example 1.8 The domain and range of the relation { ( x , y ) : x 2 + y 2 = 4 } are -2 ≤ x ≤ 2 and -2 ≤ y ≤ 2 respectively. Example 1.9 The relation R is on the set of people. The domain and range of R is the set of people who have cousins. Definition 1.14 Let R be a relation between A and B. The inverse of the relation R denoted by R -1 is a relation been B and A, defined by R -1 = { ( b , a ) : ( a , b ) ∈ R } . Example 1.10 If A = { a , b , c , d , e } and B = { 1 , 2 , 3 , 4 , 5 } , and is a relation between A and B then is a relati… \end{verbatim} ``` </details>
483. ph-14c7ac7f2663744b1c68 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md ### Plain (markdown context) 2 + y 2 = 4 } are -2 ≤ x ≤ 2 and -2 ≤ y ≤ 2 respectively. Example 1.9 The relation R is on the set of people. The domain and range of R is the set of people who have cousins. Definition 1.14 Let R be a relation between A and B. The inverse of the relation R denoted by R -1 is a relation been B and A, defined by R -1 = { ( b , a ) : ( a , b ) ∈ R } . Example 1.10 If A = { a , b , c , d , e } and B = { 1 , 2 , 3 , 4 , 5 } , and is a relation between A and B then is a relation between B and A . Definition 1.15… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{2 + y 2 = 4 \} are -2 ≤ x ≤ 2 and -2 ≤ y ≤ 2 respectively. Example 1.9 The relation R is on the set of people. The domain and range of R is the set of people who have cousins. Definition 1.14 Let R be a relation between A and B. The inverse of the relation R denoted by R -1 is a …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md:offset=24601 \begin{verbatim} 2 + y 2 = 4 } are -2 ≤ x ≤ 2 and -2 ≤ y ≤ 2 respectively. Example 1.9 The relation R is on the set of people. The domain and range of R is the set of people who have cousins. Definition 1.14 Let R be a relation between A and B. The inverse of the relation R denoted by R -1 is a relation been B and A, defined by R -1 = { ( b , a ) : ( a , b ) ∈ R } . Example 1.10 If A = { a , b , c , d , e } and B = { 1 , 2 , 3 , 4 , 5 } , and is a relation between A and B then is a relation between B and A . Definition 1.15… \end{verbatim} ``` </details>
484. ph-a6455d5019068b789026 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md ### Plain (markdown context) ple 1.9 The relation R is on the set of people. The domain and range of R is the set of people who have cousins. Definition 1.14 Let R be a relation between A and B. The inverse of the relation R denoted by R -1 is a relation been B and A, defined by R -1 = { ( b , a ) : ( a , b ) ∈ R } . Example 1.10 If A = { a , b , c , d , e } and B = { 1 , 2 , 3 , 4 , 5 } , and is a relation between A and B then is a relation between B and A . Definition 1.15 Let R be a relation between A and B, and let S be a relation … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ple 1.9 The relation R is on the set of people. The domain and range of R is the set of people who have cousins. Definition 1.14 Let R be a relation between A and B. The inverse of the relation R denoted by R -1 is a relation been B and A, defined by R -1 = \{ ( b , a ) : ( a , b…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md:offset=24664 \begin{verbatim} ple 1.9 The relation R is on the set of people. The domain and range of R is the set of people who have cousins. Definition 1.14 Let R be a relation between A and B. The inverse of the relation R denoted by R -1 is a relation been B and A, defined by R -1 = { ( b , a ) : ( a , b ) ∈ R } . Example 1.10 If A = { a , b , c , d , e } and B = { 1 , 2 , 3 , 4 , 5 } , and is a relation between A and B then is a relation between B and A . Definition 1.15 Let R be a relation between A and B, and let S be a relation … \end{verbatim} ``` </details>
485. ph-0f4d5547b98c0c389136 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md ### Plain (markdown context) ple 1.10 If A = { a , b , c , d , e } and B = { 1 , 2 , 3 , 4 , 5 } , and is a relation between A and B then is a relation between B and A . Definition 1.15 Let R be a relation between A and B, and let S be a relation between B and C. The composition of R and S , denoted by S ◦ R is a relation between A and C defined by ( a , c ) ∈ S ◦ R if there exists b ∈ B such that ( a , b ) ∈ R and ( b , c ) ∈ S . be a relation between A and B . Then, as shown above is a relation between A and B then is a relation between B and A . Definition 1.15 Let R be a relation between A and B…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md:offset=24960 \begin{verbatim} ple 1.10 If A = { a , b , c , d , e } and B = { 1 , 2 , 3 , 4 , 5 } , and is a relation between A and B then is a relation between B and A . Definition 1.15 Let R be a relation between A and B, and let S be a relation between B and C. The composition of R and S , denoted by S ◦ R is a relation between A and C defined by ( a , c ) ∈ S ◦ R if there exists b ∈ B such that ( a , b ) ∈ R and ( b , c ) ∈ S . be a relation between A and B . Then, as shown above is a relation between A and B then is a relation between B and A . Definition 1.15 Let R be a relation between A and B, and let S be a relation between B and C. The composition of R and S , denoted by S ◦ R is a relation between A and C defined by ( a , c ) ∈ S ◦ R if there exists b ∈ B such that ( a , b ) ∈ R and ( b , c ) ∈ S . be a relation between A and B . Then, as shown above is a relation between B , and A , is a relation o… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ula-not-decoded --> is a relation between A and B then is a relation between B and A . Definition 1.15 Let R be a relation between A and B, and let S be a relation between B and C. The composition of R and S , denoted by …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md:offset=25044 \begin{verbatim} ula-not-decoded --> is a relation between A and B then is a relation between B and A . Definition 1.15 Let R be a relation between A and B, and let S be a relation between B and C. The composition of R and S , denoted by S ◦ R is a relation between A and C defined by ( a , c ) ∈ S ◦ R if there exists b ∈ B such that ( a , b ) ∈ R and ( b , c ) ∈ S . be a relation between A and B . Then, as shown above is a relation between B , and A , is a relation o… \end{verbatim} ``` </details>
487. ph-7bbb47c398a40dfacc8c — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md ### Plain (markdown context) mula-not-decoded --> is a relation between B and A . Definition 1.15 Let R be a relation between A and B, and let S be a relation between B and C. The composition of R and S , denoted by S ◦ R is a relation between A and C defined by ( a , c ) ∈ S ◦ R if there exists b ∈ B such that ( a , b ) ∈ R and ( b , c ) ∈ S . be a relation between A and B . Then, as shown above is a relation between B , and A , is a relation on B , and is a relation on A . Exa… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{mula-not-decoded --> is a relation between B and A . Definition 1.15 Let R be a relation between A and B, and let S be a relation between B and C. The composition of R and S , denoted by S ◦ R is a relation between A and C defined by ( a , c ) ∈ S ◦ …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md:offset=25109 \begin{verbatim} mula-not-decoded --> is a relation between B and A . Definition 1.15 Let R be a relation between A and B, and let S be a relation between B and C. The composition of R and S , denoted by S ◦ R is a relation between A and C defined by ( a , c ) ∈ S ◦ R if there exists b ∈ B such that ( a , b ) ∈ R and ( b , c ) ∈ S . be a relation between A and B . Then, as shown above is a relation between B , and A , is a relation on B , and is a relation on A . Exa… \end{verbatim} ``` </details>
488. ph-027594f4153d8681f856 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md ### Plain (markdown context) !-- formula-not-decoded --> Definition 1.15 Let R be a relation between A and B, and let S be a relation between B and C. The composition of R and S , denoted by S ◦ R is a relation between A and C defined by ( a , c ) ∈ S ◦ R if there exists b ∈ B such that ( a , b ) ∈ R and ( b , c ) ∈ S . be a relation between A and B . Then, as shown above is a relation between B , and A , is a relation on B , and is a relation on A . Example 1.13 If R = { ( x , y ) : y = x + 5 } and S = { (… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{!-- formula-not-decoded --> Definition 1.15 Let R be a relation between A and B, and let S be a relation between B and C. The composition of R and S , denoted by S ◦ R is a relation between A and C defined by ( a , c ) ∈ S ◦ R if there exists b ∈ B such that ( a , b ) ∈ R and ( …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md:offset=25165 \begin{verbatim} !-- formula-not-decoded --> Definition 1.15 Let R be a relation between A and B, and let S be a relation between B and C. The composition of R and S , denoted by S ◦ R is a relation between A and C defined by ( a , c ) ∈ S ◦ R if there exists b ∈ B such that ( a , b ) ∈ R and ( b , c ) ∈ S . be a relation between A and B . Then, as shown above is a relation between B , and A , is a relation on B , and is a relation on A . Example 1.13 If R = { ( x , y ) : y = x + 5 } and S = { (… \end{verbatim} ``` </details>
489. ph-042f7f91121fe15e1073 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md ### Plain (markdown context) conclude that a and a are siblings, which we know is not true. Example 1.15 Let A be the set of all people and a R b if a and b have the same parents. The relation R is reflexive since everyone has the same parents as themselves. It is symmetric since if a and b have the same parents, b and a have the same parents. It is also transitive since if a and b have the same parents and b and c have the same parents, then a and c have the same parents. Example 1.16 Let A = { a , b , c , d , e } and R is not reflexive since ( e , e ) / ∈ R . It is not symmetric because ( a… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{conclude that a and a are siblings, which we know is not true. Example 1.15 Let A be the set of all people and a R b if a and b have the same parents. The relation R is reflexive since everyone has the same parents as themselves. It is symmetric since if a and b have the same pa…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md:offset=27306 \begin{verbatim} conclude that a and a are siblings, which we know is not true. Example 1.15 Let A be the set of all people and a R b if a and b have the same parents. The relation R is reflexive since everyone has the same parents as themselves. It is symmetric since if a and b have the same parents, b and a have the same parents. It is also transitive since if a and b have the same parents and b and c have the same parents, then a and c have the same parents. Example 1.16 Let A = { a , b , c , d , e } and R is not reflexive since ( e , e ) / ∈ R . It is not symmetric because ( a… \end{verbatim} ``` </details>
490. ph-2cd9a851335298bb1bd1 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md ### Plain (markdown context) , a ) ∈ R , but d /negationslash= a . It is not transitive since ( a , c ) , ( c , e ) , but ( a , e ) / . ∈ R ∈ R Example 1.17 Let R betherelation on Z defined by a R b if a -b is a multiple of 5. Certainly a -a = 0 is a multiple of 5, so R is reflexive. If a -b is a multiple of 5, then a -b = 5 k for some integer k . Hence b -a = 5( -k ) is a multiple of 5, so R is symmetric. If a -b is a multiple of 5 and b -c is a multiple of 5, then a -b = 5 k and b -c = 5 m for some integers k and m . so that a -c is a multiple of 5. Hence R is transitive. Definition 1.17 Ar… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{, a ) ∈ R , but d /negationslash= a . It is not transitive since ( a , c ) , ( c , e ) , but ( a , e ) / . ∈ R ∈ R Example 1.17 Let R betherelation on Z defined by a R b if a -b is a multiple of 5. Certainly a -a = 0 is a multiple of 5, so R is reflexive. If a -b is a multiple o…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md:offset=27990 \begin{verbatim} , a ) ∈ R , but d /negationslash= a . It is not transitive since ( a , c ) , ( c , e ) , but ( a , e ) / . ∈ R ∈ R Example 1.17 Let R betherelation on Z defined by a R b if a -b is a multiple of 5. Certainly a -a = 0 is a multiple of 5, so R is reflexive. If a -b is a multiple of 5, then a -b = 5 k for some integer k . Hence b -a = 5( -k ) is a multiple of 5, so R is symmetric. If a -b is a multiple of 5 and b -c is a multiple of 5, then a -b = 5 k and b -c = 5 m for some integers k and m . so that a -c is a multiple of 5. Hence R is transitive. Definition 1.17 Ar… \end{verbatim} ``` </details>
491. ph-02c0a644dc732675dea7 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md ### Plain (markdown context) R 1 be the relation on Z defined by R 1 = { ( m , n ) : m -n } is divisible by 5. R 1 is shown above to be an equivalence relation on the integers. Example 1.19 Let A be the set of all people. Define R 2 by a R 2 b if a and b are the same age. This is easily shown to be an equivalence relation. An equivalence relation on a set A divides A into nonempty subsets that are mutually exclusive or disjoint , meaning that no two of them have an element in common. In the first example above, the sets contain elements that are related to each other and no element in one set… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{R 1 be the relation on Z defined by R 1 = \{ ( m , n ) : m -n \} is divisible by 5. R 1 is shown above to be an equivalence relation on the integers. Example 1.19 Let A be the set of all people. Define R 2 by a R 2 b if a and b are the same age. This is easily shown to be an equiv…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md:offset=28731 \begin{verbatim} R 1 be the relation on Z defined by R 1 = { ( m , n ) : m -n } is divisible by 5. R 1 is shown above to be an equivalence relation on the integers. Example 1.19 Let A be the set of all people. Define R 2 by a R 2 b if a and b are the same age. This is easily shown to be an equivalence relation. An equivalence relation on a set A divides A into nonempty subsets that are mutually exclusive or disjoint , meaning that no two of them have an element in common. In the first example above, the sets contain elements that are related to each other and no element in one set… \end{verbatim} ``` </details>
492. ph-56baf8df401b814a6172 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md ### Plain (markdown context) two sets. (See the definition of partition below.) Notation 1.1 Let R be an equivalence relation on a set A and a ∈ A . Then [ a ] R = { x : x R a } . If the relation is understood, then [ a ] R is simply denoted by [ a ]. Let [ A ] R = { [ a ] R : a ∈ A } . Definition 1.18 Let A and I be nonempty sets and 〈 A 〉 = { Ai : i ∈ I } be a set of nonempty subsets of A. The set 〈 A 〉 is called a partition of A if both of the following are satisfied: - (a) Ai ∩ Aj = ∅ for all i /negationslash= j . Theorem 1.3 A nonempty set of subsets 〈 A 〉 of a set A is a partition of A … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{two sets. (See the definition of partition below.) Notation 1.1 Let R be an equivalence relation on a set A and a ∈ A . Then [ a ] R = \{ x : x R a \} . If the relation is understood, then [ a ] R is simply denoted by [ a ]. Let [ A ] R = \{ [ a ] R : a ∈ A \} . Definition 1.18 Let …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md:offset=29564 \begin{verbatim} two sets. (See the definition of partition below.) Notation 1.1 Let R be an equivalence relation on a set A and a ∈ A . Then [ a ] R = { x : x R a } . If the relation is understood, then [ a ] R is simply denoted by [ a ]. Let [ A ] R = { [ a ] R : a ∈ A } . Definition 1.18 Let A and I be nonempty sets and 〈 A 〉 = { Ai : i ∈ I } be a set of nonempty subsets of A. The set 〈 A 〉 is called a partition of A if both of the following are satisfied: - (a) Ai ∩ Aj = ∅ for all i /negationslash= j . Theorem 1.3 A nonempty set of subsets 〈 A 〉 of a set A is a partition of A … \end{verbatim} ``` </details>
493. ph-95228d4456515f8a6244 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md ### Plain (markdown context) t A ( if it exists ) is called the greatest element of A. For a subset B of a poset A, an element a of A is a lower bound of B if a ≤ b (or b ≥ a ) for all b in B. The element a is called a greatest lower bound (glb) of B if ( i ) a is a lower bound of B and ( ii ) if any other element a ′ of A is a lower bound of B, then a ≥ a ′ . The greatest lower bound for the entire poset A ( if it exists ) is called the least element of A. Example 1.22 Let C = { a , b , c } and X be the power set of C . Define the relation ≤ on X by T ≤ V if T ⊆ V . By definition, { a , b } … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{t A ( if it exists ) is called the greatest element of A. For a subset B of a poset A, an element a of A is a lower bound of B if a ≤ b (or b ≥ a ) for all b in B. The element a is called a greatest lower bound (glb) of B if ( i ) a is a lower bound of B and ( ii ) if any other …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md:offset=32627 \begin{verbatim} t A ( if it exists ) is called the greatest element of A. For a subset B of a poset A, an element a of A is a lower bound of B if a ≤ b (or b ≥ a ) for all b in B. The element a is called a greatest lower bound (glb) of B if ( i ) a is a lower bound of B and ( ii ) if any other element a ′ of A is a lower bound of B, then a ≥ a ′ . The greatest lower bound for the entire poset A ( if it exists ) is called the least element of A. Example 1.22 Let C = { a , b , c } and X be the power set of C . Define the relation ≤ on X by T ≤ V if T ⊆ V . By definition, { a , b } … \end{verbatim} ``` </details>
494. ph-85ad3b1d59ca7da72fe8 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md ### Plain (markdown context) ement b ∈ B, there is an element a ∈ A so that b = f ( a ) . Definition 1.27 If f : A → B, and f ( a ) = f ( a ′ ) ⇒ a = a ′ for all a , a ′ ∈ A then f is one-to-one . It is also called a monomorphism or injection . Definition 1.28 If f : A → B is one-to-one and onto, then f is called a one-to-one correspondence or bijection . If A is finite, then f is also called a permutation . Notation 1.2 If f is a permutation on the set { 1 , 2 , 3 , . . . , n } , then it can be represented in the form Thus if f ( a ) = b , f ( b ) = d , f ( c ) = a , and f ( d ) = c , we may… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ement b ∈ B, there is an element a ∈ A so that b = f ( a ) . Definition 1.27 If f : A → B, and f ( a ) = f ( a ′ ) ⇒ a = a ′ for all a , a ′ ∈ A then f is one-to-one . It is also called a monomorphism or injection . Definition 1.28 If f : A → B is one-to-one and onto, then f is …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md:offset=36760 \begin{verbatim} ement b ∈ B, there is an element a ∈ A so that b = f ( a ) . Definition 1.27 If f : A → B, and f ( a ) = f ( a ′ ) ⇒ a = a ′ for all a , a ′ ∈ A then f is one-to-one . It is also called a monomorphism or injection . Definition 1.28 If f : A → B is one-to-one and onto, then f is called a one-to-one correspondence or bijection . If A is finite, then f is also called a permutation . Notation 1.2 If f is a permutation on the set { 1 , 2 , 3 , . . . , n } , then it can be represented in the form Thus if f ( a ) = b , f ( b ) = d , f ( c ) = a , and f ( d ) = c , we may… \end{verbatim} ``` </details>
495. ph-d280bf9e674a1a020cdc — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md ### Plain (markdown context) f ( f -1 ( b )) = b , f is onto. By symmetry, f -1 Assume f : A → B is a bijection. Define the relation f -1 on B × A by f -1 ( b ) = a if f ( a ) = b . Let b ∈ B and choose a so that f ( a ) = b . This is possible since f is onto. Therefore f -1 ( b ) = a and f -1 has domain B . If f -1 ( b ) = a and f -1 ( b ) = a ′ , then f ( a ) = b and f ( a ′ ) = b . But since f is one-to-one, a = a ′ . Therefore f -1 is well defined and hence f -1 is a function. By definition f ◦ f -1 = f -1 ◦ f = I . The proof of the following theorem is left to the reader: Theorem 1.7 Let… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{f ( f -1 ( b )) = b , f is onto. By symmetry, f -1 Assume f : A → B is a bijection. Define the relation f -1 on B × A by f -1 ( b ) = a if f ( a ) = b . Let b ∈ B and choose a so that f ( a ) = b . This is possible since f is onto. Therefore f -1 ( b ) = a and f -1 has domain B …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md:offset=39129 \begin{verbatim} f ( f -1 ( b )) = b , f is onto. By symmetry, f -1 Assume f : A → B is a bijection. Define the relation f -1 on B × A by f -1 ( b ) = a if f ( a ) = b . Let b ∈ B and choose a so that f ( a ) = b . This is possible since f is onto. Therefore f -1 ( b ) = a and f -1 has domain B . If f -1 ( b ) = a and f -1 ( b ) = a ′ , then f ( a ) = b and f ( a ′ ) = b . But since f is one-to-one, a = a ′ . Therefore f -1 is well defined and hence f -1 is a function. By definition f ◦ f -1 = f -1 ◦ f = I . The proof of the following theorem is left to the reader: Theorem 1.7 Let… \end{verbatim} ``` </details>
496. ph-1e3e5c5e3ac0df0449b1 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md ### Plain (markdown context) f ◦ g ) -1 = g -1 ◦ f -1 . - (3) Give an example of a function f and sets A 1 , A 2 ⊆ A such that f ( A 1 ∩ A 2 ) /negationslash= f ( A 1 ) ∩ f ( A 2 ). - (5) Prove that if f ◦ g is onto, then f is onto. - (4) Prove that if f ◦ g is one-to-one then g is one-to-one. ## 1.4 Semigroups In the following function /star : S × S → S we shall use the notation a /star a ′ for /star (( a , a ′ )). Definition 1.32 A semigroup is a nonempty set S together with a function /star from S × S → S such that The function or operation /star with this property is called associative . … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{f ◦ g ) -1 = g -1 ◦ f -1 . - (3) Give an example of a function f and sets A 1 , A 2 ⊆ A such that f ( A 1 ∩ A 2 ) /negationslash= f ( A 1 ) ∩ f ( A 2 ). - (5) Prove that if f ◦ g is onto, then f is onto. - (4) Prove that if f ◦ g is one-to-one then g is one-to-one. \#\# 1.4 Semigr…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md:offset=42512 \begin{verbatim} f ◦ g ) -1 = g -1 ◦ f -1 . - (3) Give an example of a function f and sets A 1 , A 2 ⊆ A such that f ( A 1 ∩ A 2 ) /negationslash= f ( A 1 ) ∩ f ( A 2 ). - (5) Prove that if f ◦ g is onto, then f is onto. - (4) Prove that if f ◦ g is one-to-one then g is one-to-one. ## 1.4 Semigroups In the following function /star : S × S → S we shall use the notation a /star a ′ for /star (( a , a ′ )). Definition 1.32 A semigroup is a nonempty set S together with a function /star from S × S → S such that The function or operation /star with this property is called associative . … \end{verbatim} ``` </details>
497. ph-691637fbb7c5c7aab7ab — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md ### Plain (markdown context) tion defined in the previous definition and φ R : S → S / R defined by φ R ( s ) = s R is a homomorphism. Theorem 1.16 Let f : A → B be a homomorphism and R be the congruence a R a ′ iff f ( a ) = f ( a ′ ) , then there exists a homomorphism g : A / R → B defined by g ( a R ) = f ( a ) . Hence g ◦ φ R = f . ![Image](./AutomataTheory_chapter_1.1_artifacts/image_000005_bf80f879566ee4059c3c198a8d7b04785ae4ecab8e0f64eb8a143017edeba4ab.png) R Proof We showed in Theorem 1.9 that g is a function. /square Theorem 1.17 Let f : A / R → B be a function and S ⊆ R , so if a S … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{tion defined in the previous definition and φ R : S → S / R defined by φ R ( s ) = s R is a homomorphism. Theorem 1.16 Let f : A → B be a homomorphism and R be the congruence a R a ′ iff f ( a ) = f ( a ′ ) , then there exists a homomorphism g : A / R → B defined by g ( a R ) = …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md:offset=52622 \begin{verbatim} tion defined in the previous definition and φ R : S → S / R defined by φ R ( s ) = s R is a homomorphism. Theorem 1.16 Let f : A → B be a homomorphism and R be the congruence a R a ′ iff f ( a ) = f ( a ′ ) , then there exists a homomorphism g : A / R → B defined by g ( a R ) = f ( a ) . Hence g ◦ φ R = f . ![Image](./AutomataTheory_chapter_1.1_artifacts/image_000005_bf80f879566ee4059c3c198a8d7b04785ae4ecab8e0f64eb8a143017edeba4ab.png) R Proof We showed in Theorem 1.9 that g is a function. /square Theorem 1.17 Let f : A / R → B be a function and S ⊆ R , so if a S … \end{verbatim} ``` </details>
498. ph-8b59ddefa4661e9fa7c7 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md ### Plain (markdown context) a ′ implies a R a ′ , then there exist functions g : A / S → B and i : A / S → A / R such that f ◦ i = g. ![Image](./AutomataTheory_chapter_1.1_artifacts/image_000006_2053b4815736ff133061fdd5f00a5bfe86cf8dfe428aba6bb478524767b6009d.png) S Proof Let i : A / S → A / R be defined by i ( a S ) = a R and g : A / S → B by g ( a S ) = f ( a R ) . The function i is trivially well defined and a homomorphism. The proof that g is a function is similar to the proof of Theorem 1.9. (See Theorem 1.10). /square We already know that the set of all functions from a set A to itself… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{a ′ implies a R a ′ , then there exist functions g : A / S → B and i : A / S → A / R such that f ◦ i = g. ![Image](./AutomataTheory\_chapter\_1.1\_artifacts/image\_000006\_2053b4815736ff133061fdd5f00a5bfe86cf8dfe428aba6bb478524767b6009d.png) S Proof Let i : A / S → A / R be defined b…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md:offset=53227 \begin{verbatim} a ′ implies a R a ′ , then there exist functions g : A / S → B and i : A / S → A / R such that f ◦ i = g. ![Image](./AutomataTheory_chapter_1.1_artifacts/image_000006_2053b4815736ff133061fdd5f00a5bfe86cf8dfe428aba6bb478524767b6009d.png) S Proof Let i : A / S → A / R be defined by i ( a S ) = a R and g : A / S → B by g ( a S ) = f ( a R ) . The function i is trivially well defined and a homomorphism. The proof that g is a function is similar to the proof of Theorem 1.9. (See Theorem 1.10). /square We already know that the set of all functions from a set A to itself… \end{verbatim} ``` </details>
499. ph-1eea622cca86da83f9b4 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md ### Plain (markdown context) already know that the set of all functions from a set A to itself form a semigroup since for a ∈ A , and functions f , g , and h from A to itself, (( f ◦ ( g ◦ h ))( a ) = (( f ◦ g ) ◦ h )( a ) = f ( g ( h ( a ). Also since f , g , and h are relations we have already proven that ( f ( g h ) ( f g ) h . Conversely, given a semigroup S , and s ∈ S we can define a function φ s : S → S by φ s ( t ) = st for all t ∈ S . Let T S = { φ s : S → S for s ∈ S } . For all s , t , and u in S , ◦ ◦ = ◦ ◦ and φ st = ( φ s ◦ φ t ). Let τ : S → T S be defined by τ ( s ) = φ s . Th… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{already know that the set of all functions from a set A to itself form a semigroup since for a ∈ A , and functions f , g , and h from A to itself, (( f ◦ ( g ◦ h ))( a ) = (( f ◦ g ) ◦ h )( a ) = f ( g ( h ( a ). Also since f , g , and h are relations we have already proven that…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md:offset=53768 \begin{verbatim} already know that the set of all functions from a set A to itself form a semigroup since for a ∈ A , and functions f , g , and h from A to itself, (( f ◦ ( g ◦ h ))( a ) = (( f ◦ g ) ◦ h )( a ) = f ( g ( h ( a ). Also since f , g , and h are relations we have already proven that ( f ( g h ) ( f g ) h . Conversely, given a semigroup S , and s ∈ S we can define a function φ s : S → S by φ s ( t ) = st for all t ∈ S . Let T S = { φ s : S → S for s ∈ S } . For all s , t , and u in S , ◦ ◦ = ◦ ◦ and φ st = ( φ s ◦ φ t ). Let τ : S → T S be defined by τ ( s ) = φ s . Th… \end{verbatim} ``` </details>
500. ph-398084b9a47c42b0f6e6 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md ### Plain (markdown context) self, (( f ◦ ( g ◦ h ))( a ) = (( f ◦ g ) ◦ h )( a ) = f ( g ( h ( a ). Also since f , g , and h are relations we have already proven that ( f ( g h ) ( f g ) h . Conversely, given a semigroup S , and s ∈ S we can define a function φ s : S → S by φ s ( t ) = st for all t ∈ S . Let T S = { φ s : S → S for s ∈ S } . For all s , t , and u in S , ◦ ◦ = ◦ ◦ and φ st = ( φ s ◦ φ t ). Let τ : S → T S be defined by τ ( s ) = φ s . The function τ is a homomorphism since Theorem 1.18 Every semigroup is isomorphic to a semigroup of functions from… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{self, (( f ◦ ( g ◦ h ))( a ) = (( f ◦ g ) ◦ h )( a ) = f ( g ( h ( a ). Also since f , g , and h are relations we have already proven that ( f ( g h ) ( f g ) h . Conversely, given a semigroup S , and s ∈ S we can define a function φ s : S → S by φ s ( t ) = st for all t ∈ S . L…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md:offset=53910 \begin{verbatim} self, (( f ◦ ( g ◦ h ))( a ) = (( f ◦ g ) ◦ h )( a ) = f ( g ( h ( a ). Also since f , g , and h are relations we have already proven that ( f ( g h ) ( f g ) h . Conversely, given a semigroup S , and s ∈ S we can define a function φ s : S → S by φ s ( t ) = st for all t ∈ S . Let T S = { φ s : S → S for s ∈ S } . For all s , t , and u in S , ◦ ◦ = ◦ ◦ and φ st = ( φ s ◦ φ t ). Let τ : S → T S be defined by τ ( s ) = φ s . The function τ is a homomorphism since Theorem 1.18 Every semigroup is isomorphic to a semigroup of functions from… \end{verbatim} ``` </details>
501. ph-e81ce40349b64170aa23 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md ### Plain (markdown context) quence a 1 a 2 a 3 a 4 . . . an where ai ∈ /Sigma1 . Thus if /Sigma1 = { a , b } , then aab , a , baba , bbbbb , and baaaaa would all be strings of symbols of /Sigma1 . In addition we include an empty string denoted by λ which has no symbols in it. Definition 2.2 Let /Sigma1 ∗ denote the set of all strings of /Sigma1 including the empty string. Define the binary operation ◦ called concatenation on /Sigma1 ∗ as follows: If a 1 a 2 a 3 a 4 . . . an and b 1 b 2 b 3 b 4 . . . bm ∈ /Sigma1 ∗ then If S and T are subsets of /Sigma1 ∗ then S ◦ T = { s ◦ t : s ∈ S , t ∈ T … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{quence a 1 a 2 a 3 a 4 . . . an where ai ∈ /Sigma1 . Thus if /Sigma1 = \{ a , b \} , then aab , a , baba , bbbbb , and baaaaa would all be strings of symbols of /Sigma1 . In addition we include an empty string denoted by λ which has no symbols in it. Definition 2.2 Let /Sigma1 ∗ d…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md:offset=57585 \begin{verbatim} quence a 1 a 2 a 3 a 4 . . . an where ai ∈ /Sigma1 . Thus if /Sigma1 = { a , b } , then aab , a , baba , bbbbb , and baaaaa would all be strings of symbols of /Sigma1 . In addition we include an empty string denoted by λ which has no symbols in it. Definition 2.2 Let /Sigma1 ∗ denote the set of all strings of /Sigma1 including the empty string. Define the binary operation ◦ called concatenation on /Sigma1 ∗ as follows: If a 1 a 2 a 3 a 4 . . . an and b 1 b 2 b 3 b 4 . . . bm ∈ /Sigma1 ∗ then If S and T are subsets of /Sigma1 ∗ then S ◦ T = { s ◦ t : s ∈ S , t ∈ T … \end{verbatim} ``` </details>
502. ph-0dcc2417db42bd92b514 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md ### Plain (markdown context) the submonoid generated by a subset of a monoid described in Chapter 1. Definition 2.3 Let B be a subset of /Sigma1 ∗ then B ∗ is the set of all strings or words formed by concatenating words from B together with the empty string, i.e. B ∗ = { w 1 w 2 . . . w n : w i ∈ B } ∪ { λ } . If ∅ denotes the empty set then ∅ ∗ = { λ } . The symbol ∗ is called the Kleene star and is named after the mathematician and logician Stephen Cole Kleene. Note that /Sigma1 ∗ is consistent with this definition. Let A + be the set consisting of all finite p… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{the submonoid generated by a subset of a monoid described in Chapter 1. Definition 2.3 Let B be a subset of /Sigma1 ∗ then B ∗ is the set of all strings or words formed by concatenating words from B together with the empty string, i.e. B ∗ = \{ w 1 w 2 . . . w n : w i ∈ B \} ∪ \{ λ…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md:offset=58554 \begin{verbatim} the submonoid generated by a subset of a monoid described in Chapter 1. Definition 2.3 Let B be a subset of /Sigma1 ∗ then B ∗ is the set of all strings or words formed by concatenating words from B together with the empty string, i.e. B ∗ = { w 1 w 2 . . . w n : w i ∈ B } ∪ { λ } . If ∅ denotes the empty set then ∅ ∗ = { λ } . The symbol ∗ is called the Kleene star and is named after the mathematician and logician Stephen Cole Kleene. Note that /Sigma1 ∗ is consistent with this definition. Let A + be the set consisting of all finite p… \end{verbatim} ``` </details>
503. ph-4c9ece4db2efcf8d7b4a — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md ### Plain (markdown context) ubset of a monoid described in Chapter 1. Definition 2.3 Let B be a subset of /Sigma1 ∗ then B ∗ is the set of all strings or words formed by concatenating words from B together with the empty string, i.e. B ∗ = { w 1 w 2 . . . w n : w i ∈ B } ∪ { λ } . If ∅ denotes the empty set then ∅ ∗ = { λ } . The symbol ∗ is called the Kleene star and is named after the mathematician and logician Stephen Cole Kleene. Note that /Sigma1 ∗ is consistent with this definition. Let A + be the set consisting of all finite products of elements of a nonem… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ubset of a monoid described in Chapter 1. Definition 2.3 Let B be a subset of /Sigma1 ∗ then B ∗ is the set of all strings or words formed by concatenating words from B together with the empty string, i.e. B ∗ = \{ w 1 w 2 . . . w n : w i ∈ B \} ∪ \{ λ \} . If ∅ denotes the empty se…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md:offset=58584 \begin{verbatim} ubset of a monoid described in Chapter 1. Definition 2.3 Let B be a subset of /Sigma1 ∗ then B ∗ is the set of all strings or words formed by concatenating words from B together with the empty string, i.e. B ∗ = { w 1 w 2 . . . w n : w i ∈ B } ∪ { λ } . If ∅ denotes the empty set then ∅ ∗ = { λ } . The symbol ∗ is called the Kleene star and is named after the mathematician and logician Stephen Cole Kleene. Note that /Sigma1 ∗ is consistent with this definition. Let A + be the set consisting of all finite products of elements of a nonem… \end{verbatim} ``` </details>
504. ph-668a56d66ac0c50d07bc — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md ### Plain (markdown context) Sigma1 and the symbols ∅ , λ, ∗ , ∨ , ( , and ) . The symbol λ is used to denote the symbol ∅ ∗ . - (i) The symbol ∅ is a regular expression and for every a ∈ /Sigma1 , the symbol a is a regular expression. - (ii) If w 1 and w 2 are regular expressions, then w 1 w 2 , w 1 ∨ w 2 , w ∗ 1 , and ( w 1 ) are regular expressions. - (iii) There are no regular expressions which are not generated by ( i ) and ( ii ) . Each expression corresponds to a set with the following correspondence £ defined by The image of a regular expression is a regul… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{Sigma1 and the symbols ∅ , λ, ∗ , ∨ , ( , and ) . The symbol λ is used to denote the symbol ∅ ∗ . - (i) The symbol ∅ is a regular expression and for every a ∈ /Sigma1 , the symbol a is a regular expression. - (ii) If w 1 and w 2 are regular expressions, then w 1 w 2 , w 1 ∨ w 2 …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md:offset=61075 \begin{verbatim} Sigma1 and the symbols ∅ , λ, ∗ , ∨ , ( , and ) . The symbol λ is used to denote the symbol ∅ ∗ . - (i) The symbol ∅ is a regular expression and for every a ∈ /Sigma1 , the symbol a is a regular expression. - (ii) If w 1 and w 2 are regular expressions, then w 1 w 2 , w 1 ∨ w 2 , w ∗ 1 , and ( w 1 ) are regular expressions. - (iii) There are no regular expressions which are not generated by ( i ) and ( ii ) . Each expression corresponds to a set with the following correspondence £ defined by The image of a regular expression is a regul… \end{verbatim} ``` </details>
505. ph-3e90e6c1d973115b1b90 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md ### Plain (markdown context) ∗ , ∨ , ( , and ) . The symbol λ is used to denote the symbol ∅ ∗ . - (i) The symbol ∅ is a regular expression and for every a ∈ /Sigma1 , the symbol a is a regular expression. - (ii) If w 1 and w 2 are regular expressions, then w 1 w 2 , w 1 ∨ w 2 , w ∗ 1 , and ( w 1 ) are regular expressions. - (iii) There are no regular expressions which are not generated by ( i ) and ( ii ) . Each expression corresponds to a set with the following correspondence £ defined by The image of a regular expression is a regular language. Regular languages… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{∗ , ∨ , ( , and ) . The symbol λ is used to denote the symbol ∅ ∗ . - (i) The symbol ∅ is a regular expression and for every a ∈ /Sigma1 , the symbol a is a regular expression. - (ii) If w 1 and w 2 are regular expressions, then w 1 w 2 , w 1 ∨ w 2 , w ∗ 1 , and ( w 1 ) are regu…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.1.md:offset=61105 \begin{verbatim} ∗ , ∨ , ( , and ) . The symbol λ is used to denote the symbol ∅ ∗ . - (i) The symbol ∅ is a regular expression and for every a ∈ /Sigma1 , the symbol a is a regular expression. - (ii) If w 1 and w 2 are regular expressions, then w 1 w 2 , w 1 ∨ w 2 , w ∗ 1 , and ( w 1 ) are regular expressions. - (iii) There are no regular expressions which are not generated by ( i ) and ( ii ) . Each expression corresponds to a set with the following correspondence £ defined by The image of a regular expression is a regular language. Regular languages… \end{verbatim} ``` </details>
506. ph-ae8f6e88a99c960681c2 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.2.md ### Plain (markdown context) A : h ( a ) = ua v where only mortal symbols occur in u and v } . For each a in X, let Na be the least nonnegative integer for which h Na ( u v ) = λ . Let H = { h Na ( a ) : a ∈ X } . The fixed language L of h is the submonoid of A ∗ generated by H. The correspondence a ↔ h Na ( a ) is a one-to-one correspon- dence between X and H. Proof (1) H ∗ ⊆ L : Since h is a homomorphism it is sufficient to verify that each element h N ( a ) ( a ) of H is in L , which is confirmed by the calculation: (2) L ⊆ H ∗ : Let w be in L . Let a 1 , a 2 , a 3 , . . . , an be the subs… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{A : h ( a ) = ua v where only mortal symbols occur in u and v \} . For each a in X, let Na be the least nonnegative integer for which h Na ( u v ) = λ . Let H = \{ h Na ( a ) : a ∈ X \} . The fixed language L of h is the submonoid of A ∗ generated by H. The correspondence a ↔ h Na …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.2.md:offset=2901 \begin{verbatim} A : h ( a ) = ua v where only mortal symbols occur in u and v } . For each a in X, let Na be the least nonnegative integer for which h Na ( u v ) = λ . Let H = { h Na ( a ) : a ∈ X } . The fixed language L of h is the submonoid of A ∗ generated by H. The correspondence a ↔ h Na ( a ) is a one-to-one correspon- dence between X and H. Proof (1) H ∗ ⊆ L : Since h is a homomorphism it is sufficient to verify that each element h N ( a ) ( a ) of H is in L , which is confirmed by the calculation: (2) L ⊆ H ∗ : Let w be in L . Let a 1 , a 2 , a 3 , . . . , an be the subs… \end{verbatim} ``` </details>
507. ph-8474905c5dc36cdf7038 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.2.md ### Plain (markdown context) rds of length > 2 is not free. Corollary 2.2 If A is an alphabet having exactly n symbols, then no inclusion chain of distinct retracts of A ∗ has more than n + 1 retracts even when the retract { λ } is included. Corollary 2.3 If X is a key code and x n lies in X ∗ , then so does x. Corollary 2.4 If X is a key code and both u v and v u lie in X ∗ , then so do u and v . Let A = { a 1 , a 2 , a 3 , . . . , an } . A simple example of a longest possible inclusion chain of retracts in A ∗ is Each of these retracts, except the first, is maximal among the retracts con… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{rds of length \> 2 is not free. Corollary 2.2 If A is an alphabet having exactly n symbols, then no inclusion chain of distinct retracts of A ∗ has more than n + 1 retracts even when the retract \{ λ \} is included. Corollary 2.3 If X is a key code and x n lies in X ∗ , then so …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.2.md:offset=5650 \begin{verbatim} rds of length > 2 is not free. Corollary 2.2 If A is an alphabet having exactly n symbols, then no inclusion chain of distinct retracts of A ∗ has more than n + 1 retracts even when the retract { λ } is included. Corollary 2.3 If X is a key code and x n lies in X ∗ , then so does x. Corollary 2.4 If X is a key code and both u v and v u lie in X ∗ , then so do u and v . Let A = { a 1 , a 2 , a 3 , . . . , an } . A simple example of a longest possible inclusion chain of retracts in A ∗ is Each of these retracts, except the first, is maximal among the retracts con… \end{verbatim} ``` </details>
508. ph-e2e639efc9d1d5bb6dfb — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.2.md ### Plain (markdown context) ng the retracts contained in its predecessor. In each case the number of generators of the subretract is one less than the number of generators of its predecessor. However, maximal proper subretracts of a retract can have many fewer generators: Proposition 2.2 Let n be a positive integer and A = { a 1 , a 2 , a 3 , . . . , an } an alphabet of n symbols. Let m be any positive integer less than n. Then A ∗ contains a maximal proper retract generated by exactly m words. Proof The set of m words is a key code for which K ∗ is a maximal proper retract of A ∗ . The veri… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ng the retracts contained in its predecessor. In each case the number of generators of the subretract is one less than the number of generators of its predecessor. However, maximal proper subretracts of a retract can have many fewer generators: Proposition 2.2 Let n be a positiv…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.2.md:offset=6236 \begin{verbatim} ng the retracts contained in its predecessor. In each case the number of generators of the subretract is one less than the number of generators of its predecessor. However, maximal proper subretracts of a retract can have many fewer generators: Proposition 2.2 Let n be a positive integer and A = { a 1 , a 2 , a 3 , . . . , an } an alphabet of n symbols. Let m be any positive integer less than n. Then A ∗ contains a maximal proper retract generated by exactly m words. Proof The set of m words is a key code for which K ∗ is a maximal proper retract of A ∗ . The veri… \end{verbatim} ``` </details>
509. ph-ec63c1ed70cc107c22a1 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.2.md ### Plain (markdown context) and both u v and v u lie in X ∗ , then so do u and v . ## 2.3 Semiretracts and lattices (Optional) The intersection of two retracts of the free monoid on a finite set A need not be a retract if A contains four or more symbols. Possibly the simplest example is the following one adapted from [ 7 ]: Let A = { a , b , c , d } . The sets { ab , ac , d } and { ba , c , da } are key codes and consequently the submonoids R and R ′ that they generated are retracts of A ∗ . However, their intersection is not only not a retract; it is not even finitely generated. The set d (… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{and both u v and v u lie in X ∗ , then so do u and v . \#\# 2.3 Semiretracts and lattices (Optional) The intersection of two retracts of the free monoid on a finite set A need not be a retract if A contains four or more symbols. Possibly the simplest example is the following one a…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.2.md:offset=7974 \begin{verbatim} and both u v and v u lie in X ∗ , then so do u and v . ## 2.3 Semiretracts and lattices (Optional) The intersection of two retracts of the free monoid on a finite set A need not be a retract if A contains four or more symbols. Possibly the simplest example is the following one adapted from [ 7 ]: Let A = { a , b , c , d } . The sets { ab , ac , d } and { ba , c , da } are key codes and consequently the submonoids R and R ′ that they generated are retracts of A ∗ . However, their intersection is not only not a retract; it is not even finitely generated. The set d (… \end{verbatim} ``` </details>
510. ph-2896f5e020648338e98d — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.2.md ### Plain (markdown context) always a complete lattice. By broadening our attention slightly we obtain a similarly attractive stability result for what we call semiretracts of free monoids: Definition 2.14 By a semiretract of A ∗ , we mean an intersection of a finite number of retracts of A ∗ . Each retract of A ∗ is also a semiretract. The clearest example of a semiretract that is not a retract is the example given previously: R ∩ R ′ = ( d ( ab ) ∗ ac ) ∗ . Some pairs of retracts have as their intersection a retract: As stated above, but not to be demonstrated here, if fewer than four alpha… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{always a complete lattice. By broadening our attention slightly we obtain a similarly attractive stability result for what we call semiretracts of free monoids: Definition 2.14 By a semiretract of A ∗ , we mean an intersection of a finite number of retracts of A ∗ . Each retract…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.2.md:offset=9055 \begin{verbatim} always a complete lattice. By broadening our attention slightly we obtain a similarly attractive stability result for what we call semiretracts of free monoids: Definition 2.14 By a semiretract of A ∗ , we mean an intersection of a finite number of retracts of A ∗ . Each retract of A ∗ is also a semiretract. The clearest example of a semiretract that is not a retract is the example given previously: R ∩ R ′ = ( d ( ab ) ∗ ac ) ∗ . Some pairs of retracts have as their intersection a retract: As stated above, but not to be demonstrated here, if fewer than four alpha… \end{verbatim} ``` </details>
511. ph-7a0f2d7163d3871d66d9 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.2.md ### Plain (markdown context) [Image](./AutomataTheory_chapter_1.2_artifacts/image_000007_c7ddbf19f5bcbf685107f350d041cd0044eb2ce56089a62dbd44bbeec04cf4da.png) In this automaton, if three consecutive b s are read, then the automaton is in state s 3, which is a sink state and is not an acceptance state. This is the only way to get to s 3 and every other state is an acceptance state. Thus the language accepted by this automaton consists of all words which do not have three consecutive b s. An expression for this language is As previously mentioned, the automata that we have been discussing are c… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{[Image](./AutomataTheory\_chapter\_1.2\_artifacts/image\_000007\_c7ddbf19f5bcbf685107f350d041cd0044eb2ce56089a62dbd44bbeec04cf4da.png) In this automaton, if three consecutive b s are read, then the automaton is in state s 3, which is a sink state and is not an acceptance state. This …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.2.md:offset=19700 \begin{verbatim} [Image](./AutomataTheory_chapter_1.2_artifacts/image_000007_c7ddbf19f5bcbf685107f350d041cd0044eb2ce56089a62dbd44bbeec04cf4da.png) In this automaton, if three consecutive b s are read, then the automaton is in state s 3, which is a sink state and is not an acceptance state. This is the only way to get to s 3 and every other state is an acceptance state. Thus the language accepted by this automaton consists of all words which do not have three consecutive b s. An expression for this language is As previously mentioned, the automata that we have been discussing are c… \end{verbatim} ``` </details>
512. ph-0a017bee004262d2794f — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.2.md ### Plain (markdown context) ϒ as a set of rules, given a ∈ /Sigma1 and s ∈ Q , the rules may allow advancement to each of several states or there may not be a rule which does not allow it to go to any state after reading a in state s . In the latter case, the automaton is 'hung up' and can proceed no further. This cannot occur with a deterministic automaton. Although the definition of a nondeterministic automaton varies, we shall use the following definition: ## Definition 3.2 A nondeterministic automaton , denoted by consists of a finite alphabet /Sigma1 , a finite set Q of states, and a fu… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ϒ as a set of rules, given a ∈ /Sigma1 and s ∈ Q , the rules may allow advancement to each of several states or there may not be a rule which does not allow it to go to any state after reading a in state s . In the latter case, the automaton is 'hung up' and can proceed no furth…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.2.md:offset=20628 \begin{verbatim} ϒ as a set of rules, given a ∈ /Sigma1 and s ∈ Q , the rules may allow advancement to each of several states or there may not be a rule which does not allow it to go to any state after reading a in state s . In the latter case, the automaton is 'hung up' and can proceed no further. This cannot occur with a deterministic automaton. Although the definition of a nondeterministic automaton varies, we shall use the following definition: ## Definition 3.2 A nondeterministic automaton , denoted by consists of a finite alphabet /Sigma1 , a finite set Q of states, and a fu… \end{verbatim} ``` </details>
513. ph-6aeaf8c43ff9d1c09fd4 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.2.md ### Plain (markdown context) there may not be a rule which does not allow it to go to any state after reading a in state s . In the latter case, the automaton is 'hung up' and can proceed no further. This cannot occur with a deterministic automaton. Although the definition of a nondeterministic automaton varies, we shall use the following definition: ## Definition 3.2 A nondeterministic automaton , denoted by consists of a finite alphabet /Sigma1 , a finite set Q of states, and a function called the transition function . The set Q contains an element s 0 and a sub… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{there may not be a rule which does not allow it to go to any state after reading a in state s . In the latter case, the automaton is 'hung up' and can proceed no further. This cannot occur with a deterministic automaton. Although the definition of a nondeterministic automaton va…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.2.md:offset=20740 \begin{verbatim} there may not be a rule which does not allow it to go to any state after reading a in state s . In the latter case, the automaton is 'hung up' and can proceed no further. This cannot occur with a deterministic automaton. Although the definition of a nondeterministic automaton varies, we shall use the following definition: ## Definition 3.2 A nondeterministic automaton , denoted by consists of a finite alphabet /Sigma1 , a finite set Q of states, and a function called the transition function . The set Q contains an element s 0 and a sub… \end{verbatim} ``` </details>
514. ph-5aa22333c1d480dfb239 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.2.md ### Plain (markdown context) of P ( Q ), i.e. the set of subsets of Q , as states for the deterministic automaton which we are constructing. Some of these states may not be used since they do not occur on any path which leads to acceptance state. Hence they could be removed and greatly simplify the deterministic automaton created. However, for our purpose, we are only interested in showing that a deterministic automaton can be created. In general we have the following procedure for constructing a deterministic automaton from a nondeterministic automaton. - (1) Beg… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{of P ( Q ), i.e. the set of subsets of Q , as states for the deterministic automaton which we are constructing. Some of these states may not be used since they do not occur on any path which leads to acceptance state. Hence they could be removed and greatly simplify the determin…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.2.md:offset=24331 \begin{verbatim} of P ( Q ), i.e. the set of subsets of Q , as states for the deterministic automaton which we are constructing. Some of these states may not be used since they do not occur on any path which leads to acceptance state. Hence they could be removed and greatly simplify the deterministic automaton created. However, for our purpose, we are only interested in showing that a deterministic automaton can be created. In general we have the following procedure for constructing a deterministic automaton from a nondeterministic automaton. - (1) Beg… \end{verbatim} ``` </details>
515. ph-8841d21466fb940c45cf — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.2.md ### Plain (markdown context) rministic automaton which we are constructing. Some of these states may not be used since they do not occur on any path which leads to acceptance state. Hence they could be removed and greatly simplify the deterministic automaton created. However, for our purpose, we are only interested in showing that a deterministic automaton can be created. In general we have the following procedure for constructing a deterministic automaton from a nondeterministic automaton. - (1) Begin with the state { s 0 } where s 0 is the start state of the non… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{rministic automaton which we are constructing. Some of these states may not be used since they do not occur on any path which leads to acceptance state. Hence they could be removed and greatly simplify the deterministic automaton created. However, for our purpose, we are only in…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.2.md:offset=24397 \begin{verbatim} rministic automaton which we are constructing. Some of these states may not be used since they do not occur on any path which leads to acceptance state. Hence they could be removed and greatly simplify the deterministic automaton created. However, for our purpose, we are only interested in showing that a deterministic automaton can be created. In general we have the following procedure for constructing a deterministic automaton from a nondeterministic automaton. - (1) Begin with the state { s 0 } where s 0 is the start state of the non… \end{verbatim} ``` </details>
516. ph-ecfc7d8d630f6072bae8 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.2.md ### Plain (markdown context) bb . Assume that we have ( si , a w ) w ∈ /Sigma1 + . Thus the automaton is in state si and must still read a followed by w . The notation ( si , a w ) /turnstileleft ( s j , w ) means that the automaton has read a and moved from state si to state s j . Therefore ϒ ( si , a ) = s j . In the automaton ![Image](./AutomataTheory_chapter_1.2_artifacts/image_000018_12d71b8e061a1adc45ccfe10f6a2453c723b9c3fd128d7412c290c5ffe402835.png) we have ( s 2 , bab ) /turnstileleft ( s 3 , ab ). We also have If we have ( si , w i ) /turnstileleft ( s j , w j ) /turnstileleft · · ·… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{bb . Assume that we have ( si , a w ) w ∈ /Sigma1 + . Thus the automaton is in state si and must still read a followed by w . The notation ( si , a w ) /turnstileleft ( s j , w ) means that the automaton has read a and moved from state si to state s j . Therefore ϒ ( si , a ) = …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.2.md:offset=28202 \begin{verbatim} bb . Assume that we have ( si , a w ) w ∈ /Sigma1 + . Thus the automaton is in state si and must still read a followed by w . The notation ( si , a w ) /turnstileleft ( s j , w ) means that the automaton has read a and moved from state si to state s j . Therefore ϒ ( si , a ) = s j . In the automaton ![Image](./AutomataTheory_chapter_1.2_artifacts/image_000018_12d71b8e061a1adc45ccfe10f6a2453c723b9c3fd128d7412c290c5ffe402835.png) we have ( s 2 , bab ) /turnstileleft ( s 3 , ab ). We also have If we have ( si , w i ) /turnstileleft ( s j , w j ) /turnstileleft · · ·… \end{verbatim} ``` </details>
517. ph-4d67538e3c8c1ec263b0 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.2.md ### Plain (markdown context) onstruction, the set of states of M ′ is a subset of P ( Q ). The state s ′ 0 = E ( s 0), and F ′ is a set containing an element of F . For each element a of /Sigma1 , define ϒ ′ by ϒ ′ ( P , a ) = ⋃ p ∈ P E ( ϒ ( p , a )). We first show that M ′ is deterministic. It is certainly single valued. Further ϒ ′ ( P , a ) will always have a value even if it is the empty set. We must now show that M ( L ) = M ′ ( L ). To do this we show that for any states p and q in Q , and any word w in /Sigma1 ∗ for some P containing q . From this it will follow that for some P containing q . From this it will follow that for some P containing q . From this it will follow that for some P containing f , where f ∈ F . and it must be shown that We prove … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{), and F ′ is a set containing an element of F . For each element a of /Sigma1 , define ϒ ′ by ϒ ′ ( P , a ) = ⋃ p ∈ P E ( ϒ ( p , a )). We first show that M ′ is deterministic. It is certainly single valued. Further ϒ ′ ( P , a ) will always have a value even if it is the empty…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.2.md:offset=34570 \begin{verbatim} ), and F ′ is a set containing an element of F . For each element a of /Sigma1 , define ϒ ′ by ϒ ′ ( P , a ) = ⋃ p ∈ P E ( ϒ ( p , a )). We first show that M ′ is deterministic. It is certainly single valued. Further ϒ ′ ( P , a ) will always have a value even if it is the empty set. We must now show that M ( L ) = M ′ ( L ). To do this we show that for any states p and q in Q , and any word w in /Sigma1 ∗ for some P containing q . From this it will follow that for some P containing f , where f ∈ F . and it must be shown that We prove … \end{verbatim} ``` </details>
519. ph-3f7dc9170e95d1b7bad0 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.2.md ### Plain (markdown context) . It is certainly single valued. Further ϒ ′ ( P , a ) will always have a value even if it is the empty set. We must now show that M ( L ) = M ′ ( L ). To do this we show that for any states p and q in Q , and any word w in /Sigma1 ∗ for some P containing q . From this it will follow that for some P containing f , where f ∈ F . and it must be shown that We prove this using induction of the length of w . If | w | = 0, then w = λ , for some P containing q . Now ( p , λ ) /turnstileleft ∗ ( q , λ ) if and only… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{. It is certainly single valued. Further ϒ ′ ( P , a ) will always have a value even if it is the empty set. We must now show that M ( L ) = M ′ ( L ). To do this we show that for any states p and q in Q , and any word w in /Sigma1 ∗ for some P conta…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.2.md:offset=34747 \begin{verbatim} . It is certainly single valued. Further ϒ ′ ( P , a ) will always have a value even if it is the empty set. We must now show that M ( L ) = M ′ ( L ). To do this we show that for any states p and q in Q , and any word w in /Sigma1 ∗ for some P containing q . From this it will follow that for some P containing f , where f ∈ F . and it must be shown that We prove this using induction of the length of w . If | w | = 0, then w = λ , for some P containing q . Now ( p , λ ) /turnstileleft ∗ ( q , λ ) if and only… \end{verbatim} ``` </details>
520. ph-6b6d36d2b4f2d0921d92 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.2.md ### Plain (markdown context) ula-not-decoded --> for some P containing q . Now ( p , λ ) /turnstileleft ∗ ( q , λ ) if and only if q ∈ E ( p ); but since M ′ is deterministic and no letter is read, then P = E ( p ) and p ∈ E ( p ) . Therefore the statement is true if | w | = 0. ⇒ : Assume w = v a for some letter a and w and ( p , w ) /turnstileleft ∗ ( q , λ ) so that Assume the statement is true for all strings having nonnegative length k . We now have to prove the statement is true for any string w with length k + 1. where at the end, possibly no letters of the alphabet are read. Since ( p … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ula-not-decoded --> for some P containing q . Now ( p , λ ) /turnstileleft ∗ ( q , λ ) if and only if q ∈ E ( p ); but since M ′ is deterministic and no letter is read, then P = E ( p ) and p ∈ E ( p ) . Therefore the statement is true if | w | = 0. ⇒ : Assume w = v a for some l…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.2.md:offset=35256 \begin{verbatim} ula-not-decoded --> for some P containing q . Now ( p , λ ) /turnstileleft ∗ ( q , λ ) if and only if q ∈ E ( p ); but since M ′ is deterministic and no letter is read, then P = E ( p ) and p ∈ E ( p ) . Therefore the statement is true if | w | = 0. ⇒ : Assume w = v a for some letter a and w and ( p , w ) /turnstileleft ∗ ( q , λ ) so that Assume the statement is true for all strings having nonnegative length k . We now have to prove the statement is true for any string w with length k + 1. where at the end, possibly no letters of the alphabet are read. Since ( p … \end{verbatim} ``` </details>
521. ph-ebcc37aa267cb7610fca — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.3.md ### Plain (markdown context) states. If they do, we can always relabel them. Since we want to begin in M 1, we let s 0 be the starting state of M so that s ′′ 0 = s 0. Since we want to finish in M 2, we let the set of acceptance states be F ′ so that F ′′ = F ′ . We define the rules for ϒ ′′ as follows. If the rule If in state si and a is read, go to state s j is in ϒ and s j is not an acceptance state then include this rule in ϒ ′′ . If s j is an acceptance state then include this rule in ϒ ′′ and also include the rule Hence there is the option of going to the state s j or skipping over to s… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{states. If they do, we can always relabel them. Since we want to begin in M 1, we let s 0 be the starting state of M so that s ′′ 0 = s 0. Since we want to finish in M 2, we let the set of acceptance states be F ′ so that F ′′ = F ′ . We define the rules for ϒ ′′ as follows. If …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.3.md:offset=515 \begin{verbatim} states. If they do, we can always relabel them. Since we want to begin in M 1, we let s 0 be the starting state of M so that s ′′ 0 = s 0. Since we want to finish in M 2, we let the set of acceptance states be F ′ so that F ′′ = F ′ . We define the rules for ϒ ′′ as follows. If the rule If in state si and a is read, go to state s j is in ϒ and s j is not an acceptance state then include this rule in ϒ ′′ . If s j is an acceptance state then include this rule in ϒ ′′ and also include the rule Hence there is the option of going to the state s j or skipping over to s… \end{verbatim} ``` </details>
522. ph-65198d25e5973a74fe80 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.3.md ### Plain (markdown context) ifacts/image_000020_475e3e83df6fe2ff7c71da51a51ef85161d8f8c7b8f17a885e0294af64d160d3.png) occurs, where e 1 , e 2 , e 3 , · · · , e k are regular expressions, then replace it with the diagram ![Image](./AutomataTheory_chapter_1.3_artifacts/image_000021_ad14692ce11cff48238ca8a33d702d3aa5aa2219c1008349b9b866e1e0773544.png) ![Image](./AutomataTheory_chapter_1.3_artifacts/image_000022_09a265b2e78d8222da71fc563beaf311e119039a9d8c7470becc6958af4dbabe.png) occurs, then replace it with the diagram More generally if the diagram ![Image](./AutomataTheory_chapter_1.3_artifac… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ifacts/image\_000020\_475e3e83df6fe2ff7c71da51a51ef85161d8f8c7b8f17a885e0294af64d160d3.png) occurs, where e 1 , e 2 , e 3 , · · · , e k are regular expressions, then replace it with the diagram ![Image](./AutomataTheory\_chapter\_1.3\_artifacts/image\_000021\_ad14692ce11cff48238ca8a33d…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.3.md:offset=14616 \begin{verbatim} ifacts/image_000020_475e3e83df6fe2ff7c71da51a51ef85161d8f8c7b8f17a885e0294af64d160d3.png) occurs, where e 1 , e 2 , e 3 , · · · , e k are regular expressions, then replace it with the diagram ![Image](./AutomataTheory_chapter_1.3_artifacts/image_000021_ad14692ce11cff48238ca8a33d702d3aa5aa2219c1008349b9b866e1e0773544.png) ![Image](./AutomataTheory_chapter_1.3_artifacts/image_000022_09a265b2e78d8222da71fc563beaf311e119039a9d8c7470becc6958af4dbabe.png) occurs, then replace it with the diagram More generally if the diagram ![Image](./AutomataTheory_chapter_1.3_artifac… \end{verbatim} ``` </details>
523. ph-bee0e0e505e4397124aa — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.3.md ### Plain (markdown context) 02d3aa5aa2219c1008349b9b866e1e0773544.png) ![Image](./AutomataTheory_chapter_1.3_artifacts/image_000022_09a265b2e78d8222da71fc563beaf311e119039a9d8c7470becc6958af4dbabe.png) occurs, then replace it with the diagram More generally if the diagram ![Image](./AutomataTheory_chapter_1.3_artifacts/image_000023_644de6e8f1660091c5d4019410542b2e049c5e8bd56e5fc133c0600717a04eb8.png) occurs, where e 1 , e 2 , e 3 are regular expressions, then replace it with the diagram In particular, when e 2 = λ , then e 1 e ∗ 2 e 3 becomes e 1 e 3 so that the … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{02d3aa5aa2219c1008349b9b866e1e0773544.png) ![Image](./AutomataTheory\_chapter\_1.3\_artifacts/image\_000022\_09a265b2e78d8222da71fc563beaf311e119039a9d8c7470becc6958af4dbabe.png) occurs, then replace it with the diagram More generally if the diagram ![Ima…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.3.md:offset=14898 \begin{verbatim} 02d3aa5aa2219c1008349b9b866e1e0773544.png) ![Image](./AutomataTheory_chapter_1.3_artifacts/image_000022_09a265b2e78d8222da71fc563beaf311e119039a9d8c7470becc6958af4dbabe.png) occurs, then replace it with the diagram More generally if the diagram ![Image](./AutomataTheory_chapter_1.3_artifacts/image_000023_644de6e8f1660091c5d4019410542b2e049c5e8bd56e5fc133c0600717a04eb8.png) occurs, where e 1 , e 2 , e 3 are regular expressions, then replace it with the diagram In particular, when e 2 = λ , then e 1 e ∗ 2 e 3 becomes e 1 e 3 so that the … \end{verbatim} ``` </details>
524. ph-204827a0d2679b0372bd — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.3.md ### Plain (markdown context) d8222da71fc563beaf311e119039a9d8c7470becc6958af4dbabe.png) occurs, then replace it with the diagram More generally if the diagram ![Image](./AutomataTheory_chapter_1.3_artifacts/image_000023_644de6e8f1660091c5d4019410542b2e049c5e8bd56e5fc133c0600717a04eb8.png) occurs, where e 1 , e 2 , e 3 are regular expressions, then replace it with the diagram In particular, when e 2 = λ , then e 1 e ∗ 2 e 3 becomes e 1 e 3 so that the diagram is replaced by the diagram ![Image](./AutomataTheory_chapter_1.3_artifacts/ima… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{d8222da71fc563beaf311e119039a9d8c7470becc6958af4dbabe.png) occurs, then replace it with the diagram More generally if the diagram ![Image](./AutomataTheory\_chapter\_1.3\_artifacts/image\_000023\_644de6e8f1660091c5d4019410542b2e049c5e8bd56e5fc133c0600717a…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.3.md:offset=15014 \begin{verbatim} d8222da71fc563beaf311e119039a9d8c7470becc6958af4dbabe.png) occurs, then replace it with the diagram More generally if the diagram ![Image](./AutomataTheory_chapter_1.3_artifacts/image_000023_644de6e8f1660091c5d4019410542b2e049c5e8bd56e5fc133c0600717a04eb8.png) occurs, where e 1 , e 2 , e 3 are regular expressions, then replace it with the diagram In particular, when e 2 = λ , then e 1 e ∗ 2 e 3 becomes e 1 e 3 so that the diagram is replaced by the diagram ![Image](./AutomataTheory_chapter_1.3_artifacts/ima… \end{verbatim} ``` </details>
525. ph-14f4a6afe3dd3a5f9a23 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.3.md ### Plain (markdown context) plete the proof, we need to show that R ( i , p , j ) is regular for 1 ≤ p ≤ n + 1 . We do this using induction. If p = 1, then there are no interior states in the path so R ( i , p , j ) = { a ∈ /Sigma1 : δ ( qi , a ) = qj } if i /negationslash= j and { λ } ∪ { a ∈ /Sigma1 : δ ( qi , a ) = qj } if i = j . Hence we have a finite set of elements of /Sigma1 and possibly λ in the set so it is a regular set. Assume R ( i , k , j ) is regular. The set of words R ( i , k + 1 , j ) can be defined as where the path from qi to qj may not pass through a state qm where m ≥ k… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{plete the proof, we need to show that R ( i , p , j ) is regular for 1 ≤ p ≤ n + 1 . We do this using induction. If p = 1, then there are no interior states in the path so R ( i , p , j ) = \{ a ∈ /Sigma1 : δ ( qi , a ) = qj \} if i /negationslash= j and \{ λ \} ∪ \{ a ∈ /Sigma1 : δ …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.3.md:offset=20725 \begin{verbatim} plete the proof, we need to show that R ( i , p , j ) is regular for 1 ≤ p ≤ n + 1 . We do this using induction. If p = 1, then there are no interior states in the path so R ( i , p , j ) = { a ∈ /Sigma1 : δ ( qi , a ) = qj } if i /negationslash= j and { λ } ∪ { a ∈ /Sigma1 : δ ( qi , a ) = qj } if i = j . Hence we have a finite set of elements of /Sigma1 and possibly λ in the set so it is a regular set. Assume R ( i , k , j ) is regular. The set of words R ( i , k + 1 , j ) can be defined as where the path from qi to qj may not pass through a state qm where m ≥ k… \end{verbatim} ``` </details>
526. ph-de30f9dd597310d2cbb5 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.3.md ### Plain (markdown context) g an interior state qm where m ≥ k . Since R ( i , k + 1 , j ) is formed using union, concatenation, and Kleene star of regular states, it is regular and hence L is regular. Since we have now shown that every regular expression is accepted by an automaton and that the language accepted by an automaton is regular, we have proven Kleene's Theorem. As a result of Kleene's Theorem, we discover two new properties about the regular languages: Theorem 3.4 If L 1 and L 2 are regular languages, then and L 1 ∩ L 2 are regular languages. Proof To show /Sigma1 ∗ -L 1 is regul… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{g an interior state qm where m ≥ k . Since R ( i , k + 1 , j ) is formed using union, concatenation, and Kleene star of regular states, it is regular and hence L is regular. Since we have now shown that every regular expression is accepted by an automaton and that the language a…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.3.md:offset=21504 \begin{verbatim} g an interior state qm where m ≥ k . Since R ( i , k + 1 , j ) is formed using union, concatenation, and Kleene star of regular states, it is regular and hence L is regular. Since we have now shown that every regular expression is accepted by an automaton and that the language accepted by an automaton is regular, we have proven Kleene's Theorem. As a result of Kleene's Theorem, we discover two new properties about the regular languages: Theorem 3.4 If L 1 and L 2 are regular languages, then and L 1 ∩ L 2 are regular languages. Proof To show /Sigma1 ∗ -L 1 is regul… \end{verbatim} ``` </details>
527. ph-737c2ab629b368a8dcee — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.3.md ### Plain (markdown context) t the automaton for /Sigma1 ∗ -L 1, simply change all of the terminal states in M 1 to nonterminal states and all of the nonterminal states to terminal states. As a result, all words that were accepted because the automaton stopped in a terminal state, are no longer accepted and all words which were not accepted are now accepted since the automaton will now stop in a terminal state after reading this word. To show that L 1 ∩ L 2 is a regular language we simply use the set theory property that This is most easily seen by thinking of /Sigma1 ∗ as the universe so tha… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{t the automaton for /Sigma1 ∗ -L 1, simply change all of the terminal states in M 1 to nonterminal states and all of the nonterminal states to terminal states. As a result, all words that were accepted because the automaton stopped in a terminal state, are no longer accepted and…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.3.md:offset=22170 \begin{verbatim} t the automaton for /Sigma1 ∗ -L 1, simply change all of the terminal states in M 1 to nonterminal states and all of the nonterminal states to terminal states. As a result, all words that were accepted because the automaton stopped in a terminal state, are no longer accepted and all words which were not accepted are now accepted since the automaton will now stop in a terminal state after reading this word. To show that L 1 ∩ L 2 is a regular language we simply use the set theory property that This is most easily seen by thinking of /Sigma1 ∗ as the universe so tha… \end{verbatim} ``` </details>
528. ph-11cc6b3a80d9e5c36e40 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.3.md ### Plain (markdown context) In the graph shown below, only one element is picked from each equivalence class. ## Therefore a minimal deterministic automaton is the automaton ![Image](./AutomataTheory_chapter_1.3_artifacts/image_000057_402bf5d85ac8e549b35ed4667cc383f7893f54705f75577e9b943cfeef6c2343.png) ## Example 3.24 Let M be the deterministic automaton ![Image](./AutomataTheory_chapter_1.3_artifacts/image_000058_b9837eae8db81f54654a098c80428de31704d024be0dbeef1dcffbeca985b0c2.png) The unmarked pairs in step 1 are The unmarked pairs in the first use of step 2 are The unmarked pairs in the first use of step 2 are The unmarked pairs in the first use of step 2 are The second use of step 2 produces no new results so the equivalence classes are… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{\#\# Therefore a minimal deterministic automaton is the automaton ![Image](./AutomataTheory\_chapter\_1.3\_artifacts/image\_000057\_402bf5d85ac8e549b35ed4667cc383f7893f54705f75577e9b943cfeef6c2343.png) \#\# Example 3.24 Let M be the deterministic automaton ![Image](./AutomataTheory\_chapt…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.3.md:offset=36826 \begin{verbatim} ## Therefore a minimal deterministic automaton is the automaton ![Image](./AutomataTheory_chapter_1.3_artifacts/image_000057_402bf5d85ac8e549b35ed4667cc383f7893f54705f75577e9b943cfeef6c2343.png) ## Example 3.24 Let M be the deterministic automaton ![Image](./AutomataTheory_chapter_1.3_artifacts/image_000058_b9837eae8db81f54654a098c80428de31704d024be0dbeef1dcffbeca985b0c2.png) The unmarked pairs in step 1 are The unmarked pairs in the first use of step 2 are The second use of step 2 produces no new results so the equivalence classes are… \end{verbatim} ``` </details>
530. ph-7727db2c5e54ccb5ed2c — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.3.md ### Plain (markdown context) is minimized version of the arbitrary automaton recognizing the language L is virtually identical with the intrinsic automaton of the language. Theorem 3.5 For a given regular language L, the two minimal reduced automaton developed above accepting language L are isomorphic. Proof M = ( /Sigma1 , Q , s 0 , ϒ ′ , F ), the minimal reduced automaton developed by the collapsing method is isomorphic to the intrinsic minimal automaton. So Mi = ( /Sigma1 , Qi , [1] , ϒ i , Fi ). Define f : Q → Qi by Assume [ x ] = … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{is minimized version of the arbitrary automaton recognizing the language L is virtually identical with the intrinsic automaton of the language. Theorem 3.5 For a given regular language L, the two minimal reduced automaton developed above accepting language L are isomorphic. Proo…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.3.md:offset=37779 \begin{verbatim} is minimized version of the arbitrary automaton recognizing the language L is virtually identical with the intrinsic automaton of the language. Theorem 3.5 For a given regular language L, the two minimal reduced automaton developed above accepting language L are isomorphic. Proof M = ( /Sigma1 , Q , s 0 , ϒ ′ , F ), the minimal reduced automaton developed by the collapsing method is isomorphic to the intrinsic minimal automaton. So Mi = ( /Sigma1 , Qi , [1] , ϒ i , Fi ). Define f : Q → Qi by Assume [ x ] = … \end{verbatim} ``` </details>
531. ph-8f25f47565d259be0027 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.3.md ### Plain (markdown context) bitrary automaton recognizing the language L is virtually identical with the intrinsic automaton of the language. Theorem 3.5 For a given regular language L, the two minimal reduced automaton developed above accepting language L are isomorphic. Proof M = ( /Sigma1 , Q , s 0 , ϒ ′ , F ), the minimal reduced automaton developed by the collapsing method is isomorphic to the intrinsic minimal automaton. So Mi = ( /Sigma1 , Qi , [1] , ϒ i , Fi ). Define f : Q → Qi by Assume [ x ] = [ y ], then ϒ ( x , u ) ∈ F if… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{bitrary automaton recognizing the language L is virtually identical with the intrinsic automaton of the language. Theorem 3.5 For a given regular language L, the two minimal reduced automaton developed above accepting language L are isomorphic. Proof M = ( /Sigma1 , Q , s 0 , ϒ …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.3.md:offset=37809 \begin{verbatim} bitrary automaton recognizing the language L is virtually identical with the intrinsic automaton of the language. Theorem 3.5 For a given regular language L, the two minimal reduced automaton developed above accepting language L are isomorphic. Proof M = ( /Sigma1 , Q , s 0 , ϒ ′ , F ), the minimal reduced automaton developed by the collapsing method is isomorphic to the intrinsic minimal automaton. So Mi = ( /Sigma1 , Qi , [1] , ϒ i , Fi ). Define f : Q → Qi by Assume [ x ] = [ y ], then ϒ ( x , u ) ∈ F if… \end{verbatim} ``` </details>
532. ph-f0118d7e65e293d68cb7 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.3.md ### Plain (markdown context) the language L is virtually identical with the intrinsic automaton of the language. Theorem 3.5 For a given regular language L, the two minimal reduced automaton developed above accepting language L are isomorphic. Proof M = ( /Sigma1 , Q , s 0 , ϒ ′ , F ), the minimal reduced automaton developed by the collapsing method is isomorphic to the intrinsic minimal automaton. So Mi = ( /Sigma1 , Qi , [1] , ϒ i , Fi ). Define f : Q → Qi by Assume [ x ] = [ y ], then ϒ ( x , u ) ∈ F if and only if ϒ ( y , u ) ∈ F f… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{the language L is virtually identical with the intrinsic automaton of the language. Theorem 3.5 For a given regular language L, the two minimal reduced automaton developed above accepting language L are isomorphic. Proof M = ( /Sigma1 , Q , s 0 , ϒ ′ , F ), the minimal reduced a…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.3.md:offset=37839 \begin{verbatim} the language L is virtually identical with the intrinsic automaton of the language. Theorem 3.5 For a given regular language L, the two minimal reduced automaton developed above accepting language L are isomorphic. Proof M = ( /Sigma1 , Q , s 0 , ϒ ′ , F ), the minimal reduced automaton developed by the collapsing method is isomorphic to the intrinsic minimal automaton. So Mi = ( /Sigma1 , Qi , [1] , ϒ i , Fi ). Define f : Q → Qi by Assume [ x ] = [ y ], then ϒ ( x , u ) ∈ F if and only if ϒ ( y , u ) ∈ F f… \end{verbatim} ``` </details>
533. ph-d9c0583cc257a70c2e3a — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.3.md ### Plain (markdown context) u ) ∈ F for u , v ∈ /Sigma1 ∗ . Let f ([ x ]) = [ w ] and f ([ y ]) = ([ w ′ ]). Then w u ∈ L if and only if w ′ u ∈ L ( = Fi ). Hence [ w ] = [ w ′ ] and f is well defined. Conversely, assume f ([ x ]) = f ([ y ]) then w u ∈ L if and only if w ′ u ∈ L ( = Fi ) where ϒ ( s 0 , w ) = x and ϒ ( s 0 , w ′ ) = y . Hence ϒ ( x , u ) ∈ F if and only if ϒ ( y , u ) ∈ F and [ x ] = [ y ]. Hence f is well defined and one-to-one. Finally we must show that f ( ϒ ′ ([ x ] , a )) = ϒ i ( f ([ x ]) , a ), Let w ∈ f ([ x ] , then ϒ ( s 0 , w ) = x fo… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{u ) ∈ F for u , v ∈ /Sigma1 ∗ . Let f ([ x ]) = [ w ] and f ([ y ]) = ([ w ′ ]). Then w u ∈ L if and only if w ′ u ∈ L ( = Fi ). Hence [ w ] = [ w ′ ] and f is well defined. Conversely, assume f ([ x ]) = f ([ y ]) then w u ∈ L if and only if w ′ u ∈ L ( = Fi ) where ϒ ( s 0 , w…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.3.md:offset=38434 \begin{verbatim} u ) ∈ F for u , v ∈ /Sigma1 ∗ . Let f ([ x ]) = [ w ] and f ([ y ]) = ([ w ′ ]). Then w u ∈ L if and only if w ′ u ∈ L ( = Fi ). Hence [ w ] = [ w ′ ] and f is well defined. Conversely, assume f ([ x ]) = f ([ y ]) then w u ∈ L if and only if w ′ u ∈ L ( = Fi ) where ϒ ( s 0 , w ) = x and ϒ ( s 0 , w ′ ) = y . Hence ϒ ( x , u ) ∈ F if and only if ϒ ( y , u ) ∈ F and [ x ] = [ y ]. Hence f is well defined and one-to-one. Finally we must show that f ( ϒ ′ ([ x ] , a )) = ϒ i ( f ([ x ]) , a ), Let w ∈ f ([ x ] , then ϒ ( s 0 , w ) = x fo… \end{verbatim} ``` </details>
534. ph-deda20be53d758893e00 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.3.md ### Plain (markdown context) . Let f ([ x ]) = [ w ] and f ([ y ]) = ([ w ′ ]). Then w u ∈ L if and only if w ′ u ∈ L ( = Fi ). Hence [ w ] = [ w ′ ] and f is well defined. Conversely, assume f ([ x ]) = f ([ y ]) then w u ∈ L if and only if w ′ u ∈ L ( = Fi ) where ϒ ( s 0 , w ) = x and ϒ ( s 0 , w ′ ) = y . Hence ϒ ( x , u ) ∈ F if and only if ϒ ( y , u ) ∈ F and [ x ] = [ y ]. Hence f is well defined and one-to-one. Finally we must show that f ( ϒ ′ ([ x ] , a )) = ϒ i ( f ([ x ]) , a ), Let w ∈ f ([ x ] , then ϒ ( s 0 , w ) = x for x ∈ [ x ]. Let Let w ∈ f ([ x ] , then ϒ ( s 0 , w ) = x for x ∈ [ x ]. Let Let w ∈ f ([ x ] , then ϒ ( s 0 , w ) = x for x ∈ [ x ]. Let and [ y ] = ϒ ′ ([ x ]] , a ) . Now ϒ ( s 0 , w a ) = y , so Let w ∈ f ([ x ] , then ϒ ( s 0 , w ) = x for x ∈ [ x ]. Let and [ y ] = ϒ ′ ([ x ]] , a ) . Now ϒ ( s 0 , w a ) = y , so Let w ∈ f ([ x ] , then ϒ ( s 0 , w ) = x for x ∈ [ x ]. Let and [ y ] = ϒ ′ ([ x ]] , a ) . Now ϒ ( s 0 , w a ) = y , so and so f ( ϒ ′ ([ x ] , a )) = ϒ i ( f ([ x ]) , a ). /square Corollary 3.1 … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{]) then w u ∈ L if and only if w ′ u ∈ L ( = Fi ) where ϒ ( s 0 , w ) = x and ϒ ( s 0 , w ′ ) = y . Hence ϒ ( x , u ) ∈ F if and only if ϒ ( y , u ) ∈ F and [ x ] = [ y ]. Hence f is well defined and one-to-one. Finally we must show that f ( ϒ ′ ([ x ] , a )) = ϒ i ( f ([ x ]) ,…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.3.md:offset=38648 \begin{verbatim} ]) then w u ∈ L if and only if w ′ u ∈ L ( = Fi ) where ϒ ( s 0 , w ) = x and ϒ ( s 0 , w ′ ) = y . Hence ϒ ( x , u ) ∈ F if and only if ϒ ( y , u ) ∈ F and [ x ] = [ y ]. Hence f is well defined and one-to-one. Finally we must show that f ( ϒ ′ ([ x ] , a )) = ϒ i ( f ([ x ]) , a ), Let w ∈ f ([ x ] , then ϒ ( s 0 , w ) = x for x ∈ [ x ]. Let and [ y ] = ϒ ′ ([ x ]] , a ) . Now ϒ ( s 0 , w a ) = y , so and so f ( ϒ ′ ([ x ] , a )) = ϒ i ( f ([ x ]) , a ). /square Corollary 3.1 … \end{verbatim} ``` </details>
537. ph-18e727db819b96297029 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.3.md ### Plain (markdown context) ] = ϒ ′ ([ x ]] , a ) . Now ϒ ( s 0 , w a ) = y , so and so f ( ϒ ′ ([ x ] , a )) = ϒ i ( f ([ x ]) , a ). /square Corollary 3.1 For a given regular language, all reduced automata which accept that language are unique up to isomorphism. Instead of looking at the syntactic monoid from the intrinsic point of view, as defined above we examine it using an automaton. In particular we look at minimal automata. The transformation monoid of a deterministic automaton is the image of a homomorphism ϕ from /Sigma1 ∗ to a submonoid TM of the monoi… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{] = ϒ ′ ([ x ]] , a ) . Now ϒ ( s 0 , w a ) = y , so and so f ( ϒ ′ ([ x ] , a )) = ϒ i ( f ([ x ]) , a ). /square Corollary 3.1 For a given regular language, all reduced automata which accept that language are unique up to isomorphism. Instead of lo…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.3.md:offset=39093 \begin{verbatim} ] = ϒ ′ ([ x ]] , a ) . Now ϒ ( s 0 , w a ) = y , so and so f ( ϒ ′ ([ x ] , a )) = ϒ i ( f ([ x ]) , a ). /square Corollary 3.1 For a given regular language, all reduced automata which accept that language are unique up to isomorphism. Instead of looking at the syntactic monoid from the intrinsic point of view, as defined above we examine it using an automaton. In particular we look at minimal automata. The transformation monoid of a deterministic automaton is the image of a homomorphism ϕ from /Sigma1 ∗ to a submonoid TM of the monoi… \end{verbatim} ``` </details>
538. ph-d03e02a4fdeb6e63d43c — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md ### Plain (markdown context) if ( si , u ) /turnstileleft ∗ ( s j , λ ). In other words, if the machine is in state si and reads u , then it is in state s j . Let M be the automaton ![Image](./AutomataTheory_chapter_1.4_artifacts/image_000000_7506cbfa55a9433240114a642749aa274accce799c3ef7e315e0f0d61494460b.png) For convenience, permutation notation is used here although the functions are not usually permutations, since they are not one-to-one. Thus we have which we shall shorten to Bydefinition let ¯ λ = ( 0 1 2 3 0 1 2 3 ) . We now pe… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{if ( si , u ) /turnstileleft ∗ ( s j , λ ). In other words, if the machine is in state si and reads u , then it is in state s j . Let M be the automaton ![Image](./AutomataTheory\_chapter\_1.4\_artifacts/image\_000000\_7506cbfa55a9433240114a642749aa274accce799c3ef7e315e0f0d61494460b.…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md:offset=287 \begin{verbatim} if ( si , u ) /turnstileleft ∗ ( s j , λ ). In other words, if the machine is in state si and reads u , then it is in state s j . Let M be the automaton ![Image](./AutomataTheory_chapter_1.4_artifacts/image_000000_7506cbfa55a9433240114a642749aa274accce799c3ef7e315e0f0d61494460b.png) For convenience, permutation notation is used here although the functions are not usually permutations, since they are not one-to-one. Thus we have which we shall shorten to Bydefinition let ¯ λ = ( 0 1 2 3 0 1 2 3 ) . We now pe… \end{verbatim} ``` </details>
539. ph-d53dd3dae790e12d24d5 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md ### Plain (markdown context) if ( si , u ) /turnstileleft ∗ ( s j , λ ). In other words, if the machine is in state si and reads u , then it is in state s j . Let M be the automaton ![Image](./AutomataTheory_chapter_1.4_artifacts/image_000000_7506cbfa55a9433240114a642749aa274accce799c3ef7e315e0f0d61494460b.png) For convenience, permutation notation is used here although the functions are not usually permutations, since they are not one-to-one. Thus we have which we shall shorten to Bydefinition let ¯ λ = ( 0 1 2 3 0 1 2 3 ) . We now pe… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{if ( si , u ) /turnstileleft ∗ ( s j , λ ). In other words, if the machine is in state si and reads u , then it is in state s j . Let M be the automaton ![Image](./AutomataTheory\_chapter\_1.4\_artifacts/image\_000000\_7506cbfa55a9433240114a642749aa274accce799c3ef7e315e0f0d61494460b.…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md:offset=466 \begin{verbatim} if ( si , u ) /turnstileleft ∗ ( s j , λ ). In other words, if the machine is in state si and reads u , then it is in state s j . Let M be the automaton ![Image](./AutomataTheory_chapter_1.4_artifacts/image_000000_7506cbfa55a9433240114a642749aa274accce799c3ef7e315e0f0d61494460b.png) For convenience, permutation notation is used here although the functions are not usually permutations, since they are not one-to-one. Thus we have which we shall shorten to Bydefinition let ¯ λ = ( 0 1 2 3 0 1 2 3 ) . We now pe… \end{verbatim} ``` </details>
540. ph-72e949c1be5a2ce7e0db — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md ### Plain (markdown context) eleft ∗ ( s j , λ ). In other words, if the machine is in state si and reads u , then it is in state s j . Let M be the automaton ![Image](./AutomataTheory_chapter_1.4_artifacts/image_000000_7506cbfa55a9433240114a642749aa274accce799c3ef7e315e0f0d61494460b.png) For convenience, permutation notation is used here although the functions are not usually permutations, since they are not one-to-one. Thus we have which we shall shorten to Bydefinition let ¯ λ = ( 0 1 2 3 0 1 2 3 ) . We now perform the following pro… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{eleft ∗ ( s j , λ ). In other words, if the machine is in state si and reads u , then it is in state s j . Let M be the automaton ![Image](./AutomataTheory\_chapter\_1.4\_artifacts/image\_000000\_7506cbfa55a9433240114a642749aa274accce799c3ef7e315e0f0d61494460b.png) For convenience, permutation notation is used here although the functions are not usually permutations, since they are not one-to-one. Thus we have which we shall shorten to Bydefinition let ¯ λ = ( 0 1 2 3 0 1 2 3 ) . We now perform the following pro… \end{verbatim} ```
541. ph-95b2d9d4481c03b8033f — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md ### Plain (markdown context) e the automaton ![Image](./AutomataTheory_chapter_1.4_artifacts/image_000000_7506cbfa55a9433240114a642749aa274accce799c3ef7e315e0f0d61494460b.png) For convenience, permutation notation is used here although the functions are not usually permutations, since they are not one-to-one. Thus we have which we shall shorten to Bydefinition let ¯ λ = ( 0 1 2 3 0 1 2 3 ) . We now perform the following products: For convenience, permutation notation is used here although the functions are not usually permutations,…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md:offset=638 \begin{verbatim} e the automaton ![Image](./AutomataTheory_chapter_1.4_artifacts/image_000000_7506cbfa55a9433240114a642749aa274accce799c3ef7e315e0f0d61494460b.png) For convenience, permutation notation is used here although the functions are not usually permutations, since they are not one-to-one. Thus we have which we shall shorten to Bydefinition let ¯ λ = ( 0 1 2 3 0 1 2 3 ) . We now perform the following products: For convenience, permutation notation is used here although the functions are not usually permutations, since they are not one-to-one. Thus we have which we shall shorten to Bydefinition let ¯ λ = ( 0 1 2 3 0 1 2 3 ) . We now perform the following products: For convenience, permutation notation is used here although the functions are not usually permutations, since they are not one-to-on…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md:offset=668 \begin{verbatim} tomataTheory_chapter_1.4_artifacts/image_000000_7506cbfa55a9433240114a642749aa274accce799c3ef7e315e0f0d61494460b.png) For convenience, permutation notation is used here although the functions are not usually permutations, since they are not one-to-one. Thus we have which we shall shorten to Bydefinition let ¯ λ = ( 0 1 2 3 0 1 2 3 ) . We now perform the following products: For convenience, permutation notation is used here although the functions are not usually permutations, since they are not one-to-one. Thus we have which we shall shorten to Bydefinition let ¯ λ = ( 0 1 2 3 0 1 2 3 ) . We now perform the following products: For convenience, permutation notation is used here although the functions are not usually permutations, since they are not one-to-one. Thus we have For convenience, permutation notation is used here although the functions are not usually permutations, since they are not one-to-one. Thus we have which we shall shorten to Bydefinition let ¯ λ = ( 0 1 2 3 0 1 2 3 ) . We now perform the following products: For convenience, permutation notation is used here although the functions are not usually permutations, since they are not one-to-one. Thus we have which we shall shorten to Bydefinition let ¯ λ = ( 0 1 2 3 0 1 2 3 ) . We now perform the following products: For convenience, permutation notation is used here although the functions are not usually permutations, since they are not one-to-one. Thus we have which we shall …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md:offset=728 \begin{verbatim} 433240114a642749aa274accce799c3ef7e315e0f0d61494460b.png) For convenience, permutation notation is used here although the functions are not usually permutations, since they are not one-to-one. Thus we have which we shall shorten to Bydefinition let ¯ λ = ( 0 1 2 3 0 1 2 3 ) . We now perform the following products: For convenience, permutation notation is used here although the functions are not usually permutations, since they are not one-to-one. Thus we have which we shall shorten to Bydefinition let ¯ λ = ( 0 1 2 3 0 1 2 3 ) . We now perform the following products: For convenience, permutation notation is used here although the functions are not usually permutations, since they are not one-to-one. Thus we have which we shall shorten to For convenience, permutation notation is used here although the functions are not usually permutations, since they are not one-to-one. Thus we have which we shall shorten to Bydefinition let ¯ λ = ( 0 1 2 3 0 1 2 3 ) . We now perform the following products: For convenience, permutation notation is used here although the functions are not usually permutations, since they are not one-to-one. Thus we have which we shall shorten to Bydefinition let ¯ λ = ( 0 1 2 3 0 1 2 3 ) . We now perform the following products: then Continuing this pr… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{!-- formula-not-decoded --> For convenience, permutation notation is used here although the functions are not usually permutations, since they are not one-to-one. Thus we have which we shall shorten to Bydefinition let ¯ …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md:offset=788 \begin{verbatim} !-- formula-not-decoded --> For convenience, permutation notation is used here although the functions are not usually permutations, since they are not one-to-one. Thus we have which we shall shorten to Bydefinition let ¯ λ = ( 0 1 2 3 0 1 2 3 ) . We now perform the following products: then Continuing this pr… \end{verbatim} ``` </details>
547. ph-2069646ad4c40843e82d — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md ### Plain (markdown context) or convenience, permutation notation is used here although the functions are not usually permutations, since they are not one-to-one. Thus we have which we shall shorten to Bydefinition let ¯ λ = ( 0 1 2 3 0 1 2 3 ) . We now perform the following products: then Continuing this process and letting γ = ¯ ab , … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{or convenience, permutation notation is used here although the functions are not usually permutations, since they are not one-to-one. Thus we have which we shall shorten to Bydefinition let ¯ λ = ( 0 1 2 3 0 1 2 3 ) . We …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md:offset=818 \begin{verbatim} or convenience, permutation notation is used here although the functions are not usually permutations, since they are not one-to-one. Thus we have which we shall shorten to Bydefinition let ¯ λ = ( 0 1 2 3 0 1 2 3 ) . We now perform the following products: then Continuing this process and letting γ = ¯ ab , … \end{verbatim} ``` </details>
548. ph-209a43a96eeaa645500e — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md ### Plain (markdown context) tation is used here although the functions are not usually permutations, since they are not one-to-one. Thus we have which we shall shorten to Bydefinition let ¯ λ = ( 0 1 2 3 0 1 2 3 ) . We now perform the following products: then Continuing this process and letting γ = ¯ ab , δ = ¯ a ¯ a , ε = ¯ b ¯ b , an… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{tation is used here although the functions are not usually permutations, since they are not one-to-one. Thus we have which we shall shorten to Bydefinition let ¯ λ = ( 0 1 2 3 0 1 2 3 ) . We now perform the following prod…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md:offset=848 \begin{verbatim} tation is used here although the functions are not usually permutations, since they are not one-to-one. Thus we have which we shall shorten to Bydefinition let ¯ λ = ( 0 1 2 3 0 1 2 3 ) . We now perform the following products: then Continuing this process and letting γ = ¯ ab , δ = ¯ a ¯ a , ε = ¯ b ¯ b , an… \end{verbatim} ``` </details>
549. ph-4bbe79ebe8a4d44816c4 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md ### Plain (markdown context) δ | ε | ζ | | ¯ a | ¯ a | δ | γ | γ | γ | γ | γ | | b | b | ζ | ε | ε | ε | ε | ε | | γ | γ | γ | γ | ε | γ | γ | γ | | δ | δ | γ | γ | γ | γ | γ | γ | | ε | ε | ε | ε | ε | ε | ε | ε | | ζ | ζ | ε | ε | ε | ε | γ | γ | Example 3.25 Let M be the automaton ![Image](./AutomataTheory_chapter_1.4_artifacts/image_000001_20f8b3cae2fe46086a361c7213d433fe5e6833c128955bf285fc1c3071bbb81d.png) The table for the transformation monoid TM is seen to be | | λ | ¯ a | b | γ | δ | ε | ζ | η | θ | ϑ | ι | κ | µ | |-----|-----|-------|-----|-----|-----|… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{δ | ε | ζ | | ¯ a | ¯ a | δ | γ | γ | γ | γ | γ | | b | b | ζ | ε | ε | ε | ε | ε | | γ | γ | γ | γ | ε | γ | γ | γ | | δ | δ | γ | γ | γ | γ | γ | γ | | ε | ε | ε | ε | ε | ε | ε | ε | | ζ | ζ | ε | ε | ε | ε | γ | γ | Example 3.25 Let M be the automaton ![Image](./AutomataTheo…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md:offset=1671 \begin{verbatim} δ | ε | ζ | | ¯ a | ¯ a | δ | γ | γ | γ | γ | γ | | b | b | ζ | ε | ε | ε | ε | ε | | γ | γ | γ | γ | ε | γ | γ | γ | | δ | δ | γ | γ | γ | γ | γ | γ | | ε | ε | ε | ε | ε | ε | ε | ε | | ζ | ζ | ε | ε | ε | ε | γ | γ | Example 3.25 Let M be the automaton ![Image](./AutomataTheory_chapter_1.4_artifacts/image_000001_20f8b3cae2fe46086a361c7213d433fe5e6833c128955bf285fc1c3071bbb81d.png) The table for the transformation monoid TM is seen to be | | λ | ¯ a | b | γ | δ | ε | ζ | η | θ | ϑ | ι | κ | µ | |-----|-----|-------|-----|-----|-----|… \end{verbatim} ``` </details>
550. ph-a0b1e185128fa01005a7 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md ### Plain (markdown context) | δ | γ | γ | γ | γ | γ | | b | b | ζ | ε | ε | ε | ε | ε | | γ | γ | γ | γ | ε | γ | γ | γ | | δ | δ | γ | γ | γ | γ | γ | γ | | ε | ε | ε | ε | ε | ε | ε | ε | | ζ | ζ | ε | ε | ε | ε | γ | γ | Example 3.25 Let M be the automaton ![Image](./AutomataTheory_chapter_1.4_artifacts/image_000001_20f8b3cae2fe46086a361c7213d433fe5e6833c128955bf285fc1c3071bbb81d.png) The table for the transformation monoid TM is seen to be | | λ | ¯ a | b | γ | δ | ε | ζ | η | θ | ϑ | ι | κ | µ | |-----|-----|-------|-----|-----|-----|-----|-----|-----|-----|… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{| δ | γ | γ | γ | γ | γ | | b | b | ζ | ε | ε | ε | ε | ε | | γ | γ | γ | γ | ε | γ | γ | γ | | δ | δ | γ | γ | γ | γ | γ | γ | | ε | ε | ε | ε | ε | ε | ε | ε | | ζ | ζ | ε | ε | ε | ε | γ | γ | Example 3.25 Let M be the automaton ![Image](./AutomataTheory\_chapter\_1.4\_artifacts…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md:offset=1701 \begin{verbatim} | δ | γ | γ | γ | γ | γ | | b | b | ζ | ε | ε | ε | ε | ε | | γ | γ | γ | γ | ε | γ | γ | γ | | δ | δ | γ | γ | γ | γ | γ | γ | | ε | ε | ε | ε | ε | ε | ε | ε | | ζ | ζ | ε | ε | ε | ε | γ | γ | Example 3.25 Let M be the automaton ![Image](./AutomataTheory_chapter_1.4_artifacts/image_000001_20f8b3cae2fe46086a361c7213d433fe5e6833c128955bf285fc1c3071bbb81d.png) The table for the transformation monoid TM is seen to be | | λ | ¯ a | b | γ | δ | ε | ζ | η | θ | ϑ | ι | κ | µ | |-----|-----|-------|-----|-----|-----|-----|-----|-----|-----|… \end{verbatim} ``` </details>
551. ph-674a1a9fc5376776daa1 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md ### Plain (markdown context) here exists u , v, w ∈ /Sigma1 ∗ , v /negationslash= λ such that z = u vw and u v k w ∈ L for all k ≥ 0 . The length of the string u w is less than or equal to n. Further if M is an automaton accepting the language L and M has q states, then n < q. It is possible to have the stronger statement that z = u vw where the length of u v is less than or equal to q. Proof Let L be accepted by the automaton M = ( /Sigma1 , Q , s 0 , ϒ, F ). Let ϒ ( si , ai ) = si + 1 for i = r to t ; denote this by Since L contains a word of length m , where m > q , say w = a 1 a 2 a… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{here exists u , v, w ∈ /Sigma1 ∗ , v /negationslash= λ such that z = u vw and u v k w ∈ L for all k ≥ 0 . The length of the string u w is less than or equal to n. Further if M is an automaton accepting the language L and M has q states, then n \< q. It is possible to have the …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md:offset=8639 \begin{verbatim} here exists u , v, w ∈ /Sigma1 ∗ , v /negationslash= λ such that z = u vw and u v k w ∈ L for all k ≥ 0 . The length of the string u w is less than or equal to n. Further if M is an automaton accepting the language L and M has q states, then n < q. It is possible to have the stronger statement that z = u vw where the length of u v is less than or equal to q. Proof Let L be accepted by the automaton M = ( /Sigma1 , Q , s 0 , ϒ, F ). Let ϒ ( si , ai ) = si + 1 for i = r to t ; denote this by Since L contains a word of length m , where m > q , say w = a 1 a 2 a… \end{verbatim} ``` </details>
552. ph-7a3093fd28f9580d0714 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md ### Plain (markdown context) + 1 for i = r to t ; denote this by Since L contains a word of length m , where m > q , say w = a 1 a 2 a 3 . . . am . Note that if ( s 1 , a 1 a 2 a 3 . . . am ) /turnstileleft ∗ ( sm , λ ), then sm is an acceptance state. Since m > q , in reading w , M must pass through the same state twice. Therefore ( s 1 , a 1 a 2 a 3 . . . aj -1 ) /turnstileleft ∗ ( sk , λ ) and ( s 1 , a 1 a 2 a 3 . . . ak -1 ) /turnstileleft ∗ ( sk , λ ) = for some j < k and both Thus Also ( s j , aj a j + 2 . . . ak -1 ) /… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{+ 1 for i = r to t ; denote this by Since L contains a word of length m , where m \> q , say w = a 1 a 2 a 3 . . . am . Note that if ( s 1 , a 1 a 2 a 3 . . . am ) /turnstileleft ∗ ( sm , λ ), then sm is an acceptance state. Since m \> q , in rea…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md:offset=9102 \begin{verbatim} + 1 for i = r to t ; denote this by Since L contains a word of length m , where m > q , say w = a 1 a 2 a 3 . . . am . Note that if ( s 1 , a 1 a 2 a 3 . . . am ) /turnstileleft ∗ ( sm , λ ), then sm is an acceptance state. Since m > q , in reading w , M must pass through the same state twice. Therefore ( s 1 , a 1 a 2 a 3 . . . aj -1 ) /turnstileleft ∗ ( sk , λ ) and ( s 1 , a 1 a 2 a 3 . . . ak -1 ) /turnstileleft ∗ ( sk , λ ) = for some j < k and both Thus Also ( s j , aj a j + 2 . . . ak -1 ) /… \end{verbatim} ``` </details>
553. ph-fcd30647464003488575 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md ### Plain (markdown context) Since L contains a word of length m , where m > q , say w = a 1 a 2 a 3 . . . am . Note that if ( s 1 , a 1 a 2 a 3 . . . am ) /turnstileleft ∗ ( sm , λ ), then sm is an acceptance state. Since m > q , in reading w , M must pass through the same state twice. Therefore ( s 1 , a 1 a 2 a 3 . . . aj -1 ) /turnstileleft ∗ ( sk , λ ) and ( s 1 , a 1 a 2 a 3 . . . ak -1 ) /turnstileleft ∗ ( sk , λ ) = for some j < k and both Thus Also ( s j , aj a j + 2 . . . ak -1 ) /turnstileleft ∗ s j , so in reading … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ Since L contains a word of length m , where m \> q , say w = a 1 a 2 a 3 . . . am . Note that if ( s 1 , a 1 a 2 a 3 . . . am ) /turnstileleft ∗ ( sm , λ ), then sm is an acceptance state. Since m \> q , in reading w , M must pass through the sam…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md:offset=9138 \begin{verbatim} Since L contains a word of length m , where m > q , say w = a 1 a 2 a 3 . . . am . Note that if ( s 1 , a 1 a 2 a 3 . . . am ) /turnstileleft ∗ ( sm , λ ), then sm is an acceptance state. Since m > q , in reading w , M must pass through the same state twice. Therefore ( s 1 , a 1 a 2 a 3 . . . aj -1 ) /turnstileleft ∗ ( sk , λ ) and ( s 1 , a 1 a 2 a 3 . . . ak -1 ) /turnstileleft ∗ ( sk , λ ) = for some j < k and both Thus Also ( s j , aj a j + 2 . . . ak -1 ) /turnstileleft ∗ s j , so in reading … \end{verbatim} ``` </details>
554. ph-0661f3638d27a44f9516 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md ### Plain (markdown context) urnstileleft ∗ ( sm , λ ), then sm is an acceptance state. Since m > q , in reading w , M must pass through the same state twice. Therefore ( s 1 , a 1 a 2 a 3 . . . aj -1 ) /turnstileleft ∗ ( sk , λ ) and ( s 1 , a 1 a 2 a 3 . . . ak -1 ) /turnstileleft ∗ ( sk , λ ) = for some j < k and both Thus Also ( s j , aj a j + 2 . . . ak -1 ) /turnstileleft ∗ s j , so in reading aj a j + 2 . . . ak -1 , M returns to the same state and Letting u = a 1 a 2 . . . aj -1 , v = aj a j + 2 . . . ak -1, and w = akak … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{urnstileleft ∗ ( sm , λ ), then sm is an acceptance state. Since m \> q , in reading w , M must pass through the same state twice. Therefore ( s 1 , a 1 a 2 a 3 . . . aj -1 ) /turnstileleft ∗ ( sk , λ ) and ( s 1 , a 1 a 2 a 3 . . . ak -1 ) /turnstileleft ∗ ( sk , λ ) = for so…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md:offset=9301 \begin{verbatim} urnstileleft ∗ ( sm , λ ), then sm is an acceptance state. Since m > q , in reading w , M must pass through the same state twice. Therefore ( s 1 , a 1 a 2 a 3 . . . aj -1 ) /turnstileleft ∗ ( sk , λ ) and ( s 1 , a 1 a 2 a 3 . . . ak -1 ) /turnstileleft ∗ ( sk , λ ) = for some j < k and both Thus Also ( s j , aj a j + 2 . . . ak -1 ) /turnstileleft ∗ s j , so in reading aj a j + 2 . . . ak -1 , M returns to the same state and Letting u = a 1 a 2 . . . aj -1 , v = aj a j + 2 . . . ak -1, and w = akak … \end{verbatim} ``` </details>
555. ph-b11962866ff9a0bd040f — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md ### Plain (markdown context) k b m = a m + k b m ∈ L , which is a contradiction. Second, u = a m , v = b k , and w = b m -k . Byasimilarargument,wereachacontradiction.Third u = a m -k , v = a k b r , and w = b m -r . But then a m -k a k b r a k b r b m -r ∈ L , which is a contradiction. Hence L is not regular. /square ## Exercises For each of the following sets, determine if the set is regular. If it is, describe the set with a regular expression. If it is not a regular set, use the Pumping lemma to show that it is not. - (2) { a n b 2 n a n : n ≥ 1 } . (3) { ( ab ) n : n ≥ 1 } . (4) { a n b … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{k b m = a m + k b m ∈ L , which is a contradiction. Second, u = a m , v = b k , and w = b m -k . Byasimilarargument,wereachacontradiction.Third u = a m -k , v = a k b r , and w = b m -r . But then a m -k a k b r a k b r b m -r ∈ L , which is a contradiction. Hence L is not regul…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md:offset=10753 \begin{verbatim} k b m = a m + k b m ∈ L , which is a contradiction. Second, u = a m , v = b k , and w = b m -k . Byasimilarargument,wereachacontradiction.Third u = a m -k , v = a k b r , and w = b m -r . But then a m -k a k b r a k b r b m -r ∈ L , which is a contradiction. Hence L is not regular. /square ## Exercises For each of the following sets, determine if the set is regular. If it is, describe the set with a regular expression. If it is not a regular set, use the Pumping lemma to show that it is not. - (2) { a n b 2 n a n : n ≥ 1 } . (3) { ( ab ) n : n ≥ 1 } . (4) { a n b … \end{verbatim} ``` </details>
556. ph-5ae941dce7b887158625 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md ### Plain (markdown context) e length is greater than n and less than 2 n. Proof First assume L is infinite. By the Pumping Lemma there exists u v m w ∈ L for all m ≥ 0. Further if M is an automaton accepting the language L and M has n states, then | u w | , the length of the string u w , is less than or equal to n . Assume that after u is read, the machine is in state s . If while reading v , the machine returns to s , let v ′ be the string that is read when the machine first returns to s and v ′ x = v . Thus if we have replace it with Thus M reads the string s 0… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{e length is greater than n and less than 2 n. Proof First assume L is infinite. By the Pumping Lemma there exists u v m w ∈ L for all m ≥ 0. Further if M is an automaton accepting the language L and M has n states, then | u w | , the length of the string u w , is less than or eq…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md:offset=14803 \begin{verbatim} e length is greater than n and less than 2 n. Proof First assume L is infinite. By the Pumping Lemma there exists u v m w ∈ L for all m ≥ 0. Further if M is an automaton accepting the language L and M has n states, then | u w | , the length of the string u w , is less than or equal to n . Assume that after u is read, the machine is in state s . If while reading v , the machine returns to s , let v ′ be the string that is read when the machine first returns to s and v ′ x = v . Thus if we have replace it with Thus M reads the string s 0… \end{verbatim} ``` </details>
557. ph-09ee689f0a75ab2f35d2 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md ### Plain (markdown context) less than 2 n. Proof First assume L is infinite. By the Pumping Lemma there exists u v m w ∈ L for all m ≥ 0. Further if M is an automaton accepting the language L and M has n states, then | u w | , the length of the string u w , is less than or equal to n . Assume that after u is read, the machine is in state s . If while reading v , the machine returns to s , let v ′ be the string that is read when the machine first returns to s and v ′ x = v . Thus if we have replace it with Thus M reads the string s 0 , u ( v ′ ) n w for any nonneg… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{less than 2 n. Proof First assume L is infinite. By the Pumping Lemma there exists u v m w ∈ L for all m ≥ 0. Further if M is an automaton accepting the language L and M has n states, then | u w | , the length of the string u w , is less than or equal to n . Assume that after u …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md:offset=14833 \begin{verbatim} less than 2 n. Proof First assume L is infinite. By the Pumping Lemma there exists u v m w ∈ L for all m ≥ 0. Further if M is an automaton accepting the language L and M has n states, then | u w | , the length of the string u w , is less than or equal to n . Assume that after u is read, the machine is in state s . If while reading v , the machine returns to s , let v ′ be the string that is read when the machine first returns to s and v ′ x = v . Thus if we have replace it with Thus M reads the string s 0 , u ( v ′ ) n w for any nonneg… \end{verbatim} ``` </details>
558. ph-fdf38431a2f6fa23aede — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md ### Plain (markdown context) The PDA, beginning at the left, reads a letter at a time in the same manner as a standard automaton. The PDA may read a letter from the tape or pop (remove from the top) and read a symbol from the stack or both. Depending on its current state and the symbol(s) read, the PDA may change state, push a symbol in the stack, or both. ![Image](./AutomataTheory_chapter_1.4_artifacts/image_000011_b968bb9607a88ba814f66590a1b105c8356ddd63139d322652c16e02b93f6004.png) We now define a PDA more formally. ## Definition 3.8 A pushdown automaton is a sextuple ## Definition 3.8 A pushdown automaton is a sextuple ## Definition 3.8 A pushdown automaton is a sextuple where /Sigma1 is a finite alphabet, Q is a finite set of states, s is the i… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{andard automaton. The PDA may read a letter from the tape or pop (remove from the top) and read a symbol from the stack or both. Depending on its current state and the symbol(s) read, the PDA may change state, push a symbol in the stack, or both. ![Image](./AutomataTheory\_chapte…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md:offset=20453 \begin{verbatim} andard automaton. The PDA may read a letter from the tape or pop (remove from the top) and read a symbol from the stack or both. Depending on its current state and the symbol(s) read, the PDA may change state, push a symbol in the stack, or both. ![Image](./AutomataTheory_chapter_1.4_artifacts/image_000011_b968bb9607a88ba814f66590a1b105c8356ddd63139d322652c16e02b93f6004.png) We now define a PDA more formally. ## Definition 3.8 A pushdown automaton is a sextuple where /Sigma1 is a finite alphabet, Q is a finite set of states, s is the i… \end{verbatim} ``` </details>
560. ph-f6a846363d13ad2ac0ea — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md ### Plain (markdown context) omataTheory_chapter_1.4_artifacts/image_000011_b968bb9607a88ba814f66590a1b105c8356ddd63139d322652c16e02b93f6004.png) We now define a PDA more formally. ## Definition 3.8 A pushdown automaton is a sextuple where /Sigma1 is a finite alphabet, Q is a finite set of states, s is the initial or starting state, I is a finite of stack symbols, ϒ is the transition relation and F is the set of acceptance states. The relation ϒ is a subset of Thus the relation reads a letter from /Sigma1 λ , determines the state, and … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{omataTheory\_chapter\_1.4\_artifacts/image\_000011\_b968bb9607a88ba814f66590a1b105c8356ddd63139d322652c16e02b93f6004.png) We now define a PDA more formally. \#\# Definition 3.8 A pushdown automaton is a sextuple where /Sigma1 is…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md:offset=20715 \begin{verbatim} omataTheory_chapter_1.4_artifacts/image_000011_b968bb9607a88ba814f66590a1b105c8356ddd63139d322652c16e02b93f6004.png) We now define a PDA more formally. ## Definition 3.8 A pushdown automaton is a sextuple where /Sigma1 is a finite alphabet, Q is a finite set of states, s is the initial or starting state, I is a finite of stack symbols, ϒ is the transition relation and F is the set of acceptance states. The relation ϒ is a subset of Thus the relation reads a letter from /Sigma1 λ , determines the state, and … \end{verbatim} ``` </details>
561. ph-da171a7ab2b0baa4a828 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md ### Plain (markdown context) t reads the language L = { w c w r : w ∈ { a , b } ∗ } . - (20) Let /Sigma1 = { a , b } . Construct a pushdown automaton that reads the language L = { w : The number of a s in w is equal to twice the number of b s or the number of b s in w is equal to three times the number of a s } . - (19) Let /Sigma1 = { a , b , c } . Construct a pushdown automaton that reads the language L = { w : The number of a s in w is equal to the sum of the number of b s and c s } . - (21) Given two pushdown automata over the same alphabet /Sigma1 and accepti… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{t reads the language L = \{ w c w r : w ∈ \{ a , b \} ∗ \} . - (20) Let /Sigma1 = \{ a , b \} . Construct a pushdown automaton that reads the language L = \{ w : The number of a s in w is equal to twice the number of b s or the number of b s in w is equal to three times the number of a…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md:offset=33020 \begin{verbatim} t reads the language L = { w c w r : w ∈ { a , b } ∗ } . - (20) Let /Sigma1 = { a , b } . Construct a pushdown automaton that reads the language L = { w : The number of a s in w is equal to twice the number of b s or the number of b s in w is equal to three times the number of a s } . - (19) Let /Sigma1 = { a , b , c } . Construct a pushdown automaton that reads the language L = { w : The number of a s in w is equal to the sum of the number of b s and c s } . - (21) Given two pushdown automata over the same alphabet /Sigma1 and accepti… \end{verbatim} ``` </details>
562. ph-270068a1aebc97b67393 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md ### Plain (markdown context) w r : w ∈ { a , b } ∗ } . - (20) Let /Sigma1 = { a , b } . Construct a pushdown automaton that reads the language L = { w : The number of a s in w is equal to twice the number of b s or the number of b s in w is equal to three times the number of a s } . - (19) Let /Sigma1 = { a , b , c } . Construct a pushdown automaton that reads the language L = { w : The number of a s in w is equal to the sum of the number of b s and c s } . - (21) Given two pushdown automata over the same alphabet /Sigma1 and accepting languages L and L ′ respecti… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{w r : w ∈ \{ a , b \} ∗ \} . - (20) Let /Sigma1 = \{ a , b \} . Construct a pushdown automaton that reads the language L = \{ w : The number of a s in w is equal to twice the number of b s or the number of b s in w is equal to three times the number of a s \} . - (19) Let /Sigma1 = \{ a…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md:offset=33050 \begin{verbatim} w r : w ∈ { a , b } ∗ } . - (20) Let /Sigma1 = { a , b } . Construct a pushdown automaton that reads the language L = { w : The number of a s in w is equal to twice the number of b s or the number of b s in w is equal to three times the number of a s } . - (19) Let /Sigma1 = { a , b , c } . Construct a pushdown automaton that reads the language L = { w : The number of a s in w is equal to the sum of the number of b s and c s } . - (21) Given two pushdown automata over the same alphabet /Sigma1 and accepting languages L and L ′ respecti… \end{verbatim} ``` </details>
563. ph-5af23f327cfc1cb6ee10 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md ### Plain (markdown context) automata over the same alphabet /Sigma1 and accepting languages L and L ′ respectively, - (a) Describe how to construct a pushdown automaton /Gamma1 1 that accepts the language L ∪ L ′ . - (b) Construct a pushdown automaton /Gamma1 1 that accepts the language L ∪ L ′ where L is the language accepted by the automaton in Example 3.26 and L ′ is the language accepted by the automaton in Example 3.27. - (22) Given two pushdown automata over the same alphabet /Sigma1 and accepting la… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{automata over the same alphabet /Sigma1 and accepting languages L and L ′ respectively, - (a) Describe how to construct a pushdown automaton /Gamma1 1 that accepts the language L ∪ L ′ . - (b) Construct a pushdown automat…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md:offset=33509 \begin{verbatim} automata over the same alphabet /Sigma1 and accepting languages L and L ′ respectively, - (a) Describe how to construct a pushdown automaton /Gamma1 1 that accepts the language L ∪ L ′ . - (b) Construct a pushdown automaton /Gamma1 1 that accepts the language L ∪ L ′ where L is the language accepted by the automaton in Example 3.26 and L ′ is the language accepted by the automaton in Example 3.27. - (22) Given two pushdown automata over the same alphabet /Sigma1 and accepting la… \end{verbatim} ``` </details>
564. ph-8288c5d5cc4f56dcdc0b — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md ### Plain (markdown context) coded --> over the same alphabet /Sigma1 and accepting languages L and L ′ respectively, - (a) Describe how to construct a pushdown automaton /Gamma1 1 that accepts the language L ∪ L ′ . - (b) Construct a pushdown automaton /Gamma1 1 that accepts the language L ∪ L ′ where L is the language accepted by the automaton in Example 3.26 and L ′ is the language accepted by the automaton in Example 3.27. - (22) Given two pushdown automata over the same alphabet /Sigma1 and accepting languages L and L ′ respective… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{coded --> over the same alphabet /Sigma1 and accepting languages L and L ′ respectively, - (a) Describe how to construct a pushdown automaton /Gamma1 1 that accepts the language L ∪ L ′ . - (b) Construct a pushdown automaton /Gamma1 1 that accepts th…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md:offset=33539 \begin{verbatim} coded --> over the same alphabet /Sigma1 and accepting languages L and L ′ respectively, - (a) Describe how to construct a pushdown automaton /Gamma1 1 that accepts the language L ∪ L ′ . - (b) Construct a pushdown automaton /Gamma1 1 that accepts the language L ∪ L ′ where L is the language accepted by the automaton in Example 3.26 and L ′ is the language accepted by the automaton in Example 3.27. - (22) Given two pushdown automata over the same alphabet /Sigma1 and accepting languages L and L ′ respective… \end{verbatim} ``` </details>
565. ph-09f52a0b8e362bc33c72 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md ### Plain (markdown context) utomaton in Example 3.27 . and and - (23) Given a pushdown automaton /Gamma1 = ( N , ϒ, S , P ) over the alphabet /Sigma1 and accepting language L , 2. (a) Describe how to construct a pushdown automaton /Gamma1 3 which accepts the language L ∗ . 3. (b) Construct a pushdown automaton /Gamma1 3 that accepts the language L ∪ L ′ where L is the language accepted by the automaton in Example 3.26 and L ′ is the language accepted by the automaton in Example 3.27 . - (24) Given two pushdown automata over the same alphabet /Sigma1 and accepting… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{utomaton in Example 3.27 . and and - (23) Given a pushdown automaton /Gamma1 = ( N , ϒ, S , P ) over the alphabet /Sigma1 and accepting language L , 2. (a) Describe how to construct a pushdown automaton /Gamma1 3 which accepts the language L ∗ . 3. (b) Construct a pushdown autom…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md:offset=34428 \begin{verbatim} utomaton in Example 3.27 . and and - (23) Given a pushdown automaton /Gamma1 = ( N , ϒ, S , P ) over the alphabet /Sigma1 and accepting language L , 2. (a) Describe how to construct a pushdown automaton /Gamma1 3 which accepts the language L ∗ . 3. (b) Construct a pushdown automaton /Gamma1 3 that accepts the language L ∪ L ′ where L is the language accepted by the automaton in Example 3.26 and L ′ is the language accepted by the automaton in Example 3.27 . - (24) Given two pushdown automata over the same alphabet /Sigma1 and accepting… \end{verbatim} ``` </details>
566. ph-a031d22fe63dc3e177dc — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md ### Plain (markdown context) d and - (23) Given a pushdown automaton /Gamma1 = ( N , ϒ, S , P ) over the alphabet /Sigma1 and accepting language L , 2. (a) Describe how to construct a pushdown automaton /Gamma1 3 which accepts the language L ∗ . 3. (b) Construct a pushdown automaton /Gamma1 3 that accepts the language L ∪ L ′ where L is the language accepted by the automaton in Example 3.26 and L ′ is the language accepted by the automaton in Example 3.27 . - (24) Given two pushdown automata over the same alphabet /Sigma1 and accepting languages L and L ′ respecti… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{d and - (23) Given a pushdown automaton /Gamma1 = ( N , ϒ, S , P ) over the alphabet /Sigma1 and accepting language L , 2. (a) Describe how to construct a pushdown automaton /Gamma1 3 which accepts the language L ∗ . 3. (b) Construct a pushdown automaton /Gamma1 3 that accepts t…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md:offset=34458 \begin{verbatim} d and - (23) Given a pushdown automaton /Gamma1 = ( N , ϒ, S , P ) over the alphabet /Sigma1 and accepting language L , 2. (a) Describe how to construct a pushdown automaton /Gamma1 3 which accepts the language L ∗ . 3. (b) Construct a pushdown automaton /Gamma1 3 that accepts the language L ∪ L ′ where L is the language accepted by the automaton in Example 3.26 and L ′ is the language accepted by the automaton in Example 3.27 . - (24) Given two pushdown automata over the same alphabet /Sigma1 and accepting languages L and L ′ respecti… \end{verbatim} ``` </details>
567. ph-31db4f23fc1645954b64 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md ### Plain (markdown context) B can be replaced by other symbols while + and the integers cannot be replaced. The symbols that can be replaced by other symbols are called nonterminal symbols and the symbols that can not be replaced by other symbols are called terminal symbols . We generate an element of the language when the string consists only of terminal symbols. The rules which tell us how to replace symbols are called productions . We denote the production (or rule) which tells us that add can be replaced with A + B Thus the productions for our first example above are Thus the productions for our first example above are Thus the productions for our first example above are Below, we shall expand our rules to do arbitrary addition, subtraction, mul… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{symbols that can be replaced by other symbols are called nonterminal symbols and the symbols that can not be replaced by other symbols are called terminal symbols . We generate an element of the language when the string consists only of terminal symbols. The rules which tell us …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md:offset=58693 \begin{verbatim} symbols that can be replaced by other symbols are called nonterminal symbols and the symbols that can not be replaced by other symbols are called terminal symbols . We generate an element of the language when the string consists only of terminal symbols. The rules which tell us how to replace symbols are called productions . We denote the production (or rule) which tells us that add can be replaced with A + B Thus the productions for our first example above are Below, we shall expand our rules to do arbitrary addition, subtraction, mul… \end{verbatim} ``` </details>
569. ph-1c3be9045ce8db61fc50 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md ### Plain (markdown context) set of nonterminal symbols N, a finite set of terminal symbols /Sigma1 , an element S ∈ N, called the start symbol and a finite set of productions P , which is a relation in ( N ∪ /Sigma1 ) ∗ such that each first element in an ordered pair of P contains a symbol from N and at least one production has S as the left string in some ordered pair. Definition 4.2 If W and W ′ are elements of ( N ∪ /Sigma1 ) ∗ , W = u vw , W ′ = u v ′ w , and v → v ′ is a production, this is denoted by W ⇒ W ′ . If for n ≥ 1 , then Wn is derived from W 1 . This is denoted by W 1 ⇒ ∗ n Wn… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{set of nonterminal symbols N, a finite set of terminal symbols /Sigma1 , an element S ∈ N, called the start symbol and a finite set of productions P , which is a relation in ( N ∪ /Sigma1 ) ∗ such that each first element in an ordered pair of P contains a symbol from N and at le…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md:offset=59526 \begin{verbatim} set of nonterminal symbols N, a finite set of terminal symbols /Sigma1 , an element S ∈ N, called the start symbol and a finite set of productions P , which is a relation in ( N ∪ /Sigma1 ) ∗ such that each first element in an ordered pair of P contains a symbol from N and at least one production has S as the left string in some ordered pair. Definition 4.2 If W and W ′ are elements of ( N ∪ /Sigma1 ) ∗ , W = u vw , W ′ = u v ′ w , and v → v ′ is a production, this is denoted by W ⇒ W ′ . If for n ≥ 1 , then Wn is derived from W 1 . This is denoted by W 1 ⇒ ∗ n Wn… \end{verbatim} ``` </details>
570. ph-16236a70280fb10ae004 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md ### Plain (markdown context) is denoted by W 1 ⇒ ∗ n Wn and is called a derivation . If the number of productions in not important we simply use W 1 ⇒ ∗ Wn. The set of all strings of elements of /Sigma1 which may be generated by the set of productions P is called the language generated by the grammar /Gamma1 and is denoted by /Gamma1 ( L ) . To generate a word from the grammar /Gamma1 , we keep using productions to derive new strings until we have a string consisting only of terminal elements. Thus in our example above, and where we will denote ( add , A + B ) by … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{is denoted by W 1 ⇒ ∗ n Wn and is called a derivation . If the number of productions in not important we simply use W 1 ⇒ ∗ Wn. The set of all strings of elements of /Sigma1 which may be generated by the set of productions P is called the language generated by the grammar /Gamma…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md:offset=60103 \begin{verbatim} is denoted by W 1 ⇒ ∗ n Wn and is called a derivation . If the number of productions in not important we simply use W 1 ⇒ ∗ Wn. The set of all strings of elements of /Sigma1 which may be generated by the set of productions P is called the language generated by the grammar /Gamma1 and is denoted by /Gamma1 ( L ) . To generate a word from the grammar /Gamma1 , we keep using productions to derive new strings until we have a string consisting only of terminal elements. Thus in our example above, and where we will denote ( add , A + B ) by … \end{verbatim} ``` </details>
571. ph-c59f4ad6fd602b906a45 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md ### Plain (markdown context) alled a derivation . If the number of productions in not important we simply use W 1 ⇒ ∗ Wn. The set of all strings of elements of /Sigma1 which may be generated by the set of productions P is called the language generated by the grammar /Gamma1 and is denoted by /Gamma1 ( L ) . To generate a word from the grammar /Gamma1 , we keep using productions to derive new strings until we have a string consisting only of terminal elements. Thus in our example above, and where we will denote ( add , A + B ) by add → A + B , ( A , A + B ) by A → … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{alled a derivation . If the number of productions in not important we simply use W 1 ⇒ ∗ Wn. The set of all strings of elements of /Sigma1 which may be generated by the set of productions P is called the language generated by the grammar /Gamma1 and is denoted by /Gamma1 ( L ) .…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md:offset=60138 \begin{verbatim} alled a derivation . If the number of productions in not important we simply use W 1 ⇒ ∗ Wn. The set of all strings of elements of /Sigma1 which may be generated by the set of productions P is called the language generated by the grammar /Gamma1 and is denoted by /Gamma1 ( L ) . To generate a word from the grammar /Gamma1 , we keep using productions to derive new strings until we have a string consisting only of terminal elements. Thus in our example above, and where we will denote ( add , A + B ) by add → A + B , ( A , A + B ) by A → … \end{verbatim} ``` </details>
572. ph-ee31e40309c4acdefb33 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md ### Plain (markdown context) derive new strings until we have a string consisting only of terminal elements. Thus in our example above, and where we will denote ( add , A + B ) by add → A + B , ( A , A + B ) by A → A + B , etc. If we eliminate the production ( B , A × B ), the language generated by /Gamma1 is the set of all formal expressions of finite sums of nonnegative integers less than 10. Example 4.1 In the grammar described above, derive the expression and where we will denote ( add , A + B ) by add → A + B , ( A , A + B ) by A → A + B , etc. If we eliminate the pr…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md:offset=60494 \begin{verbatim} derive new strings until we have a string consisting only of terminal elements. Thus in our example above, and where we will denote ( add , A + B ) by add → A + B , ( A , A + B ) by A → A + B , etc. If we eliminate the production ( B , A × B ), the language generated by /Gamma1 is the set of all formal expressions of finite sums of nonnegative integers less than 10. Example 4.1 In the grammar described above, derive the expression and where we will denote ( add , A + B ) by add → A + B , ( A , A + B ) by A → A + B , etc. If we eliminate the production ( B , A × B ), the language generated by /Gamma1 is the set of all formal expressions of finite sums of nonnegative integers less than 10. Example 4.1 In the grammar described above, derive the expression and where we will denote ( add , A + B ) by add → A + B , ( A , A + B ) by A → A + B , etc. If we eliminate the production ( B , A × B ), the la…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md:offset=60524 \begin{verbatim} ve a string consisting only of terminal elements. Thus in our example above, and where we will denote ( add , A + B ) by add → A + B , ( A , A + B ) by A → A + B , etc. If we eliminate the production ( B , A × B ), the language generated by /Gamma1 is the set of all formal expressions of finite sums of nonnegative integers less than 10. Example 4.1 In the grammar described above, derive the expression and where we will denote ( add , A + B ) by add → A + B , ( A , A + B ) by A → A + B , etc. If we eliminate the production ( B , A × B ), the language generated by /Gamma1 is the set of all formal expressions of finite sums of nonnegative integers less than 10. Example 4.1 In the grammar described above, derive the expression and where we will denote ( add , A + B ) by add → A + B , ( A , A + B ) by A → A + B , etc. If we eliminate the production ( B , A × B ), the language generated by /Gamma1 is …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md:offset=60554 \begin{verbatim} terminal elements. Thus in our example above, and where we will denote ( add , A + B ) by add → A + B , ( A , A + B ) by A → A + B , etc. If we eliminate the production ( B , A × B ), the language generated by /Gamma1 is the set of all formal expressions of finite sums of nonnegative integers less than 10. Example 4.1 In the grammar described above, derive the expression and where we will denote ( add , A + B ) by add → A + B , ( A , A + B ) by A → A + B , etc. If we eliminate the production ( B , A × B ), the language generated by /Gamma1 is the set of all formal expressions of finite sums of nonnegative integers less than 10. Example 4.1 In the grammar described above, derive the expression and where we will denote ( add , A + B ) by add → A + B , ( A , A + B ) by A → A + B , etc. If we eliminate the production ( B , A × B ), the language generated by /Gamma1 is the set of all formal expres…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md:offset=60584 \begin{verbatim} ur example above, and where we will denote ( add , A + B ) by add → A + B , ( A , A + B ) by A → A + B , etc. If we eliminate the production ( B , A × B ), the language generated by /Gamma1 is the set of all formal expressions of finite sums of nonnegative integers less than 10. Example 4.1 In the grammar described above, derive the expression and where we will denote ( add , A + B ) by add → A + B , ( A , A + B ) by A → A + B , etc. If we eliminate the production ( B , A × B ), the language generated by /Gamma1 is the set of all formal expressions of finite sums of nonnegative integers less than 10. Example 4.1 In the grammar described above, derive the expression Begin with the produc… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{a-not-decoded --> and where we will denote ( add , A + B ) by add → A + B , ( A , A + B ) by A → A + B , etc. If we eliminate the production ( B , A × B ), the language generated by /Gamma1 is the set of all formal expressions of finite sums of nonne…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md:offset=60614 \begin{verbatim} a-not-decoded --> and where we will denote ( add , A + B ) by add → A + B , ( A , A + B ) by A → A + B , etc. If we eliminate the production ( B , A × B ), the language generated by /Gamma1 is the set of all formal expressions of finite sums of nonnegative integers less than 10. Example 4.1 In the grammar described above, derive the expression Begin with the produc… \end{verbatim} ``` </details>
577. ph-8c4eaa07f3bdbc805c6d — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md ### Plain (markdown context) ormula-not-decoded --> where we will denote ( add , A + B ) by add → A + B , ( A , A + B ) by A → A + B , etc. If we eliminate the production ( B , A × B ), the language generated by /Gamma1 is the set of all formal expressions of finite sums of nonnegative integers less than 10. Example 4.1 In the grammar described above, derive the expression Begin with the production to derive Then use the … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ormula-not-decoded --> where we will denote ( add , A + B ) by add → A + B , ( A , A + B ) by A → A + B , etc. If we eliminate the production ( B , A × B ), the language generated by /Gamma1 is the set of all formal expressions of finite sums of nonnegative integers less than 10…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md:offset=60644 \begin{verbatim} ormula-not-decoded --> where we will denote ( add , A + B ) by add → A + B , ( A , A + B ) by A → A + B , etc. If we eliminate the production ( B , A × B ), the language generated by /Gamma1 is the set of all formal expressions of finite sums of nonnegative integers less than 10. Example 4.1 In the grammar described above, derive the expression Begin with the production to derive Then use the … \end{verbatim} ``` </details>
578. ph-e304eb0313abf40caa86 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md ### Plain (markdown context) we will denote ( add , A + B ) by add → A + B , ( A , A + B ) by A → A + B , etc. If we eliminate the production ( B , A × B ), the language generated by /Gamma1 is the set of all formal expressions of finite sums of nonnegative integers less than 10. Example 4.1 In the grammar described above, derive the expression Begin with the production to derive Then use the production to derive Then use… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{we will denote ( add , A + B ) by add → A + B , ( A , A + B ) by A → A + B , etc. If we eliminate the production ( B , A × B ), the language generated by /Gamma1 is the set of all formal expressions of finite sums of nonnegative integers less than 10. Example 4.1 In the grammar …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md:offset=60674 \begin{verbatim} we will denote ( add , A + B ) by add → A + B , ( A , A + B ) by A → A + B , etc. If we eliminate the production ( B , A × B ), the language generated by /Gamma1 is the set of all formal expressions of finite sums of nonnegative integers less than 10. Example 4.1 In the grammar described above, derive the expression Begin with the production to derive Then use the production to derive Then use… \end{verbatim} ``` </details>
579. ph-71726996016115ae291f — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md ### Plain (markdown context) s the set of all formal expressions of finite sums of nonnegative integers less than 10. Example 4.1 In the grammar described above, derive the expression Begin with the production to derive Then use the production to derive Then use the production to derive Then use the productions to derive 2 + 4 + 7 × 6. Note that we cannot derive Ex… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{s the set of all formal expressions of finite sums of nonnegative integers less than 10. Example 4.1 In the grammar described above, derive the expression Begin with the production to derive Then use the production to derive Then use the production to derive Then use the productions to derive 2 + 4 + 7 × 6. Note that we cannot derive Ex… \end{verbatim} ```
580. ph-b3864012d2fc7b044437 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md ### Plain (markdown context) n 10. Example 4.1 In the grammar described above, derive the expression Begin with the production to derive Then use the production to derive Then use the production to derive Then use the productions to derive 2 + 4 + 7 × 6. Note that we cannot derive Example 4.2 Suppose we want a grammar which derives arithmetic expressions for the se… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{n 10. Example 4.1 In the grammar described above, derive the expression Begi…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md:offset=60920 \begin{verbatim} n 10. Example 4.1 In the grammar described above, derive the expression Begin with the production to derive Then use the production to derive Then use the production to derive Then use the productions to derive 2 + 4 + 7 × 6. Note that we cannot derive Example 4.2 Suppose we want a grammar which derives arithmetic expressions for the se… \end{verbatim} ``` </details>
581. ph-4e7666ef57f2c09828bd — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md ### Plain (markdown context) 7 , 8 , 9 } . Thus the language generated by the grammar is the set of all finite arithmetic expressions for the set of integers { 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 } . Examples would be 3 × (5 + 4) and (4 + 5) ÷ (3ˆ2), where ˆ denotes exponent. As mentioned above, we obviously want to exclude expressions such as 3 +× 6 and 3 +÷ 6 × 4 -5 . Let the set N = { S , A , B } and /Sigma1 = {+ , -, × , ÷ , ˆ , 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , ( , ) } . Wewill need the following productions: We will use the grammar to derive the arithmetic expression We will use the grammar to derive the arithmetic expression We will use the grammar to derive the arithmetic expression We begin with the production We will use the grammar to derive the arithmetic expression We begin with the production We will use the grammar to derive the arithmetic expression We begin with the production We will use the grammar to derive the arithmetic expression We begin with the production We will use the grammar to derive the arithmetic expression We begin with the production We then use the pro… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{would be 3 × (5 + 4) and (4 + 5) ÷ (3ˆ2), where ˆ denotes exponent. As mentioned above, we obviously want to exclude expressions such as 3 +× 6 and 3 +÷ 6 × 4 -5 . Let the set N = \{ S , A , B \} and /Sigma1 = \{+ , -, × , ÷ , ˆ , 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , ( , ) \} . W…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md:offset=61760 \begin{verbatim} would be 3 × (5 + 4) and (4 + 5) ÷ (3ˆ2), where ˆ denotes exponent. As mentioned above, we obviously want to exclude expressions such as 3 +× 6 and 3 +÷ 6 × 4 -5 . Let the set N = { S , A , B } and /Sigma1 = {+ , -, × , ÷ , ˆ , 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , ( , ) } . Wewill need the following productions: We will use the grammar to derive the arithmetic expression We begin with the production We then use the pro… \end{verbatim} ``` </details>
585. ph-cb3edd743d09d9a9f07b — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md ### Plain (markdown context) 5) ÷ (3ˆ2), where ˆ denotes exponent. As mentioned above, we obviously want to exclude expressions such as 3 +× 6 and 3 +÷ 6 × 4 -5 . Let the set N = { S , A , B } and /Sigma1 = {+ , -, × , ÷ , ˆ , 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , ( , ) } . Wewill need the following productions: We will use the grammar to derive the arithmetic expression We begin with the production We then use the productions and to derive The pro… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{5) ÷ (3ˆ2), where ˆ denotes exponent. As mentioned above, we obviously want to exclude expressions such as 3 +× 6 and 3 +÷ 6 × 4 -5 . Let the set N = \{ S , A , B \} and /Sigma1 = \{+ , -, × , ÷ , ˆ , 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , ( , ) \} . Wewill need the following produ…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md:offset=61790 \begin{verbatim} 5) ÷ (3ˆ2), where ˆ denotes exponent. As mentioned above, we obviously want to exclude expressions such as 3 +× 6 and 3 +÷ 6 × 4 -5 . Let the set N = { S , A , B } and /Sigma1 = {+ , -, × , ÷ , ˆ , 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , ( , ) } . Wewill need the following productions: We will use the grammar to derive the arithmetic expression We begin with the production We then use the productions and to derive The pro… \end{verbatim} ``` </details>
586. ph-4e3996d2b8244fbcf04e — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md ### Plain (markdown context) xponent. As mentioned above, we obviously want to exclude expressions such as 3 +× 6 and 3 +÷ 6 × 4 -5 . Let the set N = { S , A , B } and /Sigma1 = {+ , -, × , ÷ , ˆ , 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , ( , ) } . Wewill need the following productions: We will use the grammar to derive the arithmetic expression We begin with the production We then use the productions and to derive The productions give us to derive to… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{xponent. As mentioned above, we obviously want to exclude expressions such as 3 +× 6 and 3 +÷ 6 × 4 -5 . Let the set N = \{ S , A , B \} and /Sigma1 = \{+ , -, × , ÷ , ˆ , 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , ( , ) \} . Wewill need the following productions: We will use the grammar to derive the arithmetic expression We begin with the production We then use the productions and to derive The productions give us to derive to… \end{verbatim} ```
587. ph-c1c9a1abcbfdf51e2391 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md ### Plain (markdown context) 1 = {+ , -, × , ÷ , ˆ , 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , ( , ) } . Wewill need the following productions: We will use the grammar to derive the arithmetic expression We begin with the production We then use the productions and to derive The productions give us to derive to derive The productions give us Finally we use the productions We will use the grammar to derive the arithmetic expression We begin with the production We will use the grammar to derive the arithmetic expression We begin with the production We then use the productions and to derive The productions give us to derive to derive The productions give us Finally we use the productions We will use the grammar to derive the arithmetic expression We begin with the production We then use the productions and to derive The productions give us to derive to derive The productions give us Finally we use the productions We will use the grammar to derive the arithmetic expression We begin with the production We will use the grammar to derive the arithmetic expression We begin with the production We then use the productions and to derive The productions give us to derive to derive The productions give us Finally we use the productions We will use the grammar to derive the arithmetic expression We begin with the production We then use the productions and to derive The productions give us to derive to derive The productions give us Finally we use the productions We next use the grammar to derive the arithmetic expressio… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{llowing productions: We will use the grammar to derive the arithmetic expression We begin with the production We will use the grammar to derive the arithmetic expression We begin with the production We then use the productions and to derive The productions give us to derive to derive The productions give us Finally we use the productions We next use the grammar to derive the arithmetic expressio… \end{verbatim} ```
590. ph-f55a3b09cddc3f37e35c — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md ### Plain (markdown context) mula-not-decoded --> We will use the grammar to derive the arithmetic expression We begin with the production We then use the productions and to derive The productions give us to derive to derive The productions give us Finally we use the productions We next use the grammar to derive the arithmetic expression We will use the grammar to derive the arithmetic expression We begin with the production We then use the producti…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md:offset=62087 \begin{verbatim} mula-not-decoded --> We will use the grammar to derive the arithmetic expression We begin with the production We then use the productions and to derive The productions give us to derive to derive The productions give us Finally we use the productions We next use the grammar to derive the arithmetic expression We begin with the production We then use the productions and to derive The productions give us to derive to derive The productions give us Finally we use the productions We next use the grammar to derive the arithmetic expression We begin with the production We then use the productions <… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{mula-not-decoded --> We begin with the production We then use the productions and to derive The productions give us to derive to derive The productions give us <!-…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md:offset=62178 \begin{verbatim} mula-not-decoded --> We begin with the production We then use the productions and to derive The productions give us to derive to derive The productions give us Finally we use the productions We next use the grammar to derive the arithmetic expression We begin with the production We then use the productions <… \end{verbatim} ``` </details>
592. ph-f54a205e537eaa02b519 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md ### Plain (markdown context) mula-not-decoded --> We then use the productions and to derive The productions give us to derive to derive The productions give us Finally we use the productions We next use the grammar to derive the arithmetic expression We begin with the production We then use the productions <… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{mula-not-decoded --> We then use the productions and to derive The productions give us to derive to derive The productions give us Fin…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md:offset=62238 \begin{verbatim} mula-not-decoded --> We then use the productions and to derive The productions give us to derive to derive The productions give us Finally we use the productions We next use the grammar to derive the arithmetic expression We begin with the production We then use the productions <… \end{verbatim} ``` </details>
593. ph-e7f726ed629f4209f8cf — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md ### Plain (markdown context) rmula-not-decoded --> We then use the productions and to derive The productions give us to derive to derive The productions give us Finally we use the productions We next use the grammar to derive the arithmetic expression We begin with the production We then use the productions … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{rmula-not-decoded --> We then use the productions and to derive The productions give us to derive to derive The productions give us Finally we use the productions …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md:offset=62297 \begin{verbatim} rmula-not-decoded --> We then use the productions and to derive The productions give us to derive to derive The productions give us Finally we use the productions We next use the grammar to derive the arithmetic expression We begin with the production We then use the productions … \end{verbatim} ``` </details>
594. ph-55d8dc4baef25d76d3c6 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md ### Plain (markdown context) rmula-not-decoded --> We then use the productions and to derive The productions give us to derive to derive The productions give us Finally we use the productions We next use the grammar to derive the arithmetic expression We begin with the production We then use the productions Finally we use the production… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{rmula-not-decoded --> We then use the productions and to derive The productions give us to derive to derive The productions give us Finally we use the productions …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md:offset=62327 \begin{verbatim} rmula-not-decoded --> We then use the productions and to derive The productions give us to derive to derive The productions give us Finally we use the productions We next use the grammar to derive the arithmetic expression We begin with the production We then use the productions Finally we use the production… \end{verbatim} ``` </details>
595. ph-baa674acfef695ff06b2 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md ### Plain (markdown context) use the productions and to derive The productions give us to derive to derive The productions give us Finally we use the productions We next use the grammar to derive the arithmetic expression We begin with the production We then use the productions Finally we use the productions … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{use the productions and to derive The productions give us to derive to derive The productions give us Finally we use the productions We next use the grammar to der…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md:offset=62357 \begin{verbatim} use the productions and to derive The productions give us to derive to derive The productions give us Finally we use the productions We next use the grammar to derive the arithmetic expression We begin with the production We then use the productions Finally we use the productions … \end{verbatim} ``` </details>
596. ph-9154363c786c3c75e259 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md ### Plain (markdown context) erive The productions give us to derive to derive The productions give us Finally we use the productions We next use the grammar to derive the arithmetic expression We begin with the production We then use the productions Finally we use the productions Finally we use the productions We next use the grammar to derive the arithmetic expressio…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md:offset=62387 \begin{verbatim} erive The productions give us to derive to derive The productions give us Finally we use the productions We next use the grammar to derive the arithmetic expression We begin with the production We then use the productions Finally we use the productions Finally we use the productions We next use the grammar to derive the arithmetic expression We begin with the production We then use the productions Finally we use the productions Example 4.3 In a similar manner, we may form arithmetic e… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ctions give us Finally we use the productions We next use the grammar to derive the arithmetic expression We begin with the production…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md:offset=62449 \begin{verbatim} ctions give us Finally we use the productions We next use the grammar to derive the arithmetic expression We begin with the production We then use the productions Finally we use the productions Example 4.3 In a similar manner, we may form arithmetic e… \end{verbatim} ``` </details>
598. ph-eca8ad9aae37f7da5927 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md ### Plain (markdown context) ot-decoded --> Finally we use the productions We next use the grammar to derive the arithmetic expression We begin with the production We then use the productions Finally we use the productions Example 4.3 In a similar manner, we may form arithmetic expressions in postfix notatio… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ot-decoded --> Finally we use the productions We next use the grammar to derive the arithmetic expression We begin with the production …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md:offset=62479 \begin{verbatim} ot-decoded --> Finally we use the productions We next use the grammar to derive the arithmetic expression We begin with the production We then use the productions Finally we use the productions Example 4.3 In a similar manner, we may form arithmetic expressions in postfix notatio… \end{verbatim} ``` </details>
599. ph-243ee12556e9ed7ca649 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md ### Plain (markdown context) grammar to derive the arithmetic expression We begin with the production We then use the productions Finally we use the productions Example 4.3 In a similar manner, we may form arithmetic expressions in postfix notation. Let the set N = { S , A , B } and to derive ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{grammar to derive the arithmetic expression We begin with the production We then use the productions Fina…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_1.4.md:offset=62633 \begin{verbatim} grammar to derive the arithmetic expression We begin with the production We then use the productions Finally we use the productions Example 4.3 In a similar manner, we may form arithmetic expressions in postfix notation. Let the set N = { S , A , B } and to derive \end{verbatim} ``` </details>
600. ph-964cafd34037d80af501 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) We will need the following productions: Consider the expression 3 2 + 4 7 +× . Since our integers are all less than ten, 3 2 + represents the integer symbol 3, followed by the integer symbol 2 and the + symbol. To construct this expression we begin with the production We then use the productions Finally we use the productions Consider the expression 3 2 + 4 7 +× . Since our integers are all less than ten, 3 2 + represents the integer symbol 3, followed by the integer symbol 2 and the + symbol. To construct this expression we begin w…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=41 \begin{verbatim} We will need the following productions: Consider the expression 3 2 + 4 7 +× . Since our integers are all less than ten, 3 2 + represents the integer symbol 3, followed by the integer symbol 2 and the + symbol. To construct this expression we begin with the production We then use the productions Finally we use the productions Consider the expression 3 2 + 4 7 +× . Since our integers are all less than ten, 3 2 + represents the integer symbol 3, followed by the integer symbol 2 and the + symbol. To construct this expression we begin with the production We then use the productions Finally we use the productions Example 4.4 Agrammar may also be used t… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{We will need the following productions: Consider the expression 3 2 + 4 7 +× . Since our integers are all less than ten, 3 2 + represents the integer symbol 3, followed by the integer symbol 2 and the + symbol. To construct this expression we begin w…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=301 \begin{verbatim} We will need the following productions: Consider the expression 3 2 + 4 7 +× . Since our integers are all less than ten, 3 2 + represents the integer symbol 3, followed by the integer symbol 2 and the + symbol. To construct this expression we begin with the production We then use the productions Finally we use the productions Example 4.4 Agrammar may also be used t… \end{verbatim} ``` </details>
602. ph-daf5ea917389ea143a40 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) We will need the following productions: Consider the expression 3 2 + 4 7 +× . Since our integers are all less than ten, 3 2 + represents the integer symbol 3, followed by the integer symbol 2 and the + symbol. To construct this expression we begin with the production We then use the productions Finally we use the productions Example 4.4 Agrammar may also be used t… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{We will need the following productions: Consider the expression 3 2 + 4 7 +× . Since our integers are all less than ten, 3 2 + represents the integer symbol 3, followed by the integer symbol 2 and the + symbol. To construct this expression we begin w…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=360 \begin{verbatim} We will need the following productions: Consider the expression 3 2 + 4 7 +× . Since our integers are all less than ten, 3 2 + represents the integer symbol 3, followed by the integer symbol 2 and the + symbol. To construct this expression we begin with the production We then use the productions Finally we use the productions Example 4.4 Agrammar may also be used t… \end{verbatim} ``` </details>
603. ph-eb43cc6a61a8c9aafef8 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) We will need the following productions: Consider the expression 3 2 + 4 7 +× . Since our integers are all less than ten, 3 2 + represents the integer symbol 3, followed by the integer symbol 2 and the + symbol. To construct this expression we begin with the production We then use the productions Finally we use the productions Example 4.4 Agrammar may also be used t… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{We will need the following productions: Consider the expression 3 2 + 4 7 +× . Since our integers are all less than ten, 3 2 + represents the integer symbol 3, followed by the integer symbol 2 and the + symbol. To construct this expression we begin w…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=390 \begin{verbatim} We will need the following productions: Consider the expression 3 2 + 4 7 +× . Since our integers are all less than ten, 3 2 + represents the integer symbol 3, followed by the integer symbol 2 and the + symbol. To construct this expression we begin with the production We then use the productions Finally we use the productions Example 4.4 Agrammar may also be used t… \end{verbatim} ``` </details>
604. ph-42392a03aa09d5622b7d — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) We will need the following productions: Consider the expression 3 2 + 4 7 +× . Since our integers are all less than ten, 3 2 + represents the integer symbol 3, followed by the integer symbol 2 and the + symbol. To construct this expression we begin with the production We then use the productions Finally we use the productions Example 4.4 Agrammar may also be used t… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{We will need the following productions: Consider the expression 3 2 + 4 7 +× . Since our integers are all less than ten, 3 2 + represents the integer symbol 3, followed by the integer symbol 2 and the + symbol. To construct this expression we begin w…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=420 \begin{verbatim} We will need the following productions: Consider the expression 3 2 + 4 7 +× . Since our integers are all less than ten, 3 2 + represents the integer symbol 3, followed by the integer symbol 2 and the + symbol. To construct this expression we begin with the production We then use the productions Finally we use the productions Example 4.4 Agrammar may also be used t… \end{verbatim} ``` </details>
605. ph-406d1df3d411c2c625a5 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) We will need the following productions: Consider the expression 3 2 + 4 7 +× . Since our integers are all less than ten, 3 2 + represents the integer symbol 3, followed by the integer symbol 2 and the + symbol. To construct this expression we begin with the production We then use the productions Finally we use the productions Example 4.4 Agrammar may also be used t… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{We will need the following productions: Consider the expression 3 2 + 4 7 +× . Since our integers are all less than ten, 3 2 + represents the integer symbol 3, followed by the integer symbol 2 and the + symbol. To construct this expression we begin w…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=450 \begin{verbatim} We will need the following productions: Consider the expression 3 2 + 4 7 +× . Since our integers are all less than ten, 3 2 + represents the integer symbol 3, followed by the integer symbol 2 and the + symbol. To construct this expression we begin with the production We then use the productions Finally we use the productions Example 4.4 Agrammar may also be used t… \end{verbatim} ``` </details>
606. ph-ab533d485be48724d13c — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) the following productions: Consider the expression 3 2 + 4 7 +× . Since our integers are all less than ten, 3 2 + represents the integer symbol 3, followed by the integer symbol 2 and the + symbol. To construct this expression we begin with the production We then use the productions Finally we use the productions Example 4.4 Agrammar may also be used to derive prop… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{the following productions: Consider the expression 3 2 + 4 7 +× . Since our integers are all less than ten, 3 2 + represents the integer symbol 3, followed by the integer symbol 2 and the + symbol. To construct this expression we begin with the produ…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=512 \begin{verbatim} the following productions: Consider the expression 3 2 + 4 7 +× . Since our integers are all less than ten, 3 2 + represents the integer symbol 3, followed by the integer symbol 2 and the + symbol. To construct this expression we begin with the production We then use the productions Finally we use the productions Example 4.4 Agrammar may also be used to derive prop… \end{verbatim} ``` </details>
607. ph-41d5d28be3c553d213e9 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) !-- formula-not-decoded --> Consider the expression 3 2 + 4 7 +× . Since our integers are all less than ten, 3 2 + represents the integer symbol 3, followed by the integer symbol 2 and the + symbol. To construct this expression we begin with the production We then use the productions Finally we use the productions Example 4.4 Agrammar may also be used to derive proper sentences. These sentence… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{!-- formula-not-decoded --> Consider the expression 3 2 + 4 7 +× . Since our integers are all less than ten, 3 2 + represents the integer symbol 3, followed by the integer symbol 2 and the + symbol. To construct this expression we begin with the production Consider the expression 3 2 + 4 7 +× . Since our integers are all less than ten, 3 2 + represents the integer symbol 3, followed by the integer symbol 2 and the + symbol. To construct this expression we begin with the production We then use the productions Finally we use the productions Example 4.4 Agrammar may also be used to derive proper sentences. These sentence… \end{verbatim} ```
608. ph-67b1c9929826e47c3d41 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) he fast horse leaped over the old fence. The cowboy rode slowly into the sunset. to derive The productions give us to derive Before actually stating the grammar let us decide upon its structure. This allows us to be assured that each sentence in the grammar is a grammatically correct sentence. Each of our sentences has a noun phrase (noun p), a verb phrase (verb p), and another noun phrase. In addition the last two sentences have a preposition (prep). Therefore let the first production be In our example, the most general form of a noun phrase is an article followe… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{he fast horse leaped over the old fence. The cowboy rode slowly into the sunset. to derive The productions give us to derive Before actually stating the grammar let us decide upon its structure. This allows us to be assured that each sentence in the grammar is a grammatically co…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=1362 \begin{verbatim} he fast horse leaped over the old fence. The cowboy rode slowly into the sunset. to derive The productions give us to derive Before actually stating the grammar let us decide upon its structure. This allows us to be assured that each sentence in the grammar is a grammatically correct sentence. Each of our sentences has a noun phrase (noun p), a verb phrase (verb p), and another noun phrase. In addition the last two sentences have a preposition (prep). Therefore let the first production be In our example, the most general form of a noun phrase is an article followe… \end{verbatim} ``` </details>
609. ph-028202bf1ad188d1aa18 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) upon its structure. This allows us to be assured that each sentence in the grammar is a grammatically correct sentence. Each of our sentences has a noun phrase (noun p), a verb phrase (verb p), and another noun phrase. In addition the last two sentences have a preposition (prep). Therefore let the first production be In our example, the most general form of a noun phrase is an article followed by an adjective and then a noun. Therefore let the next production be where 'art' represents article and 'adj' represents adjective The most gen… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{upon its structure. This allows us to be assured that each sentence in the grammar is a grammatically correct sentence. Each of our sentences has a noun phrase (noun p), a verb phrase (verb p), and another noun phrase. In addition the last two sentences have a preposition (prep)…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=1541 \begin{verbatim} upon its structure. This allows us to be assured that each sentence in the grammar is a grammatically correct sentence. Each of our sentences has a noun phrase (noun p), a verb phrase (verb p), and another noun phrase. In addition the last two sentences have a preposition (prep). Therefore let the first production be In our example, the most general form of a noun phrase is an article followed by an adjective and then a noun. Therefore let the next production be where 'art' represents article and 'adj' represents adjective The most gen… \end{verbatim} ``` </details>
610. ph-f93fe1c2c4034cf331cb — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) her noun phrase. In addition the last two sentences have a preposition (prep). Therefore let the first production be In our example, the most general form of a noun phrase is an article followed by an adjective and then a noun. Therefore let the next production be where 'art' represents article and 'adj' represents adjective The most general form of a verb phrase is a verb followed by an adverb. Therefore let the next production be where 'adv' represents adverb. At this point, we know that the terminal set … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{her noun phrase. In addition the last two sentences have a preposition (prep). Therefore let the first production be In our example, the most general form of a noun phrase is an article followed by an adjective and then a noun. Therefore let the next…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=1744 \begin{verbatim} her noun phrase. In addition the last two sentences have a preposition (prep). Therefore let the first production be In our example, the most general form of a noun phrase is an article followed by an adjective and then a noun. Therefore let the next production be where 'art' represents article and 'adj' represents adjective The most general form of a verb phrase is a verb followed by an adverb. Therefore let the next production be where 'adv' represents adverb. At this point, we know that the terminal set … \end{verbatim} ``` </details>
611. ph-e4c3156603a09ec4c775 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) ode, slowly, into, sunset } . The nonterminal set N = { S , < noun p >, < verb p >, < art > , < adj >, < noun > , < adv > , < verb > , < prep > } . We next need productions which will assign values to < art > , < adj >, < noun > , < adv > , and < verb > . In some of our sentences we do not need < art > , < adjective >, < prep > , and < adv > . To solve this problem, we include the productions By assigning these symbols to the empty set, we simply erase them when t… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ode, slowly, into, sunset \} . The nonterminal set N = \{ S , \< noun p \>, \< verb p \>, \< art \> , \< adj \>, \< noun \> , \< adv \> , \< verb \> , \< prep \> \} . We next need productions which will assign values to \< art \> , \< adj \>,…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=2439 \begin{verbatim} ode, slowly, into, sunset } . The nonterminal set N = { S , < noun p >, < verb p >, < art > , < adj >, < noun > , < adv > , < verb > , < prep > } . We next need productions which will assign values to < art > , < adj >, < noun > , < adv > , and < verb > . In some of our sentences we do not need < art > , < adjective >, < prep > , and < adv > . To solve this problem, we include the productions By assigning these symbols to the empty set, we simply erase them when t… \end{verbatim} ``` </details>
612. ph-c7e9c2ae5cb9107c3ed7 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) rep > } . We next need productions which will assign values to < art > , < adj >, < noun > , < adv > , and < verb > . In some of our sentences we do not need < art > , < adjective >, < prep > , and < adv > . To solve this problem, we include the productions By assigning these symbols to the empty set, we simply erase them when they are not needed. The remainder of our productions consists of the following: To derive the sentence 'Joe chased the dog,' we begin with By assigning these symbols to the empty set, we simply erase them when they are not needed. The remainder of our productions consists of the following: To derive the sentence 'Joe chased the dog,' we begin with By assigning these symbols to the empty set, we simply erase them when they are not needed. The remainder of our productions consists of the following: To derive the sentence 'Joe chased the dog,' we begin with to derive Using the production By assigning these symbo…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=2712 \begin{verbatim} j >, < noun > , < adv > , and < verb > . In some of our sentences we do not need < art > , < adjective >, < prep > , and < adv > . To solve this problem, we include the productions By assigning these symbols to the empty set, we simply erase them when they are not needed. The remainder of our productions consists of the following: To derive the sentence 'Joe chased the dog,' we begin with to derive Using the production By assigning these symbols to the empty set, we simply erase them when they are not needed. The remainder of our productions consists of the following: To derive the sentence 'Joe chased the dog,' we begin with to derive Using the production we derive By assigning these symbols to the empty set, we simply erase them…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=2753 \begin{verbatim} and < verb > . In some of our sentences we do not need < art > , < adjective >, < prep > , and < adv > . To solve this problem, we include the productions By assigning these symbols to the empty set, we simply erase them when they are not needed. The remainder of our productions consists of the following: To derive the sentence 'Joe chased the dog,' we begin with to derive Using the production we derive By assigning these symbols to the empty set, we simply erase them when they are not needed. The remainder of our productions consists of the following: To derive the sentence 'Joe chased the dog,' we begin with to derive Using the production we derive Using we derive By assigning these symbols to the empty set, we simply erase them when they are not needed. The remainder of our prod…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=2805 \begin{verbatim} not need < art > , < adjective >, < prep > , and < adv > . To solve this problem, we include the productions By assigning these symbols to the empty set, we simply erase them when they are not needed. The remainder of our productions consists of the following: To derive the sentence 'Joe chased the dog,' we begin with to derive Using the production we derive Using we derive By assigning these symbols to the empty set, we simply erase them when they are not needed. The remainder of our productions consists of the following: To derive the sentence 'Joe chased the dog,' we begin with to derive Using the production we derive Using we derive Repeati… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{gt;, \< prep \> , and \< adv \> . To solve this problem, we include the productions By assigning these symbols to the empty set, we simply erase them when they are not needed. The remainder of our productions consists of the following: By assigning these symbols to the empty set, we simply erase them when they are not needed. The remainder of our productions consists of the following: To derive the sentence 'Joe chased the dog,' we begin with to derive Using the production we derive Using we derive Repeati… \end{verbatim} ``` </details>
617. ph-a778d4826fc9121b69c7 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) lve this problem, we include the productions By assigning these symbols to the empty set, we simply erase them when they are not needed. The remainder of our productions consists of the following: To derive the sentence 'Joe chased the dog,' we begin with to derive Using the production we derive Using we derive Repeating the process for the second < noun phrase … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{lve this problem, we include the productions By assigning these symbols to the empty set, we simply erase them when they are not needed. The remainder of our productions consists of the following: To derive the sentence '…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=2893 \begin{verbatim} lve this problem, we include the productions By assigning these symbols to the empty set, we simply erase them when they are not needed. The remainder of our productions consists of the following: To derive the sentence 'Joe chased the dog,' we begin with to derive Using the production we derive Using we derive Repeating the process for the second < noun phrase … \end{verbatim} ``` </details>
618. ph-5204ceca81d15bb25087 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) he productions By assigning these symbols to the empty set, we simply erase them when they are not needed. The remainder of our productions consists of the following: To derive the sentence 'Joe chased the dog,' we begin with to derive Using the production we derive Using we derive Repeating the process for the second < noun phrase > , we derive By assigning these symbols to the empty set, we simply erase them when they are not needed. The remainder of our productions consists of the following: To derive the sentence 'Joe chased the dog,' we begin …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=2923 \begin{verbatim} he productions By assigning these symbols to the empty set, we simply erase them when they are not needed. The remainder of our productions consists of the following: To derive the sentence 'Joe chased the dog,' we begin with to derive Using the production we derive Using we derive Repeating the process for the second < noun phrase > , we derive To derive the sentence 'Joe chased the dog,' we begin with to derive Using the production we derive Using we derive Repeating the process for the second < noun phrase > , we derive Using we derive Using we derive Using we derive 'Joe ch… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{rase them when they are not needed. The remainder of our productions consists of the following: To derive the sentence 'Joe chased the dog,' we begin with to derive Using the production To derive the sentence 'Joe chased the dog,' we begin with to derive Using the production we derive Using we derive Repeating the process for the second < noun phrase > , we derive Using we derive Using we derive Using we derive 'Joe ch… \end{verbatim} ``` </details>
620. ph-e525ad5a92742ef9af05 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) nsists of the following: To derive the sentence 'Joe chased the dog,' we begin with to derive Using the production we derive Using we derive Repeating the process for the second < noun phrase > , we derive Using we derive Using we derive Using we derive 'Joe chased the dog.' To derive the sentence 'The fast horse leaped over the t… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{nsists of the following: To derive the sentence 'Joe chased the dog,' we begin with to derive Using the production we derive Using we …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=3096 \begin{verbatim} nsists of the following: To derive the sentence 'Joe chased the dog,' we begin with to derive Using the production we derive Using we derive Repeating the process for the second < noun phrase > , we derive Using we derive Using we derive Using we derive 'Joe chased the dog.' To derive the sentence 'The fast horse leaped over the t… \end{verbatim} ``` </details>
621. ph-0331afdc15e8f3499385 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) ive Using the production we derive Using we derive Repeating the process for the second < noun phrase > , we derive Using we derive Using we derive Using we derive 'Joe chased the dog.' To derive the sentence 'The fast horse leaped over the tall fence,' we again begin with to derive Using the production we derive Using we derive Repeating the process for the second \< noun phrase \> , we derive Using the production we derive Using we derive Repeating the process for the second < noun phrase > , we derive Using we derive Using we derive Using we derive 'Joe chased the dog.' To derive the sentence 'The fast horse leaped over the tall fence,' we again begin with to derive we derive Using we derive Repeating the process for the second < noun phrase > , we derive Using we derive Using we derive Using we derive 'Joe chased the dog.' To derive the sentence 'The fast horse leaped over the tall fence,' we again begin with to derive … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{the production we derive Using we derive Repeating the process for the second \< noun phrase \> , we derive Using we derive Using …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=3289 \begin{verbatim} the production we derive Using we derive Repeating the process for the second < noun phrase > , we derive Using we derive Using we derive Using we derive 'Joe chased the dog.' To derive the sentence 'The fast horse leaped over the tall fence,' we again begin with to derive … \end{verbatim} ``` </details>
623. ph-c07bf847685f649d278e — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) ot-decoded --> we derive Using we derive Repeating the process for the second < noun phrase > , we derive Using we derive Using we derive Using we derive 'Joe chased the dog.' To derive the sentence 'The fast horse leaped over the tall fence,' we again begin with to derive … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ot-decoded --> we derive Using we derive Repeating the process for the second \< noun phrase \> , we derive Using we derive Using we derive Using we derive Using we derive Repeating the process for the second < noun phrase > , we derive Using we derive Using we derive Using we derive 'Joe chased the dog.' To derive the sentence 'The fast horse leaped over the tall fence,' we again begin with to derive … \end{verbatim} ```
624. ph-fc125f9d75ab5174414c — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) - formula-not-decoded --> Using we derive Repeating the process for the second < noun phrase > , we derive Using we derive Using we derive Using we derive 'Joe chased the dog.' To derive the sentence 'The fast horse leaped over the tall fence,' we again begin with to derive … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{- formula-not-decoded --> Using we derive Repeating the process for the second \< noun phrase \> , we derive Using we derive Using we derive Using we derive 'J…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=3349 \begin{verbatim} - formula-not-decoded --> Using we derive Repeating the process for the second < noun phrase > , we derive Using we derive Using we derive Using we derive 'Joe chased the dog.' To derive the sentence 'The fast horse leaped over the tall fence,' we again begin with to derive … \end{verbatim} ``` </details>
625. ph-e0e3164319f9c07a5aa5 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) ng we derive Repeating the process for the second < noun phrase > , we derive Using we derive Using we derive Using we derive 'Joe chased the dog.' To derive the sentence 'The fast horse leaped over the tall fence,' we again begin with to derive Using the production Repeating the process for the second \< noun phrase \> , we derive Using we derive Using we derive Using we derive 'Joe chased the dog.' To derive…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=3379 \begin{verbatim} ng we derive Repeating the process for the second < noun phrase > , we derive Using we derive Using we derive Using we derive 'Joe chased the dog.' To derive the sentence 'The fast horse leaped over the tall fence,' we again begin with to derive Using the production Repeating the process for the second < noun phrase > , we derive Using we derive Using we derive Using we derive 'Joe chased the dog.' To derive the sentence 'The fast horse leaped over the tall fence,' we again begin with to derive Using the production we deri… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{-decoded --> Repeating the process for the second \< noun phrase \> , we derive Using we derive Using we derive Using we derive 'Joe chased the dog.' To derive the sentence 'The fast horse…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=3409 \begin{verbatim} -decoded --> Repeating the process for the second < noun phrase > , we derive Using we derive Using we derive Using we derive 'Joe chased the dog.' To derive the sentence 'The fast horse leaped over the tall fence,' we again begin with to derive Using the production we deri… \end{verbatim} ``` </details>
627. ph-8f39e8e392e2b1b7a2f2 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) -decoded --> Repeating the process for the second < noun phrase > , we derive Using we derive Using we derive Using we derive 'Joe chased the dog.' To derive the sentence 'The fast horse leaped over the tall fence,' we again begin with to derive Using the production we derive Repeating the process for the second \< noun phrase \> , we derive Using we derive Using we derive Using we derive 'Joe chased the dog.' To derive the sentence 'The fast horse leaped over the tall fence,'…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=3439 \begin{verbatim} -decoded --> Repeating the process for the second < noun phrase > , we derive Using we derive Using we derive Using we derive 'Joe chased the dog.' To derive the sentence 'The fast horse leaped over the tall fence,' we again begin with to derive Using the production we derive Using we derive Using we derive Using we derive 'Joe chased the dog.' To derive the sentence 'The fast horse leaped over the tall fence,' we again begin with to derive Using the production we derive Using < art > → the < art > → The &l… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{lt; noun phrase \> , we derive Using we derive Using we derive Using we derive 'Joe chased the dog.' To derive the sentence 'The fast horse leaped over the tall fence,' we again begin with t…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=3491 \begin{verbatim} lt; noun phrase > , we derive Using we derive Using we derive Using we derive 'Joe chased the dog.' To derive the sentence 'The fast horse leaped over the tall fence,' we again begin with to derive Using the production we derive Using < art > → the < art > → The &l… \end{verbatim} ``` </details>
629. ph-28f93981488e358217cf — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) rmula-not-decoded --> Using we derive Using we derive Using we derive 'Joe chased the dog.' To derive the sentence 'The fast horse leaped over the tall fence,' we again begin with to derive Using the production we derive Using < art > → the < art > → The < adj > → fast < noun > → hors… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{rmula-not-decoded --> Using we derive Using we derive Using we derive 'Joe chased the dog.' To derive the sentence 'The fast horse leaped over the tall fence,' we again begin with to derive <!…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=3532 \begin{verbatim} rmula-not-decoded --> Using we derive Using we derive Using we derive 'Joe chased the dog.' To derive the sentence 'The fast horse leaped over the tall fence,' we again begin with to derive Using the production we derive Using < art > → the < art > → The < adj > → fast < noun > → hors… \end{verbatim} ``` </details>
630. ph-deb68e13e18b9c1ce40e — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) ula-not-decoded --> Using the production we derive Using < art > → the < art > → The < adj > → fast < noun > → horse we derive The fast horse < verb p >< prep >< noun p > . Using we derive Using we derive Using we derive we derive Using The fast horse < adv >< verb >< prep >&lt… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ula-not-decoded --> Using the production we derive Using \< art \> → the…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=3798 \begin{verbatim} ula-not-decoded --> Using the production we derive Using < art > → the < art > → The < adj > → fast < noun > → horse we derive The fast horse < verb p >< prep >< noun p > . Using we derive Using we derive Using we derive we derive Using The fast horse < adv >< verb >< prep >&lt… \end{verbatim} ``` </details>
631. ph-c1ca9cdd22675c2715c3 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) ula-not-decoded --> Using the production we derive Using < art > → the < art > → The < adj > → fast < noun > → horse we derive The fast horse < verb p >< prep >< noun p > . Using we derive Using we derive Using we derive we derive Using The fast horse < adv >< verb >< prep >< noun p >. Using the production we derive Using \< art \> → the \< art \> → The \< adj…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=3828 \begin{verbatim} ula-not-decoded --> Using the production we derive Using < art > → the < art > → The < adj > → fast < noun > → horse we derive The fast horse < verb p >< prep >< noun p > . Using we derive Using we derive Using we derive we derive Using The fast horse < adv >< verb >< prep >< noun p >. Using the production we derive Using < art > → the < art > → The < adj > → fast < noun > → horse we derive The fast horse < verb p >< prep >< noun p > . Using we derive Using we derive Using we derive we derive Using The fast horse < adv >< verb >< prep >< noun p >. The fast horse leaped < prep >< noun p >. The f… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{--> Using the production we derive Using \< art \> → the \< art \> → The \< adj \> → fast \< noun \> → horse we derive The fast horse \< verb p \>\< prep \>\< noun p …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=3933 \begin{verbatim} --> Using the production we derive Using < art > → the < art > → The < adj > → fast < noun > → horse we derive The fast horse < verb p >< prep >< noun p > . Using we derive Using we derive Using we derive we derive Using The fast horse < adv >< verb >< prep >< noun p >. The fast horse leaped < prep >< noun p >. The f… \end{verbatim} ``` </details>
633. ph-6cc11a0c2c50ea17df37 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) > we derive Using < art > → the < art > → The < adj > → fast < noun > → horse we derive The fast horse < verb p >< prep >< noun p > . Using we derive Using we derive Using we derive we derive Using The fast horse < adv >< verb >< prep >< noun p >. The fast horse leaped < prep >< noun p >. The fast horse leaped over < noun p > . Using the production Using \< art \> → the \< art \> → The \< adj \> → fast \< noun \> → horse we derive The fast horse \< verb p \>\< prep \>\< noun p \> . Using we derive Using we derive Using we derive we derive Using Using < art > → the < art > → The < adj > → fast < noun > → horse we derive The fast horse < verb p >< prep >< noun p > . Using we derive Using we derive Using we derive we derive Using The fast horse < adv >< verb >< prep >< noun p >. The fast horse leaped < prep >< noun p >. The fast horse leaped over < noun p > . Using the production The fast horse < adv >< verb >< prep >< noun p >. The fast horse leaped < prep >< noun p >. The fast horse leaped over < noun p > . Using the production The fast horse leaped over < art >< adj >< noun >. we derive The fas… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{\> → fast \< noun \> → horse we derive The fast horse \< verb p \>\< prep \>\< noun p \> . Using we derive Using we derive Using we derive we derive Using The fast horse \< adv \>\< verb \>\< p…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=4117 \begin{verbatim} > → fast < noun > → horse we derive The fast horse < verb p >< prep >< noun p > . Using we derive Using we derive Using we derive we derive Using The fast horse < adv >< verb >< prep >< noun p >. The fast horse leaped < prep >< noun p >. The fast horse leaped over < noun p > . Using the production The fast horse leaped over < art >< adj >< noun >. we derive The fas… \end{verbatim} ``` </details>
635. ph-648b17b2318766d9b30c — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) s A and B are called vertices or nodes . The vertex B is called the child of A. Note that a terminal at a vertex has no children. Such a vertex is called a leaf of the tree. The leaves of the tree, when read left to right, form the word generated by the tree. If A 0 → A 1 →··· → An forms a string of edges in the tree then there is a path of length n from A 0 to An. Example 4.5 In Example 4.1, we used productions to derive 3 + 2 + 4. To construct the tree, begin with the first production used ![Image](./AutomataTheory_chapter_2.1_artifacts/image_000002_31981663902e… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{s A and B are called vertices or nodes . The vertex B is called the child of A. Note that a terminal at a vertex has no children. Such a vertex is called a leaf of the tree. The leaves of the tree, when read left to right, form the word generated by the tree. If A 0 → A 1 →··· →…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=5512 \begin{verbatim} s A and B are called vertices or nodes . The vertex B is called the child of A. Note that a terminal at a vertex has no children. Such a vertex is called a leaf of the tree. The leaves of the tree, when read left to right, form the word generated by the tree. If A 0 → A 1 →··· → An forms a string of edges in the tree then there is a path of length n from A 0 to An. Example 4.5 In Example 4.1, we used productions to derive 3 + 2 + 4. To construct the tree, begin with the first production used ![Image](./AutomataTheory_chapter_2.1_artifacts/image_000002_31981663902e… \end{verbatim} ``` </details>
636. ph-9ff2f85c643361173d56 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) he tree, begin with the first production used ![Image](./AutomataTheory_chapter_2.1_artifacts/image_000002_31981663902e05177e805f2cb4d7f27383ad59d3f2154e51b02e31ceeed028f9.png) A + B Then use the corresponding tree ![Image](./AutomataTheory_chapter_2.1_artifacts/image_000003_f1043d9fe7637e2d30e412f9281cf31ccb07a0f9d1ab6f893a6215b29ba5b003.png) + B to form corresponding tree of the production to form the tree of the production to get the corresponding tree ![Image](./AutomataTheory_chapter_2.1_artifacts/image_000004_3c3e35a481f705225443… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{he tree, begin with the first production used ![Image](./AutomataTheory\_chapter\_2.1\_artifacts/image\_000002\_31981663902e05177e805f2cb4d7f27383ad59d3f2154e51b02e31ceeed028f9.png) A + B Then use the corresponding tree ![Image](./AutomataTheory\_chapter\_2…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=5965 \begin{verbatim} he tree, begin with the first production used ![Image](./AutomataTheory_chapter_2.1_artifacts/image_000002_31981663902e05177e805f2cb4d7f27383ad59d3f2154e51b02e31ceeed028f9.png) A + B Then use the corresponding tree ![Image](./AutomataTheory_chapter_2.1_artifacts/image_000003_f1043d9fe7637e2d30e412f9281cf31ccb07a0f9d1ab6f893a6215b29ba5b003.png) + B to form corresponding tree of the production to form the tree of the production to get the corresponding tree ![Image](./AutomataTheory_chapter_2.1_artifacts/image_000004_3c3e35a481f705225443… \end{verbatim} ``` </details>
637. ph-53e8790b561338efba3e — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) cf31ccb07a0f9d1ab6f893a6215b29ba5b003.png) + B to form corresponding tree of the production to form the tree of the production to get the corresponding tree ![Image](./AutomataTheory_chapter_2.1_artifacts/image_000004_3c3e35a481f705225443a3d758dada607e5ff67f0f5280358b7951a1168ae84e.png) + B Then use the corresponding tree in ![Image](./AutomataTheory_chapter_2.1_artifacts/image_000005_fd4940c895584fb4cd623e92ec76fe9a29c4d5b7992273cf0017919daae279af.png) A ![Image](./AutomataTheory_chapter_2.1_artifacts/image_000006_8f464c1aaa302adf8d0d… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{cf31ccb07a0f9d1ab6f893a6215b29ba5b003.png) + B to form corresponding tree of the production to form the tree of the production to get the corresponding tree ![Image](./AutomataTheory\_chapter\_2.1\_artifacts/image\_000004\_3c3e35a481f705225443a3d758dada60…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=6303 \begin{verbatim} cf31ccb07a0f9d1ab6f893a6215b29ba5b003.png) + B to form corresponding tree of the production to form the tree of the production to get the corresponding tree ![Image](./AutomataTheory_chapter_2.1_artifacts/image_000004_3c3e35a481f705225443a3d758dada607e5ff67f0f5280358b7951a1168ae84e.png) + B Then use the corresponding tree in ![Image](./AutomataTheory_chapter_2.1_artifacts/image_000005_fd4940c895584fb4cd623e92ec76fe9a29c4d5b7992273cf0017919daae279af.png) A ![Image](./AutomataTheory_chapter_2.1_artifacts/image_000006_8f464c1aaa302adf8d0d… \end{verbatim} ``` </details>
638. ph-1f588c281a7b2457ee76 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) er_2.1_artifacts/image_000004_3c3e35a481f705225443a3d758dada607e5ff67f0f5280358b7951a1168ae84e.png) + B Then use the corresponding tree in ![Image](./AutomataTheory_chapter_2.1_artifacts/image_000005_fd4940c895584fb4cd623e92ec76fe9a29c4d5b7992273cf0017919daae279af.png) A ![Image](./AutomataTheory_chapter_2.1_artifacts/image_000006_8f464c1aaa302adf8d0d32b6a0379deddea07adea2a0e601e3f8088ebf7abd69.png) + Then use the corresponding trees of the next productions to form the parse tree ![Image](./AutomataTheory_chapter_2.1_artifacts/image_00… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{er\_2.1\_artifacts/image\_000004\_3c3e35a481f705225443a3d758dada607e5ff67f0f5280358b7951a1168ae84e.png) + B Then use the corresponding tree in ![Image](./AutomataTheory\_chapter\_2.1\_artifacts/image\_000005\_fd4940c895584fb4cd623e92ec76fe9a29c4d5b7992273cf0017919daae279af.png) A ![Image](./AutomataTheory_chapter_2.1_artifacts/image_000006_8f464c1aaa302adf8d0d32b6a0379deddea07adea2a0e601e3f8088ebf7abd69.png) + Then use the corresponding trees of the next productions to form the parse tree ![Image](./AutomataTheory_chapter_2.1_artifacts/image_00… \end{verbatim} ```
639. ph-5dc32d95ba1af06f34f3 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) ng) A ![Image](./AutomataTheory_chapter_2.1_artifacts/image_000006_8f464c1aaa302adf8d0d32b6a0379deddea07adea2a0e601e3f8088ebf7abd69.png) + Then use the corresponding trees of the next productions to form the parse tree ![Image](./AutomataTheory_chapter_2.1_artifacts/image_000007_d9d70e5d6b6249912379d04fd04d942bd2aca89439bd5420f1a3db3e0cc72a63.png) Example 4.6 In Example 4.3, to derive ((2 + 3) × (4 + 5)), we use the productions ## Therefore the parse tree is the tree ![Image](./AutomataTheory_chapter_2.1_ar… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ng) A ![Image](./AutomataTheory\_chapter\_2.1\_artifacts/image\_000006\_8f464c1aaa302adf8d0d32b6a0379deddea07adea2a0e601e3f8088ebf7abd69.png) + Then use the corresponding trees of the next productions to form the parse tree ![…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=6795 \begin{verbatim} ng) A ![Image](./AutomataTheory_chapter_2.1_artifacts/image_000006_8f464c1aaa302adf8d0d32b6a0379deddea07adea2a0e601e3f8088ebf7abd69.png) + Then use the corresponding trees of the next productions to form the parse tree ![Image](./AutomataTheory_chapter_2.1_artifacts/image_000007_d9d70e5d6b6249912379d04fd04d942bd2aca89439bd5420f1a3db3e0cc72a63.png) Example 4.6 In Example 4.3, to derive ((2 + 3) × (4 + 5)), we use the productions ## Therefore the parse tree is the tree ![Image](./AutomataTheory_chapter_2.1_ar… \end{verbatim} ``` </details>
640. ph-aa59c56ec61a88ae4798 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) tomataTheory_chapter_2.1_artifacts/image_000007_d9d70e5d6b6249912379d04fd04d942bd2aca89439bd5420f1a3db3e0cc72a63.png) Example 4.6 In Example 4.3, to derive ((2 + 3) × (4 + 5)), we use the productions ## Therefore the parse tree is the tree ![Image](./AutomataTheory_chapter_2.1_artifacts/image_000008_f517b05d44932b97f57aacbeaeb3471179ae3470c7481e5271996abcb82ee993.png) 4 Example 4.7 In Example 4.4, to derive the sentence 'Joe chased the dog,' using productions to get to get < art >< adj >< noun >< verb p >< pr… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{tomataTheory\_chapter\_2.1\_artifacts/image\_000007\_d9d70e5d6b6249912379d04fd04d942bd2aca89439bd5420f1a3db3e0cc72a63.png) Example 4.6 In Example 4.3, to derive ((2 + 3) × (4 + 5)), we use the productions \#\# Therefore the parse tree is the tree ![Image](.…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=7093 \begin{verbatim} tomataTheory_chapter_2.1_artifacts/image_000007_d9d70e5d6b6249912379d04fd04d942bd2aca89439bd5420f1a3db3e0cc72a63.png) Example 4.6 In Example 4.3, to derive ((2 + 3) × (4 + 5)), we use the productions ## Therefore the parse tree is the tree ![Image](./AutomataTheory_chapter_2.1_artifacts/image_000008_f517b05d44932b97f57aacbeaeb3471179ae3470c7481e5271996abcb82ee993.png) 4 Example 4.7 In Example 4.4, to derive the sentence 'Joe chased the dog,' using productions to get to get < art >< adj >< noun >< verb p >< pr… \end{verbatim} ``` </details>
641. ph-4d0f79d8e84388783b0e — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) le 4.3, to derive ((2 + 3) × (4 + 5)), we use the productions ## Therefore the parse tree is the tree ![Image](./AutomataTheory_chapter_2.1_artifacts/image_000008_f517b05d44932b97f57aacbeaeb3471179ae3470c7481e5271996abcb82ee993.png) 4 Example 4.7 In Example 4.4, to derive the sentence 'Joe chased the dog,' using productions to get to get < art >< adj >< noun >< verb p >< prep >< noun p > Using \#\# Therefore the parse tree is the tree ![Image](./AutomataTheory\_chapter\_2.1\_artifacts/image\_000008\_f517b05d44932b97f57aacbeaeb3471179ae3470c7481e5271996abcb82ee993.png) 4 Example 4.7 In …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=7232 \begin{verbatim} le 4.3, to derive ((2 + 3) × (4 + 5)), we use the productions ## Therefore the parse tree is the tree ![Image](./AutomataTheory_chapter_2.1_artifacts/image_000008_f517b05d44932b97f57aacbeaeb3471179ae3470c7481e5271996abcb82ee993.png) 4 Example 4.7 In Example 4.4, to derive the sentence 'Joe chased the dog,' using productions to get to get < art >< adj >< noun >< verb p >< prep >< noun p > Using ## Therefore the parse tree is the tree ![Image](./AutomataTheory_chapter_2.1_artifacts/image_000008_f517b05d44932b97f57aacbeaeb3471179ae3470c7481e5271996abcb82ee993.png) 4 Example 4.7 In Example 4.4, to derive the sentence 'Joe chased the dog,' using productions to get to get < art >< adj >< noun >< verb p >< prep >< noun p > Using \#\# Therefore the parse tree is the tree ![Image](./AutomataTheory\_chapter\_2.1\_artifacts/image\_000008\_f517b05d44932b97f57aacbeaeb3471179ae3470c7481e5271996abcb82ee993.png) 4 Example 4.7 In Example 4.4, to derive the sen…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=7262 \begin{verbatim} 4 + 5)), we use the productions ## Therefore the parse tree is the tree ![Image](./AutomataTheory_chapter_2.1_artifacts/image_000008_f517b05d44932b97f57aacbeaeb3471179ae3470c7481e5271996abcb82ee993.png) 4 Example 4.7 In Example 4.4, to derive the sentence 'Joe chased the dog,' using productions to get to get < art >< adj >< noun >< verb p >< prep >< noun p > Using ## Therefore the parse tree is the tree ![Image](./AutomataTheory_chapter_2.1_artifacts/image_000008_f517b05d44932b97f57aacbeaeb3471179ae3470c7481e5271996abcb82ee993.png) 4 Example 4.7 In Example 4.4, to derive the sentence 'Joe chased the dog,' using productions to get to get < art >< adj >< noun >< verb p >< prep >< noun p > Using \#\# Therefore the parse tree is the tree ![Image](./AutomataTheory\_chapter\_2.1\_artifacts/image\_000008\_f517b05d44932b97f57aacbeaeb3471179ae3470c7481e5271996abcb82ee993.png) 4 Example 4.7 In Example 4.4, to derive the sentence 'Joe chased the dog,' us…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=7292 \begin{verbatim} s ## Therefore the parse tree is the tree ![Image](./AutomataTheory_chapter_2.1_artifacts/image_000008_f517b05d44932b97f57aacbeaeb3471179ae3470c7481e5271996abcb82ee993.png) 4 Example 4.7 In Example 4.4, to derive the sentence 'Joe chased the dog,' using productions to get to get < art >< adj >< noun >< verb p >< prep >< noun p > Using to get to get < art >< adj >< noun >< verb p >< prep >< noun p > Using we get Again using and … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{> \#\# Therefore the parse tree is the tree ![Image](./AutomataTheory\_chapter\_2.1\_artifacts/image\_000008\_f517b05d44932b97f57aacbeaeb3471179ae3470c7481e5271996abcb82ee993.png) 4 Example 4.7 In Example 4.4, to derive the sentence 'Joe chased the dog,' using productions to get to get < art >< adj >< noun >< verb p >< prep >< noun p > Using we get Again using and … \end{verbatim} ``` </details>
645. ph-9584ef97114e8d12fa4c — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) is the tree ![Image](./AutomataTheory_chapter_2.1_artifacts/image_000008_f517b05d44932b97f57aacbeaeb3471179ae3470c7481e5271996abcb82ee993.png) 4 Example 4.7 In Example 4.4, to derive the sentence 'Joe chased the dog,' using productions to get to get < art >< adj >< noun >< verb p >< prep >< noun p > Using we get Again using and we get Using to get to get …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=7352 \begin{verbatim} is the tree ![Image](./AutomataTheory_chapter_2.1_artifacts/image_000008_f517b05d44932b97f57aacbeaeb3471179ae3470c7481e5271996abcb82ee993.png) 4 Example 4.7 In Example 4.4, to derive the sentence 'Joe chased the dog,' using productions to get to get < art >< adj >< noun >< verb p >< prep >< noun p > Using we get Again using and we get Using to get to get < art >< adj >< noun >< verb p >< prep >< noun p > Using we get Again using and we get Using we get Using we … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ataTheory\_chapter\_2.1\_artifacts/image\_000008\_f517b05d44932b97f57aacbeaeb3471179ae3470c7481e5271996abcb82ee993.png) 4 Example 4.7 In Example 4.4, to derive the sentence 'Joe chased the dog,' using productions to get to get \< art \>\< adj \>\&l…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=7382 \begin{verbatim} ataTheory_chapter_2.1_artifacts/image_000008_f517b05d44932b97f57aacbeaeb3471179ae3470c7481e5271996abcb82ee993.png) 4 Example 4.7 In Example 4.4, to derive the sentence 'Joe chased the dog,' using productions to get to get < art >< adj >< noun >< verb p >< prep >< noun p > Using we get Again using and we get Using we get Using we … \end{verbatim} ``` </details>
647. ph-5467fd81dc8052d44cf7 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) eb3471179ae3470c7481e5271996abcb82ee993.png) 4 Example 4.7 In Example 4.4, to derive the sentence 'Joe chased the dog,' using productions to get to get < art >< adj >< noun >< verb p >< prep >< noun p > Using we get Again using and we get Using we get Using we get Using < noun > → Joe < noun > → dog < verb > → c… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{eb3471179ae3470c7481e5271996abcb82ee993.png) 4 Example 4.7 In Example 4.4, to derive the sentence 'Joe chased the dog,' using productions to get to get \< art \>\< adj \>\< noun \>\< verb p \>\< prep \>\< noun p \> Using to get to get < art >< adj >< noun >< verb p >< prep >< noun p > Using we get Again using and we get Using we get Using we get Using < noun > → Joe < noun > → dog < verb > → c… \end{verbatim} ```
648. ph-5bfc4d8eb78e5595b8b5 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) la-not-decoded --> we get Again using and we get Using we get Using we get Using < noun > → Joe < noun > → dog < verb > → chased art → the we have the correspondence tree for 'Joe chased the dog.' Example 4.8 In Example 4.4, to derive the sentence 'The large dog leaped over the old fence,' we use productions ![Image](./AutomataTheory_chapter_2.1_artifacts/image_000009_9702dc98d1cc7f8e99244cb0a3aeff1c3c9174cfb87e07d4422bfd6d71aa573d.png) we get Again using and we get Using we get Using we get Using \< noun \> → Joe \< noun \> → dog \< verb \> → chased art → the we have the correspondence tree for 'Joe chased the dog.' Example 4.8 In Example 4.4, to deri…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=7892 \begin{verbatim} la-not-decoded --> we get Again using and we get Using we get Using we get Using < noun > → Joe < noun > → dog < verb > → chased art → the we have the correspondence tree for 'Joe chased the dog.' Example 4.8 In Example 4.4, to derive the sentence 'The large dog leaped over the old fence,' we use productions ![Image](./AutomataTheory_chapter_2.1_artifacts/image_000009_9702dc98d1cc7f8e99244cb0a3aeff1c3c9174cfb87e07d4422bfd6d71aa573d.png) we get Using we get Using < noun > → Joe < noun > → dog < verb > → chased art → the we have the correspondence tree for 'Joe chased the dog.' Example 4.8 In Example 4.4, to derive the sentence 'The large dog leaped over the old fence,' we use productions ![Image](./AutomataTheory_chapter_2.1_artifacts/image_000009_9702dc98d1cc7f8e99244cb0a3aeff1c3c9174cfb87e07d4422bfd6d71aa573d.png) ## Thus the parse tree … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ain using and we get Using we get Using we get Using \< noun \> → Joe \< noun \> → dog \< verb \> → chased art → the we have the correspondence tree for 'Joe chased the dog.' Example 4.8 In Example 4.4, to derive the sentence 'The large d…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=7922 \begin{verbatim} ain using and we get Using we get Using we get Using < noun > → Joe < noun > → dog < verb > → chased art → the we have the correspondence tree for 'Joe chased the dog.' Example 4.8 In Example 4.4, to derive the sentence 'The large dog leaped over the old fence,' we use productions ![Image](./AutomataTheory_chapter_2.1_artifacts/image_000009_9702dc98d1cc7f8e99244cb0a3aeff1c3c9174cfb87e07d4422bfd6d71aa573d.png) ## Thus the parse tree … \end{verbatim} ``` </details>
650. ph-a32c412d5b13ba259d4d — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) we get Using we get Using < noun > → Joe < noun > → dog < verb > → chased art → the we have the correspondence tree for 'Joe chased the dog.' Example 4.8 In Example 4.4, to derive the sentence 'The large dog leaped over the old fence,' we use productions ![Image](./AutomataTheory_chapter_2.1_artifacts/image_000009_9702dc98d1cc7f8e99244cb0a3aeff1c3c9174cfb87e07d4422bfd6d71aa573d.png) ## Thus the parse tree is ![Image](./AutomataTheor… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ we get Using we get Using \< noun \> → Joe \< noun \> → dog \< verb \> → chased art → the we have the correspondence tree for 'Joe chased the dog.' Example 4.8 In Example 4.4, to derive the sentence 'The large dog leaped over the old fenc…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=7952 \begin{verbatim} we get Using we get Using < noun > → Joe < noun > → dog < verb > → chased art → the we have the correspondence tree for 'Joe chased the dog.' Example 4.8 In Example 4.4, to derive the sentence 'The large dog leaped over the old fence,' we use productions ![Image](./AutomataTheory_chapter_2.1_artifacts/image_000009_9702dc98d1cc7f8e99244cb0a3aeff1c3c9174cfb87e07d4422bfd6d71aa573d.png) ## Thus the parse tree is ![Image](./AutomataTheor… \end{verbatim} ``` </details>
651. ph-bde9d1e0ab7dbcba1261 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) we get Using we get Using < noun > → Joe < noun > → dog < verb > → chased art → the we have the correspondence tree for 'Joe chased the dog.' Example 4.8 In Example 4.4, to derive the sentence 'The large dog leaped over the old fence,' we use productions ![Image](./AutomataTheory_chapter_2.1_artifacts/image_000009_9702dc98d1cc7f8e99244cb0a3aeff1c3c9174cfb87e07d4422bfd6d71aa573d.png) ## Thus the parse tree is ![Image](./AutomataTheory_chapter_2.1_artifacts/image… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{we get Using we get Using \< noun \> → Joe \< noun \> → dog \< verb \> → chased art → the we have the correspondence tree for 'Joe chased the dog.' Example 4.8 In Example 4.4, to derive the sentence 'The large dog leaped over the old fence,' we use productions ![Imag…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=7982 \begin{verbatim} we get Using we get Using < noun > → Joe < noun > → dog < verb > → chased art → the we have the correspondence tree for 'Joe chased the dog.' Example 4.8 In Example 4.4, to derive the sentence 'The large dog leaped over the old fence,' we use productions ![Image](./AutomataTheory_chapter_2.1_artifacts/image_000009_9702dc98d1cc7f8e99244cb0a3aeff1c3c9174cfb87e07d4422bfd6d71aa573d.png) ## Thus the parse tree is ![Image](./AutomataTheory_chapter_2.1_artifacts/image… \end{verbatim} ``` </details>
652. ph-9f0756374bb7fb2e008e — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) can only be used when a is on the left-hand side of A and b is on the right-hand side. It therefore cannot be used whenever A appears and so it is dependent on the context in which A appears. Such a grammar is called a context-sensitive grammar . In the following examples, we consider context-free grammars which generate more abstract languages: Example 4.9 Let /Gamma1 = ( N , /Sigma1 , S , P ) be the grammar defined by N = { S , A , B } , /Sigma1 = { a , b } and P be the set of productions Using the production S → AB , we derive AB . Next using the productions A … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{can only be used when a is on the left-hand side of A and b is on the right-hand side. It therefore cannot be used whenever A appears and so it is dependent on the context in which A appears. Such a grammar is called a context-sensitive grammar . In the following examples, we co…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=9150 \begin{verbatim} can only be used when a is on the left-hand side of A and b is on the right-hand side. It therefore cannot be used whenever A appears and so it is dependent on the context in which A appears. Such a grammar is called a context-sensitive grammar . In the following examples, we consider context-free grammars which generate more abstract languages: Example 4.9 Let /Gamma1 = ( N , /Sigma1 , S , P ) be the grammar defined by N = { S , A , B } , /Sigma1 = { a , b } and P be the set of productions Using the production S → AB , we derive AB . Next using the productions A … \end{verbatim} ``` </details>
653. ph-1274a5446569dc3f3790 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) the context in which A appears. Such a grammar is called a context-sensitive grammar . In the following examples, we consider context-free grammars which generate more abstract languages: Example 4.9 Let /Gamma1 = ( N , /Sigma1 , S , P ) be the grammar defined by N = { S , A , B } , /Sigma1 = { a , b } and P be the set of productions Using the production S → AB , we derive AB . Next using the productions A → a and B → λ , we derive a . If we use the productions in order, we derive b . We can also generate aabbb , aaaa , aaab , and bbbb… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{the context in which A appears. Such a grammar is called a context-sensitive grammar . In the following examples, we consider context-free grammars which generate more abstract languages: Example 4.9 Let /Gamma1 = ( N , /Sigma1 , S , P ) be the grammar defined by N = \{ S , A , B…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=9311 \begin{verbatim} the context in which A appears. Such a grammar is called a context-sensitive grammar . In the following examples, we consider context-free grammars which generate more abstract languages: Example 4.9 Let /Gamma1 = ( N , /Sigma1 , S , P ) be the grammar defined by N = { S , A , B } , /Sigma1 = { a , b } and P be the set of productions Using the production S → AB , we derive AB . Next using the productions A → a and B → λ , we derive a . If we use the productions in order, we derive b . We can also generate aabbb , aaaa , aaab , and bbbb… \end{verbatim} ``` </details>
654. ph-c5eab3abb4814a5e8cb6 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) e derive AB . Next using the productions A → a and B → λ , we derive a . If we use the productions in order, we derive b . We can also generate aabbb , aaaa , aaab , and bbbbb . In fact, we can generate a m b n for all nonnegative integers m , n . Hence the expression for the language generated by /Gamma1 is a ∗ b ∗ . Example 4.10 Let /Gamma1 ′ = ( N , /Sigma1 , S , P ) be the grammar defined by N = { S , A } , /Sigma1 = { a , b } and P be the set of productions Using the productions S → aAb and A → λ we derive ab . Using the productio… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{e derive AB . Next using the productions A → a and B → λ , we derive a . If we use the productions in order, we derive b . We can also generate aabbb , aaaa , aaab , and bbbbb . In fact, we can generate a m b n for all nonnegative integers m , n . He…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=9711 \begin{verbatim} e derive AB . Next using the productions A → a and B → λ , we derive a . If we use the productions in order, we derive b . We can also generate aabbb , aaaa , aaab , and bbbbb . In fact, we can generate a m b n for all nonnegative integers m , n . Hence the expression for the language generated by /Gamma1 is a ∗ b ∗ . Example 4.10 Let /Gamma1 ′ = ( N , /Sigma1 , S , P ) be the grammar defined by N = { S , A } , /Sigma1 = { a , b } and P be the set of productions Using the productions S → aAb and A → λ we derive ab . Using the productio… \end{verbatim} ``` </details>
655. ph-0f56d39646cab89dd1fb — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) mula-not-decoded --> in order, we derive b . We can also generate aabbb , aaaa , aaab , and bbbbb . In fact, we can generate a m b n for all nonnegative integers m , n . Hence the expression for the language generated by /Gamma1 is a ∗ b ∗ . Example 4.10 Let /Gamma1 ′ = ( N , /Sigma1 , S , P ) be the grammar defined by N = { S , A } , /Sigma1 = { a , b } and P be the set of productions Using the productions S → aAb and A → λ we derive ab . Using the productions in order, we derive aabb or a 2 b 2 . Using the productions in order, we derive b . We can also generate aabbb , aaaa , aaab , and bbbbb . In fact, we can generate a m b n for all nonnegative integers m , n . Hence the expression for the language generated by /Gamma1 is a ∗ b ∗ . Example 4.10 Let /Gamma1 ′ = ( N , /S…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=9819 \begin{verbatim} mula-not-decoded --> in order, we derive b . We can also generate aabbb , aaaa , aaab , and bbbbb . In fact, we can generate a m b n for all nonnegative integers m , n . Hence the expression for the language generated by /Gamma1 is a ∗ b ∗ . Example 4.10 Let /Gamma1 ′ = ( N , /Sigma1 , S , P ) be the grammar defined by N = { S , A } , /Sigma1 = { a , b } and P be the set of productions Using the productions S → aAb and A → λ we derive ab . Using the productions in order, we derive aabb or a 2 b 2 . Using the productions Using the productions S → aAb and A → λ we derive ab . Using the productions in order, we derive aabb or a 2 b 2 . Using the productions in order, we derive aaabbb or a 3 b 3 . It is easily seen that the language … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{d bbbbb . In fact, we can generate a m b n for all nonnegative integers m , n . Hence the expression for the language generated by /Gamma1 is a ∗ b ∗ . Example 4.10 Let /Gamma1 ′ = ( N , /Sigma1 , S , P ) be the grammar defined by N = \{ S , A \} , /Sigma1 = \{ a , b \} and P be the…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=9910 \begin{verbatim} d bbbbb . In fact, we can generate a m b n for all nonnegative integers m , n . Hence the expression for the language generated by /Gamma1 is a ∗ b ∗ . Example 4.10 Let /Gamma1 ′ = ( N , /Sigma1 , S , P ) be the grammar defined by N = { S , A } , /Sigma1 = { a , b } and P be the set of productions Using the productions S → aAb and A → λ we derive ab . Using the productions in order, we derive aabb or a 2 b 2 . Using the productions in order, we derive aaabbb or a 3 b 3 . It is easily seen that the language … \end{verbatim} ``` </details>
657. ph-53b5a86a99c68588e92b — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) -> in order, we derive aabb or a 2 b 2 . Using the productions in order, we derive aaabbb or a 3 b 3 . It is easily seen that the language generated by /Gamma1 ′ is { a n b n : n is a positive integer } . Note that this is not the same as a ∗ b ∗ since this would also include a m b n where m and n are not equal. Example 4.11 Let /Gamma1 ′′ = ( N , /Sigma1 , S , P ) be the grammar defined by N = { S , A , B } , /Sigma1 = { a , b } and P be the set of productions It can be shown that the expression for the language generated by /Gamma1 ′… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{-> in order, we derive aabb or a 2 b 2 . Using the productions in order, we derive aaabbb or a 3 b 3 . It is easily seen that the language generated by /Gamma1 ′ is \{ a n b n : n is a positive integer \} . Note that this is not the same as a ∗ b ∗ sin…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=10345 \begin{verbatim} -> in order, we derive aabb or a 2 b 2 . Using the productions in order, we derive aaabbb or a 3 b 3 . It is easily seen that the language generated by /Gamma1 ′ is { a n b n : n is a positive integer } . Note that this is not the same as a ∗ b ∗ since this would also include a m b n where m and n are not equal. Example 4.11 Let /Gamma1 ′′ = ( N , /Sigma1 , S , P ) be the grammar defined by N = { S , A , B } , /Sigma1 = { a , b } and P be the set of productions It can be shown that the expression for the language generated by /Gamma1 ′… \end{verbatim} ``` </details>
658. ph-a4eb3faf6325cd926295 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) 4.5 A context-free grammar /Gamma1 = ( N , /Sigma1 , S , P ) is called a regular grammar if every production p ∈ P has the form n → w where w is the empty word λ or the string w contains at most one nonterminal symbol and it occurs at the end of the string if at all. Therefore w could be of the form aacA , ab , λ or bA , where a , b , and c are terminals and A is a nonterminal. However, w could not be of the form aAb , aAB , or Aa . The production n → abcA could be replaced by the productions Also it is possible w could contain no terminal and one nonterminal so w… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{4.5 A context-free grammar /Gamma1 = ( N , /Sigma1 , S , P ) is called a regular grammar if every production p ∈ P has the form n → w where w is the empty word λ or the string w contains at most one nonterminal symbol and it occurs at the end of the string if at all. Therefore w…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=11622 \begin{verbatim} 4.5 A context-free grammar /Gamma1 = ( N , /Sigma1 , S , P ) is called a regular grammar if every production p ∈ P has the form n → w where w is the empty word λ or the string w contains at most one nonterminal symbol and it occurs at the end of the string if at all. Therefore w could be of the form aacA , ab , λ or bA , where a , b , and c are terminals and A is a nonterminal. However, w could not be of the form aAb , aAB , or Aa . The production n → abcA could be replaced by the productions Also it is possible w could contain no terminal and one nonterminal so w… \end{verbatim} ``` </details>
659. ph-e9551e8d238bb2b46458 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) where a , b , and c are terminals and A is a nonterminal. However, w could not be of the form aAb , aAB , or Aa . The production n → abcA could be replaced by the productions Also it is possible w could contain no terminal and one nonterminal so we have B → C , but if this is followed by C → t D , where t is a terminal, then we can combine the two productions to get B → t D . Hence it is no restriction to require each production to be one of the following forms: where A , B , and C are nonterminal elements, and a and b are terminal ele… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{where a , b , and c are terminals and A is a nonterminal. However, w could not be of the form aAb , aAB , or Aa . The production n → abcA could be replaced by the productions Also it is possible w could contain no terminal and one nonterminal so we h…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=11946 \begin{verbatim} where a , b , and c are terminals and A is a nonterminal. However, w could not be of the form aAb , aAB , or Aa . The production n → abcA could be replaced by the productions Also it is possible w could contain no terminal and one nonterminal so we have B → C , but if this is followed by C → t D , where t is a terminal, then we can combine the two productions to get B → t D . Hence it is no restriction to require each production to be one of the following forms: where A , B , and C are nonterminal elements, and a and b are terminal ele… \end{verbatim} ``` </details>
660. ph-991120386058757908a8 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) the string w has the form xY, x or λ where x ∈ /Sigma1 and Y ∈ N. Theorem 4.1 A language is generated by a linear regular grammar if and only if it is generated by a regular grammar. Proof Obviously every language that is generated by a linear regular grammar is generated by a regular grammar. To show every regular grammar is generated by a linear regular grammar, we divide the proof into two parts. We first show the language of a regular grammar can be generated by productions of the forms where A , B , C , and D are nonterminals and a , b are terminals. Let /Gam… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{the string w has the form xY, x or λ where x ∈ /Sigma1 and Y ∈ N. Theorem 4.1 A language is generated by a linear regular grammar if and only if it is generated by a regular grammar. Proof Obviously every language that is generated by a linear regular grammar is generated by a r…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=12778 \begin{verbatim} the string w has the form xY, x or λ where x ∈ /Sigma1 and Y ∈ N. Theorem 4.1 A language is generated by a linear regular grammar if and only if it is generated by a regular grammar. Proof Obviously every language that is generated by a linear regular grammar is generated by a regular grammar. To show every regular grammar is generated by a linear regular grammar, we divide the proof into two parts. We first show the language of a regular grammar can be generated by productions of the forms where A , B , C , and D are nonterminals and a , b are terminals. Let /Gam… \end{verbatim} ``` </details>
661. ph-a0b8a17bf18caa2f7662 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) n -1 B . So any word of L will be created by the grammar /Gamma1 ′ . Conversely if A ⇒ ∗ a 1 a 2 a 3 . . . an -1 B is formed by productions A → a 1 A 1 , A 1 → a 2 A 2 , . . . , An -1 → an -2 An -2 , An → an -1 B , then there must be a production A → a 1 a 2 a 3 . . . an -1 B in /Gamma1 since the symbols A 1 , A 2 , . . . , An are symbols which appear only in forming A → a 1 a 2 a 3 . . . an -1 B . Hence we can now assume that a regular grammar can be formed using only productions of the form where A , B , C , and D are nonterminals and a , b are terminals. We wan… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{n -1 B . So any word of L will be created by the grammar /Gamma1 ′ . Conversely if A ⇒ ∗ a 1 a 2 a 3 . . . an -1 B is formed by productions A → a 1 A 1 , A 1 → a 2 A 2 , . . . , An -1 → an -2 An -2 , An → an -1 B , then there must be a production A → a 1 a 2 a 3 . . . an -1 B in…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=13858 \begin{verbatim} n -1 B . So any word of L will be created by the grammar /Gamma1 ′ . Conversely if A ⇒ ∗ a 1 a 2 a 3 . . . an -1 B is formed by productions A → a 1 A 1 , A 1 → a 2 A 2 , . . . , An -1 → an -2 An -2 , An → an -1 B , then there must be a production A → a 1 a 2 a 3 . . . an -1 B in /Gamma1 since the symbols A 1 , A 2 , . . . , An are symbols which appear only in forming A → a 1 a 2 a 3 . . . an -1 B . Hence we can now assume that a regular grammar can be formed using only productions of the form where A , B , C , and D are nonterminals and a , b are terminals. We wan… \end{verbatim} ``` </details>
662. ph-f2ed5e0dfd3981e83740 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) where A , B , C , and D are nonterminals and a , b are terminals. We want to show that we can form a regular grammar without productions of the form C → D where C and D are both nonterminals. Call this a 1-production. Let /Gamma1 be a regular grammar formed using the productions above and L be the language generated by /Gamma1 . Assume that we have productions of the form above. Let /Gamma1 ′′ be the grammar with all 1-productions deleted and insert the production occurred in L . If where A , B , C , and D are nonterminals and a , b are terminals. We want to show that we can form a regular grammar without productions of the form C → D where C and D are both nonterminals. Call this a 1-production. Let /Gamma1 be a regular grammar…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=14358 \begin{verbatim} where A , B , C , and D are nonterminals and a , b are terminals. We want to show that we can form a regular grammar without productions of the form C → D where C and D are both nonterminals. Call this a 1-production. Let /Gamma1 be a regular grammar formed using the productions above and L be the language generated by /Gamma1 . Assume that we have productions of the form above. Let /Gamma1 ′′ be the grammar with all 1-productions deleted and insert the production occurred in L . If occurred in L . If occurred in L , insert t… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{where A , B , C , and D are nonterminals and a , b are terminals. We want to show that we can form a regular grammar without productions of the form C → D where C and D are both nonterminals. Call this a 1-production. Let /Gamma1 be a regular grammar formed using the productions…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=14388 \begin{verbatim} where A , B , C , and D are nonterminals and a , b are terminals. We want to show that we can form a regular grammar without productions of the form C → D where C and D are both nonterminals. Call this a 1-production. Let /Gamma1 be a regular grammar formed using the productions above and L be the language generated by /Gamma1 . Assume that we have productions of the form above. Let /Gamma1 ′′ be the grammar with all 1-productions deleted and insert the production occurred in L . If occurred in L , insert t… \end{verbatim} ``` </details>
664. ph-e53c14f5ee58850b0513 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) are terminals. We want to show that we can form a regular grammar without productions of the form C → D where C and D are both nonterminals. Call this a 1-production. Let /Gamma1 be a regular grammar formed using the productions above and L be the language generated by /Gamma1 . Assume that we have productions of the form above. Let /Gamma1 ′′ be the grammar with all 1-productions deleted and insert the production occurred in L . If occurred in L , insert the production A 1 → b in L ′′ . If occurred in L . If occurred in L , insert the production A 1 → b in L ′′ . If occurred in L . If occurred in L , insert the production A 1 → b in L ′′ . If occurred in L , insert the production A 1 → λ in L ′′ . Let L ′′ be the langua… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{e form C → D where C and D are both nonterminals. Call this a 1-production. Let /Gamma1 be a regular grammar formed using the productions above and L be the language generated by /Gamma1 . Assume that we have productions of the form above. Let /Gamma1 ′′ be the grammar with all …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=14530 \begin{verbatim} e form C → D where C and D are both nonterminals. Call this a 1-production. Let /Gamma1 be a regular grammar formed using the productions above and L be the language generated by /Gamma1 . Assume that we have productions of the form above. Let /Gamma1 ′′ be the grammar with all 1-productions deleted and insert the production occurred in L . If occurred in L , insert the production A 1 → b in L ′′ . If occurred in L , insert the production A 1 → λ in L ′′ . Let L ′′ be the langua… \end{verbatim} ``` </details>
666. ph-ec7a1f4d2d0b7d1a55db — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) adding letters to words rather than removing them. Suppose that we consider the string that has been read rather than the string left to read. In the example above, at state s 1, we have read a . At state s 2 we have read ab . At state s 3 we have read abb , and at state s 4 we have read abbc . Thus at each state we are adding a letter. Consider the grammar /Gamma1 = ( N , /Sigma1 , s 0 , P ), where N = { s 0 , s 1 , s 2 , s 3 , s 4 } , /Sigma1 = { a , b , c } , and P is the set of productions where we have a production s 4 → λ only if s 4 is a terminal state. It … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{adding letters to words rather than removing them. Suppose that we consider the string that has been read rather than the string left to read. In the example above, at state s 1, we have read a . At state s 2 we have read ab . At state s 3 we have read abb , and at state s 4 we …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=17236 \begin{verbatim} adding letters to words rather than removing them. Suppose that we consider the string that has been read rather than the string left to read. In the example above, at state s 1, we have read a . At state s 2 we have read ab . At state s 3 we have read abb , and at state s 4 we have read abbc . Thus at each state we are adding a letter. Consider the grammar /Gamma1 = ( N , /Sigma1 , s 0 , P ), where N = { s 0 , s 1 , s 2 , s 3 , s 4 } , /Sigma1 = { a , b , c } , and P is the set of productions where we have a production s 4 → λ only if s 4 is a terminal state. It … \end{verbatim} ``` </details>
667. ph-13f18b0379e879fbc7c9 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) productions B → at and t → λ . Obviously this does not change the language of the grammar. Let M = ( /Sigma1 , Q , s 0 , T , F ) be the automaton in which Q is the set of nonterminals together with the additional nonterminal t , s 0 = S . The set F is defined by F ( a , A ) = B if and only if A → aB is in P . The state B ∈ T if B → λ . Example 4.14 Let /Gamma1 = ( N , /Sigma1 , S , P ) be the grammar defined by N = { S , A , B , C } , /Sigma1 = { a , b , c } , and P be the set of productions The corresponding automaton is ![Image](./AutomataTheory_chapter_2.1_arti… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{productions B → at and t → λ . Obviously this does not change the language of the grammar. Let M = ( /Sigma1 , Q , s 0 , T , F ) be the automaton in which Q is the set of nonterminals together with the additional nonterminal t , s 0 = S . The set F is defined by F ( a , A ) = B …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=22122 \begin{verbatim} productions B → at and t → λ . Obviously this does not change the language of the grammar. Let M = ( /Sigma1 , Q , s 0 , T , F ) be the automaton in which Q is the set of nonterminals together with the additional nonterminal t , s 0 = S . The set F is defined by F ( a , A ) = B if and only if A → aB is in P . The state B ∈ T if B → λ . Example 4.14 Let /Gamma1 = ( N , /Sigma1 , S , P ) be the grammar defined by N = { S , A , B , C } , /Sigma1 = { a , b , c } , and P be the set of productions The corresponding automaton is ![Image](./AutomataTheory_chapter_2.1_arti… \end{verbatim} ``` </details>
668. ph-6f25685f1fe54d1b7d65 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) ample 4.14 Let /Gamma1 = ( N , /Sigma1 , S , P ) be the grammar defined by N = { S , A , B , C } , /Sigma1 = { a , b , c } , and P be the set of productions The corresponding automaton is ![Image](./AutomataTheory_chapter_2.1_artifacts/image_000013_ac32653a06fc360c3b9dc744f85b0a235f681dfcc5c31676a83b1622c8ac2b74.png) Example 4.15 Let /Gamma1 = ( N , T , S , P ) be the grammar defined by N = { S , A , B , C } , T = { a , b , c } , and P be the set of productions The corresponding automaton is ![Image](./AutomataTheory_chapter_2.1_artifa… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ample 4.14 Let /Gamma1 = ( N , /Sigma1 , S , P ) be the grammar defined by N = \{ S , A , B , C \} , /Sigma1 = \{ a , b , c \} , and P be the set of productions The corresponding automaton is ![Image](./AutomataTheory\_chapter\_2.1\_artifacts/image\_000013\_a…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=22464 \begin{verbatim} ample 4.14 Let /Gamma1 = ( N , /Sigma1 , S , P ) be the grammar defined by N = { S , A , B , C } , /Sigma1 = { a , b , c } , and P be the set of productions The corresponding automaton is ![Image](./AutomataTheory_chapter_2.1_artifacts/image_000013_ac32653a06fc360c3b9dc744f85b0a235f681dfcc5c31676a83b1622c8ac2b74.png) Example 4.15 Let /Gamma1 = ( N , T , S , P ) be the grammar defined by N = { S , A , B , C } , T = { a , b , c } , and P be the set of productions The corresponding automaton is ![Image](./AutomataTheory_chapter_2.1_artifa… \end{verbatim} ``` </details>
669. ph-1232ab3cacf0ee3d8166 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) ammar in Example 4.9, construct a parse tree for abbb . - (2) Using the grammar in Example 4.10, construct a parse tree for aaabbb . - (3) Using the grammar in Example 4.11, construct a parse tree for babaab . - (4) In Example 4.4, derive the statement 'The cowboy rode slowly into the sunset' and construct the correspondence parse tree. - (5) Find the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B } , T = { a , b } and the set of productions P given by - (6) Find the language generated by the grammar /Gamma1 = ( N , T , S … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ammar in Example 4.9, construct a parse tree for abbb . - (2) Using the grammar in Example 4.10, construct a parse tree for aaabbb . - (3) Using the grammar in Example 4.11, construct a parse tree for babaab . - (4) In Example 4.4, derive the statement 'The cowboy rode slowly in…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=24973 \begin{verbatim} ammar in Example 4.9, construct a parse tree for abbb . - (2) Using the grammar in Example 4.10, construct a parse tree for aaabbb . - (3) Using the grammar in Example 4.11, construct a parse tree for babaab . - (4) In Example 4.4, derive the statement 'The cowboy rode slowly into the sunset' and construct the correspondence parse tree. - (5) Find the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B } , T = { a , b } and the set of productions P given by - (6) Find the language generated by the grammar /Gamma1 = ( N , T , S … \end{verbatim} ``` </details>
670. ph-7dc1f556f22387d949da — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) tree for babaab . - (4) In Example 4.4, derive the statement 'The cowboy rode slowly into the sunset' and construct the correspondence parse tree. - (5) Find the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B } , T = { a , b } and the set of productions P given by - (6) Find the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B } , T = { a , b } and the set of productions P given by - (7) Find the language generated by the grammar /Gamma1 = ( N , T , S , P… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{tree for babaab . - (4) In Example 4.4, derive the statement 'The cowboy rode slowly into the sunset' and construct the correspondence parse tree. - (5) Find the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = \{ S , A , B \} , T = \{ a , b \} and the se…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=25164 \begin{verbatim} tree for babaab . - (4) In Example 4.4, derive the statement 'The cowboy rode slowly into the sunset' and construct the correspondence parse tree. - (5) Find the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B } , T = { a , b } and the set of productions P given by - (6) Find the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B } , T = { a , b } and the set of productions P given by - (7) Find the language generated by the grammar /Gamma1 = ( N , T , S , P… \end{verbatim} ``` </details>
671. ph-23e65a466897016b1277 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) ammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B } , T = { a , b } and the set of productions P given by - (6) Find the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B } , T = { a , b } and the set of productions P given by - (7) Find the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B } , T = { a , b } and the set of productions P given by - (8) Find the language generated by the grammar /Gamma1 = ( N , T , S , P … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ammar /Gamma1 = ( N , T , S , P ) defined by N = \{ S , A , B \} , T = \{ a , b \} and the set of productions P given by - (6) Find the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = \{ S , A , B \} , T = \{ a , b \} and the set…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=25355 \begin{verbatim} ammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B } , T = { a , b } and the set of productions P given by - (6) Find the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B } , T = { a , b } and the set of productions P given by - (7) Find the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B } , T = { a , b } and the set of productions P given by - (8) Find the language generated by the grammar /Gamma1 = ( N , T , S , P … \end{verbatim} ``` </details>
672. ph-54e48736eb28fccfbe96 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) r /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B } , T = { a , b } and the set of productions P given by - (7) Find the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B } , T = { a , b } and the set of productions P given by - (8) Find the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B , C } , T = { a , b } and the set of productions P given by - (9) Find the grammar which generates the language ww r where w is a strin… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{r /Gamma1 = ( N , T , S , P ) defined by N = \{ S , A , B \} , T = \{ a , b \} and the set of productions P given by - (7) Find the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = \{ S , A , B \} , T = \{ a , b \} and the set of …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=25550 \begin{verbatim} r /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B } , T = { a , b } and the set of productions P given by - (7) Find the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B } , T = { a , b } and the set of productions P given by - (8) Find the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B , C } , T = { a , b } and the set of productions P given by - (9) Find the grammar which generates the language ww r where w is a strin… \end{verbatim} ``` </details>
673. ph-d1dfd758316555d35f5c — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) 22) Construct a grammar which generates the language expressed by ((aa ∗ b) ∨ bb ∗ a)ac ∗ . - (21) Construct a grammar which generates the language expressed by (a ∨ b) ∗ (aa ∨ bb)(a ∨ b) ∗ . - (23) Construct a grammar to generate arithmetic expressions for positive integers less than ten in prefix notation. - (24) Find an automaton which accepts the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B } , T = { a , b } and the set of productions P given by - (25) Find an automaton which accepts the language generated by the gra… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{22) Construct a grammar which generates the language expressed by ((aa ∗ b) ∨ bb ∗ a)ac ∗ . - (21) Construct a grammar which generates the language expressed by (a ∨ b) ∗ (aa ∨ bb)(a ∨ b) ∗ . - (23) Construct a grammar to generate arithmetic expressions for positive integers les…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=27338 \begin{verbatim} 22) Construct a grammar which generates the language expressed by ((aa ∗ b) ∨ bb ∗ a)ac ∗ . - (21) Construct a grammar which generates the language expressed by (a ∨ b) ∗ (aa ∨ bb)(a ∨ b) ∗ . - (23) Construct a grammar to generate arithmetic expressions for positive integers less than ten in prefix notation. - (24) Find an automaton which accepts the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B } , T = { a , b } and the set of productions P given by - (25) Find an automaton which accepts the language generated by the gra… \end{verbatim} ``` </details>
674. ph-28bf5e2fbcc5e44ca8b6 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) to generate arithmetic expressions for positive integers less than ten in prefix notation. - (24) Find an automaton which accepts the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B } , T = { a , b } and the set of productions P given by - (25) Find an automaton which accepts the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B } , T = { a , b } and the set of productions P given by - (26) Find an automaton which accepts the language generated by the gramm… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{to generate arithmetic expressions for positive integers less than ten in prefix notation. - (24) Find an automaton which accepts the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = \{ S , A , B \} , T = \{ a , b \} and the set of productions P given by …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=27557 \begin{verbatim} to generate arithmetic expressions for positive integers less than ten in prefix notation. - (24) Find an automaton which accepts the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B } , T = { a , b } and the set of productions P given by - (25) Find an automaton which accepts the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B } , T = { a , b } and the set of productions P given by - (26) Find an automaton which accepts the language generated by the gramm… \end{verbatim} ``` </details>
675. ph-6aabbc5c08795648e9ef — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) } , T = { a , b } and the set of productions P given by - (25) Find an automaton which accepts the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B } , T = { a , b } and the set of productions P given by - (26) Find an automaton which accepts the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B , C } , T = { a , b } and the set of productions P given by - (27) Find an automaton which accepts the language generated by the grammar… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{\} , T = \{ a , b \} and the set of productions P given by - (25) Find an automaton which accepts the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = \{ S , A , B \} , T = \{ a , b \} and the set of productions P given by - (25) Find an automaton which accepts the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B } , T = { a , b } and the set of productions P given by - (26) Find an automaton which accepts the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B , C } , T = { a , b } and the set of productions P given by - (27) Find an automaton which accepts the language generated by the grammar… \end{verbatim} ```
676. ph-4c8abea3c41cf1079ca4 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) T = { a , b } and the set of productions P given by - (26) Find an automaton which accepts the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B , C } , T = { a , b } and the set of productions P given by - (27) Find an automaton which accepts the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B , C } , T = { a , b } and the set of productions P given by - (28) Find an automaton which accepts the language generated by the grammar… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{T = \{ a , b \} and the set of productions P given by - (26) Find an automaton which accepts the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = \{ S , A , B , C \} , T = \{ a , b \} and the set of productions P given by - (26) Find an automaton which accepts the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B , C } , T = { a , b } and the set of productions P given by - (27) Find an automaton which accepts the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B , C } , T = { a , b } and the set of productions P given by - (28) Find an automaton which accepts the language generated by the grammar… \end{verbatim} ```
677. ph-7ffc4ceaa678095fc4c1 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) T = { a , b } and the set of productions P given by - (27) Find an automaton which accepts the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B , C } , T = { a , b } and the set of productions P given by - (28) Find an automaton which accepts the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B , C } , T = { a , b } and the set of productions P given by - (29) Find an automaton which accepts the language generated by the grammar… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{T = \{ a , b \} and the set of productions P given by - (27) Find an automaton which accepts the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = \{ S , A , B , C \} , T = \{ a , b \} and the set of productions P given by - (27) Find an automaton which accepts the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B , C } , T = { a , b } and the set of productions P given by - (28) Find an automaton which accepts the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B , C } , T = { a , b } and the set of productions P given by - (29) Find an automaton which accepts the language generated by the grammar… \end{verbatim} ```
678. ph-1475f9ff5815e2f7094c — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) T = { a , b } and the set of productions P given by - (28) Find an automaton which accepts the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B , C } , T = { a , b } and the set of productions P given by - (29) Find an automaton which accepts the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B , C } , T = { a , b } and the set of productions P given by - (30) Construct a grammar which generates the language accepted by the auto… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{T = \{ a , b \} and the set of productions P given by - (28) Find an automaton which accepts the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = \{ S , A , B , C \} , T = \{ a , b \} and the set of productions P given by - (28) Find an automaton which accepts the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B , C } , T = { a , b } and the set of productions P given by - (29) Find an automaton which accepts the language generated by the grammar /Gamma1 = ( N , T , S , P ) defined by N = { S , A , B , C } , T = { a , b } and the set of productions P given by - (30) Construct a grammar which generates the language accepted by the auto… \end{verbatim} ```
679. ph-9206150488edef2872fb — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) mmar which generates the language accepted by the automaton ![Image](./AutomataTheory_chapter_2.1_artifacts/image_000020_5ba4c1145345b9e4f95b7318fabe020415ce2e81fc410d3e76b4c5fcf05435e5.png) ![Image](./AutomataTheory_chapter_2.1_artifacts/image_000021_509a5bc5dfb461de92a1976ff376d17f9cac1f8cef51044298d0e63707d471c8.png) ## 4.2 Chomsky normal form and Greibach normal form Definition 4.8 A context-free grammar /Gamma1 is in Chomsky normal form if each of its productions is either of the form where A, B, and C are nonterminals and a is a … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{mmar which generates the language accepted by the automaton ![Image](./AutomataTheory\_chapter\_2.1\_artifacts/image\_000020\_5ba4c1145345b9e4f95b7318fabe020415ce2e81fc410d3e76b4c5fcf05435e5.png) ![Image](./AutomataTheory\_chapter\_2.1\_artifacts/image\_000021\_509a5bc5dfb461de92a1976ff37…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=30162 \begin{verbatim} mmar which generates the language accepted by the automaton ![Image](./AutomataTheory_chapter_2.1_artifacts/image_000020_5ba4c1145345b9e4f95b7318fabe020415ce2e81fc410d3e76b4c5fcf05435e5.png) ![Image](./AutomataTheory_chapter_2.1_artifacts/image_000021_509a5bc5dfb461de92a1976ff376d17f9cac1f8cef51044298d0e63707d471c8.png) ## 4.2 Chomsky normal form and Greibach normal form Definition 4.8 A context-free grammar /Gamma1 is in Chomsky normal form if each of its productions is either of the form where A, B, and C are nonterminals and a is a … \end{verbatim} ``` </details>
680. ph-287c9ac93728d1f17777 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) age accepted by the automaton ![Image](./AutomataTheory_chapter_2.1_artifacts/image_000020_5ba4c1145345b9e4f95b7318fabe020415ce2e81fc410d3e76b4c5fcf05435e5.png) ![Image](./AutomataTheory_chapter_2.1_artifacts/image_000021_509a5bc5dfb461de92a1976ff376d17f9cac1f8cef51044298d0e63707d471c8.png) ## 4.2 Chomsky normal form and Greibach normal form Definition 4.8 A context-free grammar /Gamma1 is in Chomsky normal form if each of its productions is either of the form where A, B, and C are nonterminals and a is a terminal. Definition 4.9 A con… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{age accepted by the automaton ![Image](./AutomataTheory\_chapter\_2.1\_artifacts/image\_000020\_5ba4c1145345b9e4f95b7318fabe020415ce2e81fc410d3e76b4c5fcf05435e5.png) ![Image](./AutomataTheory\_chapter\_2.1\_artifacts/image\_000021\_509a5bc5dfb461de92a1976ff376d17f9cac1f8cef51044298d0e6370…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=30192 \begin{verbatim} age accepted by the automaton ![Image](./AutomataTheory_chapter_2.1_artifacts/image_000020_5ba4c1145345b9e4f95b7318fabe020415ce2e81fc410d3e76b4c5fcf05435e5.png) ![Image](./AutomataTheory_chapter_2.1_artifacts/image_000021_509a5bc5dfb461de92a1976ff376d17f9cac1f8cef51044298d0e63707d471c8.png) ## 4.2 Chomsky normal form and Greibach normal form Definition 4.8 A context-free grammar /Gamma1 is in Chomsky normal form if each of its productions is either of the form where A, B, and C are nonterminals and a is a terminal. Definition 4.9 A con… \end{verbatim} ``` </details>
681. ph-53d6f91ca731035132cd — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) tifacts/image_000021_509a5bc5dfb461de92a1976ff376d17f9cac1f8cef51044298d0e63707d471c8.png) ## 4.2 Chomsky normal form and Greibach normal form Definition 4.8 A context-free grammar /Gamma1 is in Chomsky normal form if each of its productions is either of the form where A, B, and C are nonterminals and a is a terminal. Definition 4.9 A context-free grammar /Gamma1 is in Greibach normal form if each of its productions is of the form where a is a terminal and W is a possibly empty string of nonterminals. Wesha… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{tifacts/image\_000021\_509a5bc5dfb461de92a1976ff376d17f9cac1f8cef51044298d0e63707d471c8.png) \#\# 4.2 Chomsky normal form and Greibach normal form Definition 4.8 A context-free grammar /Gamma1 is in Chomsky normal form if each of its productions is either of the form where A, B, and C are nonterminals and a is a terminal. Definition 4.9 A context-free grammar /Gamma1 is in Greibach normal form if each of its productions is of the form where a is a terminal and W is a possibly empty string of nonterminals. Wesha… \end{verbatim} ```
682. ph-1c9939edc804834be2be — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) ∈ ( N ∪ T ) ∗ , be a derivation in /Gamma1 with n steps, then W can be expressed as W 1 W 2 where U ⇒ ∗ W 1 , V ⇒ ∗ W 2 are derivations in /Gamma1 , both containing at most n steps. Proof The proof of this lemma uses induction on the number of steps in the production. Assume there is one step. Then only one nonterminal is replaced using a production. Assume it is the production A → w B w ′ . Either A is in the string U or is in the string V . Without loss of generality assume it is in U , so so letting W 1 … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{∈ ( N ∪ T ) ∗ , be a derivation in /Gamma1 with n steps, then W can be expressed as W 1 W 2 where U ⇒ ∗ W 1 , V ⇒ ∗ W 2 are derivations in /Gamma1 , both containing at most n steps. Proof The proof of this lemma uses induction on the number of steps in the production. Assume the…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=31869 \begin{verbatim} ∈ ( N ∪ T ) ∗ , be a derivation in /Gamma1 with n steps, then W can be expressed as W 1 W 2 where U ⇒ ∗ W 1 , V ⇒ ∗ W 2 are derivations in /Gamma1 , both containing at most n steps. Proof The proof of this lemma uses induction on the number of steps in the production. Assume there is one step. Then only one nonterminal is replaced using a production. Assume it is the production A → w B w ′ . Either A is in the string U or is in the string V . Without loss of generality assume it is in U , so so letting W 1 … \end{verbatim} ``` </details>
683. ph-69532fdde02f91f82e45 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) on in /Gamma1 with n steps, then W can be expressed as W 1 W 2 where U ⇒ ∗ W 1 , V ⇒ ∗ W 2 are derivations in /Gamma1 , both containing at most n steps. Proof The proof of this lemma uses induction on the number of steps in the production. Assume there is one step. Then only one nonterminal is replaced using a production. Assume it is the production A → w B w ′ . Either A is in the string U or is in the string V . Without loss of generality assume it is in U , so so letting W 1 = U ′ and W 2 = V we are done… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{on in /Gamma1 with n steps, then W can be expressed as W 1 W 2 where U ⇒ ∗ W 1 , V ⇒ ∗ W 2 are derivations in /Gamma1 , both containing at most n steps. Proof The proof of this lemma uses induction on the number of steps in the production. Assume there is one step. Then only one…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=31899 \begin{verbatim} on in /Gamma1 with n steps, then W can be expressed as W 1 W 2 where U ⇒ ∗ W 1 , V ⇒ ∗ W 2 are derivations in /Gamma1 , both containing at most n steps. Proof The proof of this lemma uses induction on the number of steps in the production. Assume there is one step. Then only one nonterminal is replaced using a production. Assume it is the production A → w B w ′ . Either A is in the string U or is in the string V . Without loss of generality assume it is in U , so so letting W 1 = U ′ and W 2 = V we are done… \end{verbatim} ``` </details>
684. ph-9457576e174422ce4386 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) en W can be expressed as W 1 W 2 where U ⇒ ∗ W 1 , V ⇒ ∗ W 2 are derivations in /Gamma1 , both containing at most n steps. Proof The proof of this lemma uses induction on the number of steps in the production. Assume there is one step. Then only one nonterminal is replaced using a production. Assume it is the production A → w B w ′ . Either A is in the string U or is in the string V . Without loss of generality assume it is in U , so so letting W 1 = U ′ and W 2 = V we are done. Assume the lemma is true for… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{en W can be expressed as W 1 W 2 where U ⇒ ∗ W 1 , V ⇒ ∗ W 2 are derivations in /Gamma1 , both containing at most n steps. Proof The proof of this lemma uses induction on the number of steps in the production. Assume there is one step. Then only one nonterminal is replaced using…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=31929 \begin{verbatim} en W can be expressed as W 1 W 2 where U ⇒ ∗ W 1 , V ⇒ ∗ W 2 are derivations in /Gamma1 , both containing at most n steps. Proof The proof of this lemma uses induction on the number of steps in the production. Assume there is one step. Then only one nonterminal is replaced using a production. Assume it is the production A → w B w ′ . Either A is in the string U or is in the string V . Without loss of generality assume it is in U , so so letting W 1 = U ′ and W 2 = V we are done. Assume the lemma is true for… \end{verbatim} ``` </details>
685. ph-c5c6098cfef55b7aa280 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) W 1 = U ′ and W 2 = V we are done. Assume the lemma is true for all derivations with less than k steps. Assume UV ⇒ ∗ W contains k steps . As above assume the first step is UV ⇒ U ′ V where U ⇒ U ′ . Note that U ′ V ⇒ ∗ W uses only k -1steps. By induction there are derivations U ′ ⇒ ∗ W 1 , V ⇒ ∗ W 2 containing at most k steps. Therefore U ⇒ U ′ , U ′ ⇒ ∗ W 1 , V ⇒ ∗ W 2 are the required derivations. /square One of the results of this lemma is that we can get from UV ⇒ ∗ W by the derivations where W = W 1 W 2 since if X α ⇒ X β is a derivation, then so are X α V ⇒… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{W 1 = U ′ and W 2 = V we are done. Assume the lemma is true for all derivations with less than k steps. Assume UV ⇒ ∗ W contains k steps . As above assume the first step is UV ⇒ U ′ V where U ⇒ U ′ . Note that U ′ V ⇒ ∗ W uses only k -1steps. By induction there are derivations U…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=32470 \begin{verbatim} W 1 = U ′ and W 2 = V we are done. Assume the lemma is true for all derivations with less than k steps. Assume UV ⇒ ∗ W contains k steps . As above assume the first step is UV ⇒ U ′ V where U ⇒ U ′ . Note that U ′ V ⇒ ∗ W uses only k -1steps. By induction there are derivations U ′ ⇒ ∗ W 1 , V ⇒ ∗ W 2 containing at most k steps. Therefore U ⇒ U ′ , U ′ ⇒ ∗ W 1 , V ⇒ ∗ W 2 are the required derivations. /square One of the results of this lemma is that we can get from UV ⇒ ∗ W by the derivations where W = W 1 W 2 since if X α ⇒ X β is a derivation, then so are X α V ⇒… \end{verbatim} ``` </details>
686. ph-c18a60d6b9c5e27ca2f7 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) n /Gamma1 can be replaced with the original productions in /Gamma1 used to define it. To show L ⊆ L ′ , let w ∈ L . Using induction on the number of steps in the derivation, we show that if S ⇒ ∗ w using productions in P , then S ⇒ ∗ w using productions in P ′ . If n = 1 then S ⇒ w is obviously a production in P ′ since w /negationslash= λ . Assume n = k and S ⇒ ∗ w is a derivation in /Gamma1 containing k steps. Let S ⇒ A 1 A 2 A 3 . . . Am be the first derivation where Ai ∈ N ∪ T . Therefore By Lemma 4.2, there exist derivations Ai ⇒ ∗ w i in /Gamma1 , for 1 ≤ i … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{n /Gamma1 can be replaced with the original productions in /Gamma1 used to define it. To show L ⊆ L ′ , let w ∈ L . Using induction on the number of steps in the derivation, we show that if S ⇒ ∗ w using productions in P , then S ⇒ ∗ w using productions in P ′ . If n = 1 then S …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=36405 \begin{verbatim} n /Gamma1 can be replaced with the original productions in /Gamma1 used to define it. To show L ⊆ L ′ , let w ∈ L . Using induction on the number of steps in the derivation, we show that if S ⇒ ∗ w using productions in P , then S ⇒ ∗ w using productions in P ′ . If n = 1 then S ⇒ w is obviously a production in P ′ since w /negationslash= λ . Assume n = k and S ⇒ ∗ w is a derivation in /Gamma1 containing k steps. Let S ⇒ A 1 A 2 A 3 . . . Am be the first derivation where Ai ∈ N ∪ T . Therefore By Lemma 4.2, there exist derivations Ai ⇒ ∗ w i in /Gamma1 , for 1 ≤ i … \end{verbatim} ``` </details>
687. ph-0b6ecd61fb21c4972cb6 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) , then remove the trivial productions and include A 1 → B . Conversely, assume S ⇒ ∗ w occurs in /Gamma1 . We use induction on the number of trivial derivations to show that there is a derivation S ⇒ ∗ w in /Gamma1 ′ . Obviously if there is no trivial production then the derivation is in /Gamma1 ′ . Assume there are k trivial productions in the derivation. Assume that the derivation is a leftmost derivation of w . Assume S ⇒ ∗ w has the form By construction, /Gamma1 ( L ′ ) ⊆ /Gamma1 ( L ). where A 1 → A 2 → A 3 →··· → Am is the last sequence of trivial production… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{, then remove the trivial productions and include A 1 → B . Conversely, assume S ⇒ ∗ w occurs in /Gamma1 . We use induction on the number of trivial derivations to show that there is a derivation S ⇒ ∗ w in /Gamma1 ′ . Obviously if there is no trivial production then the derivat…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=39070 \begin{verbatim} , then remove the trivial productions and include A 1 → B . Conversely, assume S ⇒ ∗ w occurs in /Gamma1 . We use induction on the number of trivial derivations to show that there is a derivation S ⇒ ∗ w in /Gamma1 ′ . Obviously if there is no trivial production then the derivation is in /Gamma1 ′ . Assume there are k trivial productions in the derivation. Assume that the derivation is a leftmost derivation of w . Assume S ⇒ ∗ w has the form By construction, /Gamma1 ( L ′ ) ⊆ /Gamma1 ( L ). where A 1 → A 2 → A 3 →··· → Am is the last sequence of trivial production… \end{verbatim} ``` </details>
688. ph-02d51c053805a22f8f0e — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) amma1 ′ . Obviously if there is no trivial production then the derivation is in /Gamma1 ′ . Assume there are k trivial productions in the derivation. Assume that the derivation is a leftmost derivation of w . Assume S ⇒ ∗ w has the form By construction, /Gamma1 ( L ′ ) ⊆ /Gamma1 ( L ). where A 1 → A 2 → A 3 →··· → Am is the last sequence of trivial productions in the derivation, and Am → w ′ . Then there are derivations V 1 ⇒ ∗ w 1 , V 2 ⇒ ∗ w 2 in /Gamma1 , and has less trivial productions and all productions are in /Gamma1 ∪ /Gamma1 … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{amma1 ′ . Obviously if there is no trivial production then the derivation is in /Gamma1 ′ . Assume there are k trivial productions in the derivation. Assume that the derivation is a leftmost derivation of w . Assume S ⇒ ∗ w has the form By construction, /Gamma1 ( L ′ ) ⊆ /Gamma1…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=39281 \begin{verbatim} amma1 ′ . Obviously if there is no trivial production then the derivation is in /Gamma1 ′ . Assume there are k trivial productions in the derivation. Assume that the derivation is a leftmost derivation of w . Assume S ⇒ ∗ w has the form By construction, /Gamma1 ( L ′ ) ⊆ /Gamma1 ( L ). where A 1 → A 2 → A 3 →··· → Am is the last sequence of trivial productions in the derivation, and Am → w ′ . Then there are derivations V 1 ⇒ ∗ w 1 , V 2 ⇒ ∗ w 2 in /Gamma1 , and has less trivial productions and all productions are in /Gamma1 ∪ /Gamma1 … \end{verbatim} ``` </details>
689. ph-cb03b14945ef91259337 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) , . . . , Am with A ′ 1 , A ′ 2 , A ′ 3 , . . . , A ′ m where A ′ i = Ai if Ai is a nonterminal and A ′ i = Xai if Ai is a terminal. Thus if we have V 1 a 1 V 2 a 2 V 3 a 3 . . . VnanVn + 1 where Vi ∈ N ∗ and ai is a terminal, replace it with V 1 Xa 1 V 2 Xa 2 V 3 Xa 3 . . . Vn Xan Vn + 1 and add productions Xai → ai for 1 ≤ i ≤ n . Let /Gamma1 ′ = ( N , T , S , P ′ ) be the new grammar formed. We need to show that /Gamma1 ( L ′ ) = /Gamma1 ( L ). Clearly /Gamma1 ( L ) ⊆ /Gamma1 ( L ′ ) since in /Gamma1 can be replaced by in /Gamma1 ′ … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{, . . . , Am with A ′ 1 , A ′ 2 , A ′ 3 , . . . , A ′ m where A ′ i = Ai if Ai is a nonterminal and A ′ i = Xai if Ai is a terminal. Thus if we have V 1 a 1 V 2 a 2 V 3 a 3 . . . VnanVn + 1 where Vi ∈ N ∗ and ai is a terminal, replace it with V 1 Xa 1 V 2 Xa 2 V 3 Xa 3 . . . Vn …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=40824 \begin{verbatim} , . . . , Am with A ′ 1 , A ′ 2 , A ′ 3 , . . . , A ′ m where A ′ i = Ai if Ai is a nonterminal and A ′ i = Xai if Ai is a terminal. Thus if we have V 1 a 1 V 2 a 2 V 3 a 3 . . . VnanVn + 1 where Vi ∈ N ∗ and ai is a terminal, replace it with V 1 Xa 1 V 2 Xa 2 V 3 Xa 3 . . . Vn Xan Vn + 1 and add productions Xai → ai for 1 ≤ i ≤ n . Let /Gamma1 ′ = ( N , T , S , P ′ ) be the new grammar formed. We need to show that /Gamma1 ( L ′ ) = /Gamma1 ( L ). Clearly /Gamma1 ( L ) ⊆ /Gamma1 ( L ′ ) since in /Gamma1 can be replaced by in /Gamma1 ′ … \end{verbatim} ``` </details>
690. ph-0ee5430de170f73794f6 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) e A ′ i = Ai if Ai is a nonterminal and A ′ i = Xai if Ai is a terminal. Thus if we have V 1 a 1 V 2 a 2 V 3 a 3 . . . VnanVn + 1 where Vi ∈ N ∗ and ai is a terminal, replace it with V 1 Xa 1 V 2 Xa 2 V 3 Xa 3 . . . Vn Xan Vn + 1 and add productions Xai → ai for 1 ≤ i ≤ n . Let /Gamma1 ′ = ( N , T , S , P ′ ) be the new grammar formed. We need to show that /Gamma1 ( L ′ ) = /Gamma1 ( L ). Clearly /Gamma1 ( L ) ⊆ /Gamma1 ( L ′ ) since in /Gamma1 can be replaced by in /Gamma1 ′ . Conversely assume that in the derivation in /Gamma1 can be replaced by in /Gamma1 ′ . Conversely assume that in the derivation in /Gamma1 can be replaced by in /Gamma1 ′ . Conversely assume that in the derivation where U ⇒ ∗ w 1 and A ⇒ ∗ w and Vi ⇒ v i are productions in /Gamma1 where <!-… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{V 1 a 1 V 2 a 2 V 3 a 3 . . . VnanVn + 1 where Vi ∈ N ∗ and ai is a terminal, replace it with V 1 Xa 1 V 2 Xa 2 V 3 Xa 3 . . . Vn Xan Vn + 1 and add productions Xai → ai for 1 ≤ i ≤ n . Let /Gamma1 ′ = ( N , T , S , P ′ ) be the new grammar formed. We need to show that /Gamma1 (…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=40973 \begin{verbatim} V 1 a 1 V 2 a 2 V 3 a 3 . . . VnanVn + 1 where Vi ∈ N ∗ and ai is a terminal, replace it with V 1 Xa 1 V 2 Xa 2 V 3 Xa 3 . . . Vn Xan Vn + 1 and add productions Xai → ai for 1 ≤ i ≤ n . Let /Gamma1 ′ = ( N , T , S , P ′ ) be the new grammar formed. We need to show that /Gamma1 ( L ′ ) = /Gamma1 ( L ). Clearly /Gamma1 ( L ) ⊆ /Gamma1 ( L ′ ) since in /Gamma1 can be replaced by in /Gamma1 ′ . Conversely assume that in the derivation where U ⇒ ∗ w 1 and A ⇒ ∗ w and Vi ⇒ v i are productions in /Gamma1 where <!-… \end{verbatim} ``` </details>
692. ph-4f79a9fe5c1d74790431 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) 2 Xa 2 V 3 Xa 3 . . . Vn Xan Vn + 1 and add productions Xai → ai for 1 ≤ i ≤ n . Let /Gamma1 ′ = ( N , T , S , P ′ ) be the new grammar formed. We need to show that /Gamma1 ( L ′ ) = /Gamma1 ( L ). Clearly /Gamma1 ( L ) ⊆ /Gamma1 ( L ′ ) since in /Gamma1 can be replaced by in /Gamma1 ′ . Conversely assume that in the derivation where U ⇒ ∗ w 1 and A ⇒ ∗ w and Vi ⇒ v i are productions in /Gamma1 where is a production which is in /Gamma1 ′ and not in /Gamma1 and the derivation is … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{2 Xa 2 V 3 Xa 3 . . . Vn Xan Vn + 1 and add productions Xai → ai for 1 ≤ i ≤ n . Let /Gamma1 ′ = ( N , T , S , P ′ ) be the new grammar formed. We need to show that /Gamma1 ( L ′ ) = /Gamma1 ( L ). Clearly /Gamma1 ( L ) ⊆ /Gamma1 ( L ′ ) since in /Ga…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=41079 \begin{verbatim} 2 Xa 2 V 3 Xa 3 . . . Vn Xan Vn + 1 and add productions Xai → ai for 1 ≤ i ≤ n . Let /Gamma1 ′ = ( N , T , S , P ′ ) be the new grammar formed. We need to show that /Gamma1 ( L ′ ) = /Gamma1 ( L ). Clearly /Gamma1 ( L ) ⊆ /Gamma1 ( L ′ ) since in /Gamma1 can be replaced by in /Gamma1 ′ . Conversely assume that in the derivation where U ⇒ ∗ w 1 and A ⇒ ∗ w and Vi ⇒ v i are productions in /Gamma1 where is a production which is in /Gamma1 ′ and not in /Gamma1 and the derivation is … \end{verbatim} ``` </details>
693. ph-07fcbc27c3c49e22bd01 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) P ′ ) be the new grammar formed. We need to show that /Gamma1 ( L ′ ) = /Gamma1 ( L ). Clearly /Gamma1 ( L ) ⊆ /Gamma1 ( L ′ ) since in /Gamma1 can be replaced by in /Gamma1 ′ . Conversely assume that in the derivation where U ⇒ ∗ w 1 and A ⇒ ∗ w and Vi ⇒ v i are productions in /Gamma1 where is a production which is in /Gamma1 ′ and not in /Gamma1 and the derivation is This may be replaced by … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{P ′ ) be the new grammar formed. We need to show that /Gamma1 ( L ′ ) = /Gamma1 ( L ). Clearly /Gamma1 ( L ) ⊆ /Gamma1 ( L ′ ) since in /Gamma1 can be replaced by in /Gamma1 ′ . Conversely assume that in the derivation <!…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=41189 \begin{verbatim} P ′ ) be the new grammar formed. We need to show that /Gamma1 ( L ′ ) = /Gamma1 ( L ). Clearly /Gamma1 ( L ) ⊆ /Gamma1 ( L ′ ) since in /Gamma1 can be replaced by in /Gamma1 ′ . Conversely assume that in the derivation where U ⇒ ∗ w 1 and A ⇒ ∗ w and Vi ⇒ v i are productions in /Gamma1 where is a production which is in /Gamma1 ′ and not in /Gamma1 and the derivation is This may be replaced by … \end{verbatim} ``` </details>
694. ph-bfe249eea7787ca12648 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) ed. We need to show that /Gamma1 ( L ′ ) = /Gamma1 ( L ). Clearly /Gamma1 ( L ) ⊆ /Gamma1 ( L ′ ) since in /Gamma1 can be replaced by in /Gamma1 ′ . Conversely assume that in the derivation where U ⇒ ∗ w 1 and A ⇒ ∗ w and Vi ⇒ v i are productions in /Gamma1 where is a production which is in /Gamma1 ′ and not in /Gamma1 and the derivation is This may be replaced by We have a derivation for S ⇒ … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ed. We need to show that /Gamma1 ( L ′ ) = /Gamma1 ( L ). Clearly /Gamma1 ( L ) ⊆ /Gamma1 ( L ′ ) since in /Gamma1 can be replaced by in /Gamma1 ′ . Conversely assume that in the derivation wh…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=41219 \begin{verbatim} ed. We need to show that /Gamma1 ( L ′ ) = /Gamma1 ( L ). Clearly /Gamma1 ( L ) ⊆ /Gamma1 ( L ′ ) since in /Gamma1 can be replaced by in /Gamma1 ′ . Conversely assume that in the derivation where U ⇒ ∗ w 1 and A ⇒ ∗ w and Vi ⇒ v i are productions in /Gamma1 where is a production which is in /Gamma1 ′ and not in /Gamma1 and the derivation is This may be replaced by We have a derivation for S ⇒ … \end{verbatim} ``` </details>
695. ph-4ef942659c2c1d51cdec — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) ). Clearly /Gamma1 ( L ) ⊆ /Gamma1 ( L ′ ) since in /Gamma1 can be replaced by in /Gamma1 ′ . Conversely assume that in the derivation where U ⇒ ∗ w 1 and A ⇒ ∗ w and Vi ⇒ v i are productions in /Gamma1 where is a production which is in /Gamma1 ′ and not in /Gamma1 and the derivation is This may be replaced by We have a derivation for S ⇒ ∗ w 1 ww 2 in /Gamma1 . Hence /Gamma1 ′ ⊆ /Gamma1 . /sq… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{). Clearly /Gamma1 ( L ) ⊆ /Gamma1 ( L ′ ) since in /Gamma1 can be replaced by in /Gamma1 ′ . Conversely assume that in the derivation where U ⇒ ∗ w 1 and A ⇒ ∗ w and Vi ⇒ v i are productions …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=41274 \begin{verbatim} ). Clearly /Gamma1 ( L ) ⊆ /Gamma1 ( L ′ ) since in /Gamma1 can be replaced by in /Gamma1 ′ . Conversely assume that in the derivation where U ⇒ ∗ w 1 and A ⇒ ∗ w and Vi ⇒ v i are productions in /Gamma1 where is a production which is in /Gamma1 ′ and not in /Gamma1 and the derivation is This may be replaced by We have a derivation for S ⇒ ∗ w 1 ww 2 in /Gamma1 . Hence /Gamma1 ′ ⊆ /Gamma1 . /sq… \end{verbatim} ``` </details>
696. ph-0f343c9e28a805ba111f — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) replaced by We have a derivation for S ⇒ ∗ w 1 ww 2 in /Gamma1 . Hence /Gamma1 ′ ⊆ /Gamma1 . /square From the above lemmas we are now able to prove that a context-free grammar /Gamma1 whose language does not contain the empty word can be expressed in Chomsky normal form . Lemma 4.7 If /Gamma1 ( L ) , the language generated by /Gamma1 = ( N , T , S , P ) , does not contain the empty word, then there exists a grammar in which every production has either the form where A, B, and C are nonterminals and a is a t… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{replaced by We have a derivation for S ⇒ ∗ w 1 ww 2 in /Gamma1 . Hence /Gamma1 ′ ⊆ /Gamma1 . /square From the above lemmas we are now able to prove that a context-free grammar /Gamma1 whose language does not contain the empty word can be expressed in…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=41761 \begin{verbatim} replaced by We have a derivation for S ⇒ ∗ w 1 ww 2 in /Gamma1 . Hence /Gamma1 ′ ⊆ /Gamma1 . /square From the above lemmas we are now able to prove that a context-free grammar /Gamma1 whose language does not contain the empty word can be expressed in Chomsky normal form . Lemma 4.7 If /Gamma1 ( L ) , the language generated by /Gamma1 = ( N , T , S , P ) , does not contain the empty word, then there exists a grammar in which every production has either the form where A, B, and C are nonterminals and a is a t… \end{verbatim} ``` </details>
697. ph-98d1c2c88033c570e894 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) decoded --> We have a derivation for S ⇒ ∗ w 1 ww 2 in /Gamma1 . Hence /Gamma1 ′ ⊆ /Gamma1 . /square From the above lemmas we are now able to prove that a context-free grammar /Gamma1 whose language does not contain the empty word can be expressed in Chomsky normal form . Lemma 4.7 If /Gamma1 ( L ) , the language generated by /Gamma1 = ( N , T , S , P ) , does not contain the empty word, then there exists a grammar in which every production has either the form where A, B, and C are nonterminals and a is a terminal such that /Gamma1 ( L… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{decoded --> We have a derivation for S ⇒ ∗ w 1 ww 2 in /Gamma1 . Hence /Gamma1 ′ ⊆ /Gamma1 . /square From the above lemmas we are now able to prove that a context-free grammar /Gamma1 whose language does not contain the empty word can be expressed in Chomsky normal form . Lemma …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=41791 \begin{verbatim} decoded --> We have a derivation for S ⇒ ∗ w 1 ww 2 in /Gamma1 . Hence /Gamma1 ′ ⊆ /Gamma1 . /square From the above lemmas we are now able to prove that a context-free grammar /Gamma1 whose language does not contain the empty word can be expressed in Chomsky normal form . Lemma 4.7 If /Gamma1 ( L ) , the language generated by /Gamma1 = ( N , T , S , P ) , does not contain the empty word, then there exists a grammar in which every production has either the form where A, B, and C are nonterminals and a is a terminal such that /Gamma1 ( L… \end{verbatim} ``` </details>
698. ph-3807ee0e0e1af6916818 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) Gamma1 ( L ) = /Gamma1 ( L ′ ) . Proof By the previous lemma, in which every production has either the form A → A 1 A 2 A 3 . . . Am where A , A 1 , A 2 , A 3 , . . . , Am are nonterminals or A → a where A is a nonterminal and a is a terminal. We construct a new grammar by replacing every production of the form A → A 1 A 2 A 3 . . . Am by the set of productions A → A 1 X 1 , X 1 → A 2 X 2 , . . . , Xm -2 → Am -1 Am , where each replacement of a production in /Gamma1 uses a new set of symbols. is a derivation in /Gamma1 ′ , /Gamma1 ( L ) ⊆ /Gamma1 ( L ′ ). Converse… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{Gamma1 ( L ) = /Gamma1 ( L ′ ) . Proof By the previous lemma, in which every production has either the form A → A 1 A 2 A 3 . . . Am where A , A 1 , A 2 , A 3 , . . . , Am are nonterminals or A → a where A is a nonterminal and a is a terminal. We construct a new grammar by repla…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=42387 \begin{verbatim} Gamma1 ( L ) = /Gamma1 ( L ′ ) . Proof By the previous lemma, in which every production has either the form A → A 1 A 2 A 3 . . . Am where A , A 1 , A 2 , A 3 , . . . , Am are nonterminals or A → a where A is a nonterminal and a is a terminal. We construct a new grammar by replacing every production of the form A → A 1 A 2 A 3 . . . Am by the set of productions A → A 1 X 1 , X 1 → A 2 X 2 , . . . , Xm -2 → Am -1 Am , where each replacement of a production in /Gamma1 uses a new set of symbols. is a derivation in /Gamma1 ′ , /Gamma1 ( L ) ⊆ /Gamma1 ( L ′ ). Converse… \end{verbatim} ``` </details>
699. ph-ef20f97bc07e23039b48 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) is a derivation in /Gamma1 ′ , /Gamma1 ( L ) ⊆ /Gamma1 ( L ′ ). Conversely, if S ⇒ ∗ w in /Gamma1 ′ contains no productions which are not in /Gamma1 , then w ∈ /Gamma1 ( L ). If it does, let Wm bethelast term in the derivation containing a symbol in /Gamma1 ′ which is not in /Gamma1 so we have Wm ⇒ Wm + 1 ⇒ ∗ w and Wm ⇒ Wm + 1 has the form U ′ Xm -2 V ⇒ UAm -1 AmV . Therefore the derivation uses the set or of productions A → A 1 X 1 , X 1 → A 2 X 2 , . . . , Xm -2 → Am -1 Am and has the form where U ′ = UA ′ 1 A ′ 2 A ′ 3 · · · A ′ m -2 and Ai ⇒ ∗ Ai is a derivati… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{is a derivation in /Gamma1 ′ , /Gamma1 ( L ) ⊆ /Gamma1 ( L ′ ). Conversely, if S ⇒ ∗ w in /Gamma1 ′ contains no productions which are not in /Gamma1 , then w ∈ /Gamma1 ( L ). If it does, let Wm bethelast term in the derivation containing a symbol in /Gamma1 ′ which is not in /Ga…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=42917 \begin{verbatim} is a derivation in /Gamma1 ′ , /Gamma1 ( L ) ⊆ /Gamma1 ( L ′ ). Conversely, if S ⇒ ∗ w in /Gamma1 ′ contains no productions which are not in /Gamma1 , then w ∈ /Gamma1 ( L ). If it does, let Wm bethelast term in the derivation containing a symbol in /Gamma1 ′ which is not in /Gamma1 so we have Wm ⇒ Wm + 1 ⇒ ∗ w and Wm ⇒ Wm + 1 has the form U ′ Xm -2 V ⇒ UAm -1 AmV . Therefore the derivation uses the set or of productions A → A 1 X 1 , X 1 → A 2 X 2 , . . . , Xm -2 → Am -1 Am and has the form where U ′ = UA ′ 1 A ′ 2 A ′ 3 · · · A ′ m -2 and Ai ⇒ ∗ Ai is a derivati… \end{verbatim} ``` </details>
700. ph-4f375a98a8afee870d7e — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) ons, we have no bound on how many derivations may occur before the first terminal symbol appears at the left of the string. For example, using the production A → Aa , we can generate the string Aa n for arbitrary n , using n derivations without beginning a string with a terminal symbol. We can eliminate this particular problem by eliminating the productions of the form A → Aa . This is called elimination of left recursion . In a grammar /Gamma1 with no λ productions or trivial productions, let be productions in which the right-hand side of the production begins wi… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ons, we have no bound on how many derivations may occur before the first terminal symbol appears at the left of the string. For example, using the production A → Aa , we can generate the string Aa n for arbitrary n , using n derivations without beginning a string with a terminal…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=45041 \begin{verbatim} ons, we have no bound on how many derivations may occur before the first terminal symbol appears at the left of the string. For example, using the production A → Aa , we can generate the string Aa n for arbitrary n , using n derivations without beginning a string with a terminal symbol. We can eliminate this particular problem by eliminating the productions of the form A → Aa . This is called elimination of left recursion . In a grammar /Gamma1 with no λ productions or trivial productions, let be productions in which the right-hand side of the production begins wi… \end{verbatim} ``` </details>
701. ph-0c4130820c7d15f13428 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) e string. For example, using the production A → Aa , we can generate the string Aa n for arbitrary n , using n derivations without beginning a string with a terminal symbol. We can eliminate this particular problem by eliminating the productions of the form A → Aa . This is called elimination of left recursion . In a grammar /Gamma1 with no λ productions or trivial productions, let be productions in which the right-hand side of the production begins with an A and be productions in which the right-hand side of the production does not be… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{e string. For example, using the production A → Aa , we can generate the string Aa n for arbitrary n , using n derivations without beginning a string with a terminal symbol. We can eliminate this particular problem by eliminating the productions of the form A → Aa . This is call…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=45155 \begin{verbatim} e string. For example, using the production A → Aa , we can generate the string Aa n for arbitrary n , using n derivations without beginning a string with a terminal symbol. We can eliminate this particular problem by eliminating the productions of the form A → Aa . This is called elimination of left recursion . In a grammar /Gamma1 with no λ productions or trivial productions, let be productions in which the right-hand side of the production begins with an A and be productions in which the right-hand side of the production does not be… \end{verbatim} ``` </details>
702. ph-91ecdaaa78c6fa61a448 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) Vi A ′ and A ′ → Vi . - (2) Form productions A → Ui A ′ for 1 ≤ i ≤ n . Lemma 4.8 /Gamma1 ( L ) = /Gamma1 ′ ( L ) . Proof Let a derivation beginning with A have the form assuming A → Ui A ′ for 1 ≤ i ≤ n and we have A ⇒ AV (1) ⇒ AV (2) V (1) ⇒ ∗ AV ( k ) . . . V (2) V (1) ⇒ U ( i ) V ( k ) . . . V (2) V (1) where V ( j ) ∈ { V 1 , V 2 , . . . , Vn } for all 1 ≤ j ≤ k and U ( i ) ∈ { U 1 , U 2 , . . . , Um } . Therefore using leftmost derivation, any production containing A will have the form But can be replaced by But can be replaced by But can be replaced by Placing w on the left and W… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{oductions A → Ui A ′ for 1 ≤ i ≤ n . Lemma 4.8 /Gamma1 ( L ) = /Gamma1 ′ ( L ) . Proof Let a derivation beginning with A have the form assuming A → Ui A ′ for 1 ≤ i ≤ n and we have A ⇒ AV (1) ⇒ AV (2) V (1) ⇒ ∗ AV ( k ) . . . V (2) V (1) ⇒ U ( i ) V ( k ) . . . V (2) V (1) where…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=46018 \begin{verbatim} oductions A → Ui A ′ for 1 ≤ i ≤ n . Lemma 4.8 /Gamma1 ( L ) = /Gamma1 ′ ( L ) . Proof Let a derivation beginning with A have the form assuming A → Ui A ′ for 1 ≤ i ≤ n and we have A ⇒ AV (1) ⇒ AV (2) V (1) ⇒ ∗ AV ( k ) . . . V (2) V (1) ⇒ U ( i ) V ( k ) . . . V (2) V (1) where V ( j ) ∈ { V 1 , V 2 , . . . , Vn } for all 1 ≤ j ≤ k and U ( i ) ∈ { U 1 , U 2 , . . . , Um } . Therefore using leftmost derivation, any production containing A will have the form But can be replaced by Placing w on the left and W… \end{verbatim} ``` </details>
704. ph-a12cad12593ffb779566 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) amma1 ( L ) = /Gamma1 ′ ( L ) . Proof Let a derivation beginning with A have the form assuming A → Ui A ′ for 1 ≤ i ≤ n and we have A ⇒ AV (1) ⇒ AV (2) V (1) ⇒ ∗ AV ( k ) . . . V (2) V (1) ⇒ U ( i ) V ( k ) . . . V (2) V (1) where V ( j ) ∈ { V 1 , V 2 , . . . , Vn } for all 1 ≤ j ≤ k and U ( i ) ∈ { U 1 , U 2 , . . . , Um } . Therefore using leftmost derivation, any production containing A will have the form But can be replaced by Placing w on the left and W on the right of each term, we have, w AW ⇒ ∗ w U… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{amma1 ( L ) = /Gamma1 ′ ( L ) . Proof Let a derivation beginning with A have the form assuming A → Ui A ′ for 1 ≤ i ≤ n and we have A ⇒ AV (1) ⇒ AV (2) V (1) ⇒ ∗ AV ( k ) . . . V (2) V (1) ⇒ U ( i ) V ( k ) . . . V (2) V (1) where V ( j ) ∈ \{ V 1 , V 2 , . . . , Vn \} for all 1 ≤…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=46068 \begin{verbatim} amma1 ( L ) = /Gamma1 ′ ( L ) . Proof Let a derivation beginning with A have the form assuming A → Ui A ′ for 1 ≤ i ≤ n and we have A ⇒ AV (1) ⇒ AV (2) V (1) ⇒ ∗ AV ( k ) . . . V (2) V (1) ⇒ U ( i ) V ( k ) . . . V (2) V (1) where V ( j ) ∈ { V 1 , V 2 , . . . , Vn } for all 1 ≤ j ≤ k and U ( i ) ∈ { U 1 , U 2 , . . . , Um } . Therefore using leftmost derivation, any production containing A will have the form But can be replaced by Placing w on the left and W on the right of each term, we have, w AW ⇒ ∗ w U… \end{verbatim} ``` </details>
705. ph-46546a1dfe5248c926b2 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) ree grammar can be expressed in Greibach normal form can be proved by first expressing the grammar in Chomsky normal form. We shall not do so however so that the development for Chomsky normal form may be omitted if desired. Using the above lemmas, we are about to take a giant leap toward proving that every context-free grammar can be expressed in Greibach normal form. Lemma 4.10 Any context-free grammar which does not generate λ can be expressed so that each of its productions is of the form where a is a terminal and W is a string which is empty or consists of a … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ree grammar can be expressed in Greibach normal form can be proved by first expressing the grammar in Chomsky normal form. We shall not do so however so that the development for Chomsky normal form may be omitted if desired. Using the above lemmas, we are about to take a giant l…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=47409 \begin{verbatim} ree grammar can be expressed in Greibach normal form can be proved by first expressing the grammar in Chomsky normal form. We shall not do so however so that the development for Chomsky normal form may be omitted if desired. Using the above lemmas, we are about to take a giant leap toward proving that every context-free grammar can be expressed in Greibach normal form. Lemma 4.10 Any context-free grammar which does not generate λ can be expressed so that each of its productions is of the form where a is a terminal and W is a string which is empty or consists of a … \end{verbatim} ``` </details>
706. ph-2eff7282856c6ab91ce8 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) m. Lemma 4.10 Any context-free grammar which does not generate λ can be expressed so that each of its productions is of the form where a is a terminal and W is a string which is empty or consists of a string of terminals and/or nonterminals. Proof We first order the nonterminals beginning with S , the start symbol. For simplicity, let the nonterminals be A 1 , A 2 , A 3 , . . . , Am . Our first goal is to change every production so that it is either in the form where a is a terminal and W is a string which is empty or consists of a str… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{m. Lemma 4.10 Any context-free grammar which does not generate λ can be expressed so that each of its productions is of the form where a is a terminal and W is a string which is empty or consists of a string of terminals and/or nonterminals. Proof We…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=47778 \begin{verbatim} m. Lemma 4.10 Any context-free grammar which does not generate λ can be expressed so that each of its productions is of the form where a is a terminal and W is a string which is empty or consists of a string of terminals and/or nonterminals. Proof We first order the nonterminals beginning with S , the start symbol. For simplicity, let the nonterminals be A 1 , A 2 , A 3 , . . . , Am . Our first goal is to change every production so that it is either in the form where a is a terminal and W is a string which is empty or consists of a str… \end{verbatim} ``` </details>
707. ph-7d92c11ffdf50253adce — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) where a is a terminal and W is a string which is empty or consists of a string of terminals and/or nonterminals. Proof We first order the nonterminals beginning with S , the start symbol. For simplicity, let the nonterminals be A 1 , A 2 , A 3 , . . . , Am . Our first goal is to change every production so that it is either in the form where a is a terminal and W is a string which is empty or consists of a string of terminals and/or nonterminals, or in the form where i < j and Y consists of a string of terminals and/or nonterminals. … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{where a is a terminal and W is a string which is empty or consists of a string of terminals and/or nonterminals. Proof We first order the nonterminals beginning with S , the start symbol. For simplicity, let the nonterminals be A 1 , A 2 , A 3 , . . . , Am . Our first goal is to…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=47937 \begin{verbatim} where a is a terminal and W is a string which is empty or consists of a string of terminals and/or nonterminals. Proof We first order the nonterminals beginning with S , the start symbol. For simplicity, let the nonterminals be A 1 , A 2 , A 3 , . . . , Am . Our first goal is to change every production so that it is either in the form where a is a terminal and W is a string which is empty or consists of a string of terminals and/or nonterminals, or in the form where i < j and Y consists of a string of terminals and/or nonterminals. … \end{verbatim} ``` </details>
708. ph-63fd56b71bb78426d3b7 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) y Ai where i < k . We now prove the statement for i = k . In each case where Ak → AjY is a production for k > j , use the procedure in Lemma 4.9 to eliminate Aj . When Ak → AkY is a production, use the process of elimination of left recursion to remove Ak from the right-hand side. Therefore by induction we have every production so that it is either in the form where a is a terminal and W is a string which is empty or consists of a string of terminals and/or nonterminals, or in the form where i < j and Y consists of a string of terminals and/or nonterminal… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{y Ai where i \< k . We now prove the statement for i = k . In each case where Ak → AjY is a production for k \> j , use the procedure in Lemma 4.9 to eliminate Aj . When Ak → AkY is a production, use the process of elimination of left recursion to remove Ak from the right-ha…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=49025 \begin{verbatim} y Ai where i < k . We now prove the statement for i = k . In each case where Ak → AjY is a production for k > j , use the procedure in Lemma 4.9 to eliminate Aj . When Ak → AkY is a production, use the process of elimination of left recursion to remove Ak from the right-hand side. Therefore by induction we have every production so that it is either in the form where a is a terminal and W is a string which is empty or consists of a string of terminals and/or nonterminals, or in the form where i < j and Y consists of a string of terminals and/or nonterminal… \end{verbatim} ``` </details>
709. ph-211712d79ccb90c17e5b — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) nterminals. Anyproduction with Am onthe left-hand side must have the form Am → aW since there is no nonterminal larger than Am . If there is a production of the form Am -1 → AmW ′ , use the procedures in Lemma 4.9 to eliminate Am . The result is a production of the form Am → bW ′′ . Assume k is the largest value so Ak → AjY is a production where k < j . Again using the procedures in Lemma 4.9 to eliminate Aj , we have a procedure of the form Ak → aW . When the process is completed, we have where a is a terminal and W is a string which is empty or consists of a … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{nterminals. Anyproduction with Am onthe left-hand side must have the form Am → aW since there is no nonterminal larger than Am . If there is a production of the form Am -1 → AmW ′ , use the procedures in Lemma 4.9 to eliminate Am . The result is a production of the form Am → bW …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=49619 \begin{verbatim} nterminals. Anyproduction with Am onthe left-hand side must have the form Am → aW since there is no nonterminal larger than Am . If there is a production of the form Am -1 → AmW ′ , use the procedures in Lemma 4.9 to eliminate Am . The result is a production of the form Am → bW ′′ . Assume k is the largest value so Ak → AjY is a production where k < j . Again using the procedures in Lemma 4.9 to eliminate Aj , we have a procedure of the form Ak → aW . When the process is completed, we have where a is a terminal and W is a string which is empty or consists of a … \end{verbatim} ``` </details>
710. ph-a60e8e293b846139345e — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) i , it is impossible to have a production of the form Bi → Bj W . Therefore productions with Bi on the left have the form Bi → aW or Bi → Aj W . Repeating the process above we can change these to the form Bi → aW , and the lemma is proved. /square Theorem 4.5 Every context-free grammar whose language does not contain λ can be expressed in Greibach normal form. Proof Weoutline the proof. The details are left to the reader. Since we already know that every production can be written in the form where a is a terminal and W is a string which is empty or consists of a s… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{i , it is impossible to have a production of the form Bi → Bj W . Therefore productions with Bi on the left have the form Bi → aW or Bi → Aj W . Repeating the process above we can change these to the form Bi → aW , and the lemma is proved. /square Theorem 4.5 Every context-free …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=50394 \begin{verbatim} i , it is impossible to have a production of the form Bi → Bj W . Therefore productions with Bi on the left have the form Bi → aW or Bi → Aj W . Repeating the process above we can change these to the form Bi → aW , and the lemma is proved. /square Theorem 4.5 Every context-free grammar whose language does not contain λ can be expressed in Greibach normal form. Proof Weoutline the proof. The details are left to the reader. Since we already know that every production can be written in the form where a is a terminal and W is a string which is empty or consists of a s… \end{verbatim} ``` </details>
711. ph-aebd9e13f91d776bf84f — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) /Gamma1 with B on the left. Let /Gamma1 ′ be the grammar with production A → UBV removed and the productions A → UWiV for 1 ≤ i ≤ m added, then /Gamma1 ( L ) = /Gamma1 ′ ( L ),' prove /Gamma1 ( L ) ⊆ /Gamma1 ′ ( L ). - (2) Prove Theorem 4.5 'Every context-free grammar can be expressed in Greibach normal form.' - (3) Complete the proof of Lemma 4.8. - (4) Let /Gamma1 ′′ = ( N , /Sigma1 , S , P ) be the grammar defined by N = { S , A , B } , /Sigma1 = { a , b } , and P be the set of productions Express this grammar in Chomsky normal form. - (5) Express the previous … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{/Gamma1 with B on the left. Let /Gamma1 ′ be the grammar with production A → UBV removed and the productions A → UWiV for 1 ≤ i ≤ m added, then /Gamma1 ( L ) = /Gamma1 ′ ( L ),' prove /Gamma1 ( L ) ⊆ /Gamma1 ′ ( L ). - (2) Prove Theorem 4.5 'Every context-free grammar can be exp…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=51539 \begin{verbatim} /Gamma1 with B on the left. Let /Gamma1 ′ be the grammar with production A → UBV removed and the productions A → UWiV for 1 ≤ i ≤ m added, then /Gamma1 ( L ) = /Gamma1 ′ ( L ),' prove /Gamma1 ( L ) ⊆ /Gamma1 ′ ( L ). - (2) Prove Theorem 4.5 'Every context-free grammar can be expressed in Greibach normal form.' - (3) Complete the proof of Lemma 4.8. - (4) Let /Gamma1 ′′ = ( N , /Sigma1 , S , P ) be the grammar defined by N = { S , A , B } , /Sigma1 = { a , b } , and P be the set of productions Express this grammar in Chomsky normal form. - (5) Express the previous … \end{verbatim} ``` </details>
712. ph-aef9b3fb6cae99fed5fa — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) ext-free grammar can be expressed in Greibach normal form.' - (3) Complete the proof of Lemma 4.8. - (4) Let /Gamma1 ′′ = ( N , /Sigma1 , S , P ) be the grammar defined by N = { S , A , B } , /Sigma1 = { a , b } , and P be the set of productions Express this grammar in Chomsky normal form. - (5) Express the previous grammar in Greibach normal form. - (6) Let /Gamma1 = ( N , T , S , P ) be the grammar with N = { S } , T = { a , b } , and P contain the productions Express this grammar in Chomsky normal form. - (7) Express the previous gr… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ext-free grammar can be expressed in Greibach normal form.' - (3) Complete the proof of Lemma 4.8. - (4) Let /Gamma1 ′′ = ( N , /Sigma1 , S , P ) be the grammar defined by N = \{ S , A , B \} , /Sigma1 = \{ a , b \} , and P be the set of productions Expr…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=51792 \begin{verbatim} ext-free grammar can be expressed in Greibach normal form.' - (3) Complete the proof of Lemma 4.8. - (4) Let /Gamma1 ′′ = ( N , /Sigma1 , S , P ) be the grammar defined by N = { S , A , B } , /Sigma1 = { a , b } , and P be the set of productions Express this grammar in Chomsky normal form. - (5) Express the previous grammar in Greibach normal form. - (6) Let /Gamma1 = ( N , T , S , P ) be the grammar with N = { S } , T = { a , b } , and P contain the productions Express this grammar in Chomsky normal form. - (7) Express the previous gr… \end{verbatim} ``` </details>
713. ph-5632616cd5a7d43e8c79 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) this grammar in Chomsky normal form. - (5) Express the previous grammar in Greibach normal form. - (6) Let /Gamma1 = ( N , T , S , P ) be the grammar with N = { S } , T = { a , b } , and P contain the productions Express this grammar in Chomsky normal form. - (7) Express the previous grammar in Greibach normal form. - (8) Let /Gamma1 ′′ = ( N , /Sigma1 , S , P ) be the grammar defined by N = { S , A , B } , /Sigma1 = { a , b } , and P be the set of productions Express this grammar in Chomsky normal form. - (9) Express the previous gram… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{this grammar in Chomsky normal form. - (5) Express the previous grammar in Greibach normal form. - (6) Let /Gamma1 = ( N , T , S , P ) be the grammar with N = \{ S \} , T = \{ a , b \} , and P contain the productions Express this grammar in Chomsky norma…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=52076 \begin{verbatim} this grammar in Chomsky normal form. - (5) Express the previous grammar in Greibach normal form. - (6) Let /Gamma1 = ( N , T , S , P ) be the grammar with N = { S } , T = { a , b } , and P contain the productions Express this grammar in Chomsky normal form. - (7) Express the previous grammar in Greibach normal form. - (8) Let /Gamma1 ′′ = ( N , /Sigma1 , S , P ) be the grammar defined by N = { S , A , B } , /Sigma1 = { a , b } , and P be the set of productions Express this grammar in Chomsky normal form. - (9) Express the previous gram… \end{verbatim} ``` </details>
714. ph-cc6569b52389f9e0c5a5 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) e first part of the production with the second part, so does the PDA. The stack then resembles the strings derived in the grammar except that the terminals on the left of the derived string (top of the stack) are then removed as they occur in the stack and compared with the letters on the tape. As before a word is accepted if the word has been read and the stack is empty. Example 4.17 Let /Gamma1 = ( N , T , S , P ) be the grammar with N = { S } , T = { a , b } , and P contain the productions which generates the language { ww R : w ∈ T ∗ } . This has the PDA ![Ima… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{e first part of the production with the second part, so does the PDA. The stack then resembles the strings derived in the grammar except that the terminals on the left of the derived string (top of the stack) are then removed as they occur in the stack and compared with the lett…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=56256 \begin{verbatim} e first part of the production with the second part, so does the PDA. The stack then resembles the strings derived in the grammar except that the terminals on the left of the derived string (top of the stack) are then removed as they occur in the stack and compared with the letters on the tape. As before a word is accepted if the word has been read and the stack is empty. Example 4.17 Let /Gamma1 = ( N , T , S , P ) be the grammar with N = { S } , T = { a , b } , and P contain the productions which generates the language { ww R : w ∈ T ∗ } . This has the PDA ![Ima… \end{verbatim} ``` </details>
715. ph-9ce126f9f417d44d535c — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) | | t | pop a | Sa | abba | t | read b | a | a | | t | read a | Sa | bba | t | pop a | a | λ | | t | pop S | a | bba | t | read a | λ | λ | | t | push bSb | bSba | bba | t | accept | λ | λ | Example 4.18 Let /Gamma1 = ( N , T , S , P ) be the grammar with N = { S } , T = { a , b } , and P contain the productions which generates the language { w : w ∈ A ∗ and contains an even number of a s and an even number of b s. } . This has the PDA ![Image](./AutomataTheory_chapter_2.1_artifacts/image_000027_70fd125189dbb28a3982406b175df13baf1d6fd0ab24cca159fd284d49179241.png)… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{| | t | pop a | Sa | abba | t | read b | a | a | | t | read a | Sa | bba | t | pop a | a | λ | | t | pop S | a | bba | t | read a | λ | λ | | t | push bSb | bSba | bba | t | accept | λ | λ | Example 4.18 Let /Gamma1 = ( N , T , S , P ) be the grammar with N = \{ S \} , T = \{ a , b…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=57608 \begin{verbatim} | | t | pop a | Sa | abba | t | read b | a | a | | t | read a | Sa | bba | t | pop a | a | λ | | t | pop S | a | bba | t | read a | λ | λ | | t | push bSb | bSba | bba | t | accept | λ | λ | Example 4.18 Let /Gamma1 = ( N , T , S , P ) be the grammar with N = { S } , T = { a , b } , and P contain the productions which generates the language { w : w ∈ A ∗ and contains an even number of a s and an even number of b s. } . This has the PDA ![Image](./AutomataTheory_chapter_2.1_artifacts/image_000027_70fd125189dbb28a3982406b175df13baf1d6fd0ab24cca159fd284d49179241.png)… \end{verbatim} ``` </details>
716. ph-d1480b3e5b05bdff7d15 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) n condition ( ab , s , BaaB ) gives notation ( ab , s , BaaB ) /turnstileleft ( b , t , aaB ) and the transition (( a , s , λ ) , ( t , D )) when the PDA is in condition ( ab , s , BaaB ) gives notation ( ab , s , BaaB ) /turnstileleft ( b , t , DBaaB ). We say that ( s , u , V ) /turnstileleft ∗ ( t , v, W ) if the PDA can be changed from ( s , u , V ) to ( t , v, W ) in a finite number of transitions. Lemma 4.11 Let /Gamma1 ( L ) be a context-free language. There exists a PDA that accepts L where /Sigma1 is a finite alphabet, Q is a finite set of states, s is th… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{n condition ( ab , s , BaaB ) gives notation ( ab , s , BaaB ) /turnstileleft ( b , t , aaB ) and the transition (( a , s , λ ) , ( t , D )) when the PDA is in condition ( ab , s , BaaB ) gives notation ( ab , s , BaaB ) /turnstileleft ( b , t , DBaaB ). We say that ( s , u , V …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=60391 \begin{verbatim} n condition ( ab , s , BaaB ) gives notation ( ab , s , BaaB ) /turnstileleft ( b , t , aaB ) and the transition (( a , s , λ ) , ( t , D )) when the PDA is in condition ( ab , s , BaaB ) gives notation ( ab , s , BaaB ) /turnstileleft ( b , t , DBaaB ). We say that ( s , u , V ) /turnstileleft ∗ ( t , v, W ) if the PDA can be changed from ( s , u , V ) to ( t , v, W ) in a finite number of transitions. Lemma 4.11 Let /Gamma1 ( L ) be a context-free language. There exists a PDA that accepts L where /Sigma1 is a finite alphabet, Q is a finite set of states, s is th… \end{verbatim} ``` </details>
717. ph-4df14d3b30bab4b4c3b8 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) hat ( s , u , V ) /turnstileleft ∗ ( t , v, W ) if the PDA can be changed from ( s , u , V ) to ( t , v, W ) in a finite number of transitions. Lemma 4.11 Let /Gamma1 ( L ) be a context-free language. There exists a PDA that accepts L where /Sigma1 is a finite alphabet, Q is a finite set of states, s is the initial or starting state, I is a finite of stack symbols, ϒ is the transition relation, and F is the set of acceptance states. The relation ϒ is a subset of Proof As previously mentioned, we shall assume that the PDA has two states… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{hat ( s , u , V ) /turnstileleft ∗ ( t , v, W ) if the PDA can be changed from ( s , u , V ) to ( t , v, W ) in a finite number of transitions. Lemma 4.11 Let /Gamma1 ( L ) be a context-free language. There exists a PDA that accepts L where /Sigma1 i…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=60654 \begin{verbatim} hat ( s , u , V ) /turnstileleft ∗ ( t , v, W ) if the PDA can be changed from ( s , u , V ) to ( t , v, W ) in a finite number of transitions. Lemma 4.11 Let /Gamma1 ( L ) be a context-free language. There exists a PDA that accepts L where /Sigma1 is a finite alphabet, Q is a finite set of states, s is the initial or starting state, I is a finite of stack symbols, ϒ is the transition relation, and F is the set of acceptance states. The relation ϒ is a subset of Proof As previously mentioned, we shall assume that the PDA has two states… \end{verbatim} ``` </details>
718. ph-51aa0936bdf11c91e34a — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) ins with a nonterminal or is empty, then ( t , α, S ) /turnstileleft ∗ ( t , λ, β ). Hence if S ⇒ ∗ α in /Gamma1 where α ∈ T ∗ , then ( s , α, λ ) /turnstileleft ( t , α, S ) /turnstileleft ∗ ( t , λ, λ ), and α is accepted by the PDA. We prove this using induction on the length of the derivation. Suppose n = 0, but then we have S ⇒ ∗ S , so α = λ , β = S , and ( t , λ, S ) /turnstileleft ∗ ( t , λ, S ) gives us ( t , α, S ) /turnstileleft ∗ ( t , λ, β ). Now assume S ⇒ ∗ γ in k + 1 steps. Say Then there is a first nonterminal B in the string mk and a production B… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ins with a nonterminal or is empty, then ( t , α, S ) /turnstileleft ∗ ( t , λ, β ). Hence if S ⇒ ∗ α in /Gamma1 where α ∈ T ∗ , then ( s , α, λ ) /turnstileleft ( t , α, S ) /turnstileleft ∗ ( t , λ, λ ), and α is accepted by the PDA. We prove this using induction on the length…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=63652 \begin{verbatim} ins with a nonterminal or is empty, then ( t , α, S ) /turnstileleft ∗ ( t , λ, β ). Hence if S ⇒ ∗ α in /Gamma1 where α ∈ T ∗ , then ( s , α, λ ) /turnstileleft ( t , α, S ) /turnstileleft ∗ ( t , λ, λ ), and α is accepted by the PDA. We prove this using induction on the length of the derivation. Suppose n = 0, but then we have S ⇒ ∗ S , so α = λ , β = S , and ( t , λ, S ) /turnstileleft ∗ ( t , λ, S ) gives us ( t , α, S ) /turnstileleft ∗ ( t , λ, β ). Now assume S ⇒ ∗ γ in k + 1 steps. Say Then there is a first nonterminal B in the string mk and a production B… \end{verbatim} ``` </details>
719. ph-5e5f36fc541dbabf652f — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) ut read in passing from state p to state q and leaving the stack as it was in state p . The productions consist of the following four types. - (1) For each q T , the production S s , λ, q . - (2) For each transition (( p , a , B ) , ( q , D )) ∈ ϒ , where B , D ∈ I ∪ { λ } , the productions p , B , t a q , D , t for all t Q . 3. ∈ →〈 〉 4. 〈 〉 → 〈 〉 ∈ 5. { } 1 2 Bn ∈ - (3) For each transition (( p , a , D ) , ( q , B 1 B 2 . . . Bn )) ∈ ϒ , where D ∈ I ∪ λ , B , B , . . . , C , the productions for all q 1 , q 2 , . . . , qn -1 , t ∈ Q . - (4) For each q ∈ Q , the p… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ut read in passing from state p to state q and leaving the stack as it was in state p . The productions consist of the following four types. - (1) For each q T , the production S s , λ, q . - (2) For each transition (( p , a , B ) , ( q , D )) ∈ ϒ , where B , D ∈ I ∪ \{ λ \} , the…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=66838 \begin{verbatim} ut read in passing from state p to state q and leaving the stack as it was in state p . The productions consist of the following four types. - (1) For each q T , the production S s , λ, q . - (2) For each transition (( p , a , B ) , ( q , D )) ∈ ϒ , where B , D ∈ I ∪ { λ } , the productions p , B , t a q , D , t for all t Q . 3. ∈ →〈 〉 4. 〈 〉 → 〈 〉 ∈ 5. { } 1 2 Bn ∈ - (3) For each transition (( p , a , D ) , ( q , B 1 B 2 . . . Bn )) ∈ ϒ , where D ∈ I ∪ λ , B , B , . . . , C , the productions for all q 1 , q 2 , . . . , qn -1 , t ∈ Q . - (4) For each q ∈ Q , the p… \end{verbatim} ``` </details>
720. ph-5f6b51ff08fb2e3df05b — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) h other nonterminals. Hence a word in the language of the grammar cannot be generated without these productions. Lemma 4.12 A language M ( L ) accepted by a pushdown automaton M = ( /Sigma1 , Q , s , I , ϒ, F ) , is a context-free language. Proof Using the grammar /Gamma1 = ( N , /Sigma1 , S , P ), where the nonterminals and productions are described above, we show that /Gamma1 generates the same language as accepted by M . We first show that for p , q ∈ Q , B ∈ I ∪ { λ } and w ∈ A ∗ , that Thus for t ∈ Q , 〈 s , λ, t 〉 ⇒ ∗ w if and only if ( s , w, λ ) /turnstile… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{h other nonterminals. Hence a word in the language of the grammar cannot be generated without these productions. Lemma 4.12 A language M ( L ) accepted by a pushdown automaton M = ( /Sigma1 , Q , s , I , ϒ, F ) , is a context-free language. Proof Using the grammar /Gamma1 = ( N …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=68915 \begin{verbatim} h other nonterminals. Hence a word in the language of the grammar cannot be generated without these productions. Lemma 4.12 A language M ( L ) accepted by a pushdown automaton M = ( /Sigma1 , Q , s , I , ϒ, F ) , is a context-free language. Proof Using the grammar /Gamma1 = ( N , /Sigma1 , S , P ), where the nonterminals and productions are described above, we show that /Gamma1 generates the same language as accepted by M . We first show that for p , q ∈ Q , B ∈ I ∪ { λ } and w ∈ A ∗ , that Thus for t ∈ Q , 〈 s , λ, t 〉 ⇒ ∗ w if and only if ( s , w, λ ) /turnstile… \end{verbatim} ``` </details>
721. ph-c7ad9d53023944dfecb9 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) nstileleft ∗ ( p , λ, λ ) which is obvious. Assume n = k > 1, then the first production can only be of type (2) or type (3). If it is type (2), we have 〈 p , B , q 〉 → a 〈 r , D , q 〉 for p , r ∈ Q , where (( p , a , B ) , ( r , D )) ∈ ϒ . Henceletting w = a v , ( p , w, B ) /turnstileleft ( q , v, D ) and by induction if 〈 r , D , q 〉 ⇒ ∗ v then ( q , v, D ) /turnstileleft ∗ ( q , λ, λ ). Therefore ( p , w, B ) /turnstileleft ∗ ( q , λ, λ ). If the first production is of type (3), we have and (( p , a , B ) , ( q , B 1 B 2 . . . Bn )) ∈ ϒ . So if w = a v , ( p… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{nstileleft ∗ ( p , λ, λ ) which is obvious. Assume n = k \> 1, then the first production can only be of type (2) or type (3). If it is type (2), we have 〈 p , B , q 〉 → a 〈 r , D , q 〉 for p , r ∈ Q , where (( p , a , B ) , ( r , D )) ∈ ϒ . Henceletting w = a v , ( p , w, B ) …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=69980 \begin{verbatim} nstileleft ∗ ( p , λ, λ ) which is obvious. Assume n = k > 1, then the first production can only be of type (2) or type (3). If it is type (2), we have 〈 p , B , q 〉 → a 〈 r , D , q 〉 for p , r ∈ Q , where (( p , a , B ) , ( r , D )) ∈ ϒ . Henceletting w = a v , ( p , w, B ) /turnstileleft ( q , v, D ) and by induction if 〈 r , D , q 〉 ⇒ ∗ v then ( q , v, D ) /turnstileleft ∗ ( q , λ, λ ). Therefore ( p , w, B ) /turnstileleft ∗ ( q , λ, λ ). If the first production is of type (3), we have and (( p , a , B ) , ( q , B 1 B 2 . . . Bn )) ∈ ϒ . So if w = a v , ( p… \end{verbatim} ``` </details>
722. ph-6b9a3830b55d975bb82b — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) ileleft ∗ ( q , λ, λ ). Therefore ( p , w, B ) /turnstileleft ∗ ( q , λ, λ ). If the first production is of type (3), we have and (( p , a , B ) , ( q , B 1 B 2 . . . Bn )) ∈ ϒ . So if w = a v , ( p , w, B ) /turnstileleft ( q , v, B 1 B 2 . . . Bn ). For convenience of notation, let q = qn . Let 〈 qi -1 , Bi , qi 〉 ⇒ ∗ ui so that w = au 1 u 2 . . . un and v = u 1 u 2 . . . un . By induction, ( qi -1 , ui , Bi ) /turnstileleft ∗ ( qi , λ, λ ). Therefore we have so that ( p , w, B ) ( q , λ, λ ). We now show that if ( p , w, B ) /turnst… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ileleft ∗ ( q , λ, λ ). Therefore ( p , w, B ) /turnstileleft ∗ ( q , λ, λ ). If the first production is of type (3), we have and (( p , a , B ) , ( q , B 1 B 2 . . . Bn )) ∈ ϒ . So if w = a v , ( p , w, B ) /turnstileleft ( q , v, B 1 B 2 . . . Bn )…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=70352 \begin{verbatim} ileleft ∗ ( q , λ, λ ). Therefore ( p , w, B ) /turnstileleft ∗ ( q , λ, λ ). If the first production is of type (3), we have and (( p , a , B ) , ( q , B 1 B 2 . . . Bn )) ∈ ϒ . So if w = a v , ( p , w, B ) /turnstileleft ( q , v, B 1 B 2 . . . Bn ). For convenience of notation, let q = qn . Let 〈 qi -1 , Bi , qi 〉 ⇒ ∗ ui so that w = au 1 u 2 . . . un and v = u 1 u 2 . . . un . By induction, ( qi -1 , ui , Bi ) /turnstileleft ∗ ( qi , λ, λ ). Therefore we have so that ( p , w, B ) ( q , λ, λ ). We now show that if ( p , w, B ) /turnst… \end{verbatim} ``` </details>
723. ph-6f6ad060536bf787891b — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) o that ( p , w, B ) ( q , λ, λ ). We now show that if ( p , w, B ) /turnstileleft ∗ ( q , λ, λ ) then 〈 p , B , q 〉 ⇒ ∗ w . We use induction on the number of steps in ( p , w, B ) /turnstileleft ∗ ( q , λ, λ ). If there are 0 steps, then p = q and w = B = λ . This corresponds to 〈 p , λ, p 〉 ⇒ λ which is one of the productions. Therefore the statement is true for 0 steps. /turnstileleft ∗ Assume ( p , w, B ) /turnstileleft ∗ ( q , λ, λ ) in k + 1 steps. First assume that we have w = a v and where (( p , a , B ) , ( r , D )) ∈ ϒ , and B , D ∈ I ∪ { λ } , giving pro… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{o that ( p , w, B ) ( q , λ, λ ). We now show that if ( p , w, B ) /turnstileleft ∗ ( q , λ, λ ) then 〈 p , B , q 〉 ⇒ ∗ w . We use induction on the number of steps in ( p , w, B ) /turnstileleft ∗ ( q , λ, λ ). If there are 0 steps, then p = q and w = B = λ . This corresponds to…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=70883 \begin{verbatim} o that ( p , w, B ) ( q , λ, λ ). We now show that if ( p , w, B ) /turnstileleft ∗ ( q , λ, λ ) then 〈 p , B , q 〉 ⇒ ∗ w . We use induction on the number of steps in ( p , w, B ) /turnstileleft ∗ ( q , λ, λ ). If there are 0 steps, then p = q and w = B = λ . This corresponds to 〈 p , λ, p 〉 ⇒ λ which is one of the productions. Therefore the statement is true for 0 steps. /turnstileleft ∗ Assume ( p , w, B ) /turnstileleft ∗ ( q , λ, λ ) in k + 1 steps. First assume that we have w = a v and where (( p , a , B ) , ( r , D )) ∈ ϒ , and B , D ∈ I ∪ { λ } , giving pro… \end{verbatim} ``` </details>
724. ph-2805eaf7bd991256b936 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) ∗ ( q , λ, λ ) in k + 1 steps. First assume that we have w = a v and where (( p , a , B ) , ( r , D )) ∈ ϒ , and B , D ∈ I ∪ { λ } , giving productions 〈 p , B , q 〉 → a 〈 r , D , q 〉 . Since ( q , v, D ) /turnstileleft ∗ ( q , λ, λ ) by induction hypothesis, 〈 r , D , q 〉 ⇒ ∗ v . Therefore 〈 p , B , q 〉 ⇒ a 〈 r , D , q 〉 ⇒ ∗ a v = w and we are finished. Next assume w = a v and the first step is ( p , w, B ) /turnstileleft ( q , v, B 1 B 2 . . . Bn ) so we have and each Bi is eventually removed from the sta… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{∗ ( q , λ, λ ) in k + 1 steps. First assume that we have w = a v and where (( p , a , B ) , ( r , D )) ∈ ϒ , and B , D ∈ I ∪ \{ λ \} , giving productions 〈 p , B , q 〉 → a 〈 r , D , q 〉 . Since ( q , v, D ) /turnstileleft ∗ ( q , λ, λ ) by induction hy…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=71312 \begin{verbatim} ∗ ( q , λ, λ ) in k + 1 steps. First assume that we have w = a v and where (( p , a , B ) , ( r , D )) ∈ ϒ , and B , D ∈ I ∪ { λ } , giving productions 〈 p , B , q 〉 → a 〈 r , D , q 〉 . Since ( q , v, D ) /turnstileleft ∗ ( q , λ, λ ) by induction hypothesis, 〈 r , D , q 〉 ⇒ ∗ v . Therefore 〈 p , B , q 〉 ⇒ a 〈 r , D , q 〉 ⇒ ∗ a v = w and we are finished. Next assume w = a v and the first step is ( p , w, B ) /turnstileleft ( q , v, B 1 B 2 . . . Bn ) so we have and each Bi is eventually removed from the sta… \end{verbatim} ``` </details>
725. ph-c579f49c7fba401a3ba9 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) . First assume that we have w = a v and where (( p , a , B ) , ( r , D )) ∈ ϒ , and B , D ∈ I ∪ { λ } , giving productions 〈 p , B , q 〉 → a 〈 r , D , q 〉 . Since ( q , v, D ) /turnstileleft ∗ ( q , λ, λ ) by induction hypothesis, 〈 r , D , q 〉 ⇒ ∗ v . Therefore 〈 p , B , q 〉 ⇒ a 〈 r , D , q 〉 ⇒ ∗ a v = w and we are finished. Next assume w = a v and the first step is ( p , w, B ) /turnstileleft ( q , v, B 1 B 2 . . . Bn ) so we have and each Bi is eventually removed from the stack in order so that there are… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{. First assume that we have w = a v and where (( p , a , B ) , ( r , D )) ∈ ϒ , and B , D ∈ I ∪ \{ λ \} , giving productions 〈 p , B , q 〉 → a 〈 r , D , q 〉 . Since ( q , v, D ) /turnstileleft ∗ ( q , λ, λ ) by induction hypothesis, 〈 r , D , q 〉 ⇒ ∗ v…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md:offset=71342 \begin{verbatim} . First assume that we have w = a v and where (( p , a , B ) , ( r , D )) ∈ ϒ , and B , D ∈ I ∪ { λ } , giving productions 〈 p , B , q 〉 → a 〈 r , D , q 〉 . Since ( q , v, D ) /turnstileleft ∗ ( q , λ, λ ) by induction hypothesis, 〈 r , D , q 〉 ⇒ ∗ v . Therefore 〈 p , B , q 〉 ⇒ a 〈 r , D , q 〉 ⇒ ∗ a v = w and we are finished. Next assume w = a v and the first step is ( p , w, B ) /turnstileleft ( q , v, B 1 B 2 . . . Bn ) so we have and each Bi is eventually removed from the stack in order so that there are… \end{verbatim} ``` </details>
726. ph-9c3a0089b4a7ecb82018 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.1.md ### Plain (markdown context) v, D ) /turnstileleft ∗ ( q , λ, λ ) by induction hypothesis, 〈 r , D , q 〉 ⇒ ∗ v . Therefore 〈 p , B , q 〉 ⇒ a 〈 r , D , q 〉 ⇒ ∗ a v = w and we are finished. Next assume w = a v and the first step is ( p , w, B ) /turnstileleft ( q , v, B 1 B 2 . . . Bn ) so we have and each Bi is eventually removed from the stack in order so that there are states q , q , . . . , where q and v v v . . . v v such that 1 2 qn -1 qn qn = = 1 2 n -1 n ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{v, D ) /turnstileleft ∗ ( q , λ, λ ) by induction hypothesis, 〈 r , D , q 〉 ⇒ ∗ v . Therefore 〈 p , B , q 〉 ⇒ a 〈 r , D , q 〉 ⇒ ∗ a v = w and we are finished. Next assume w = a v and the first step is ( p , w, B ) /turnstileleft ( q , v, B 1 B 2 . . . Bn ) so we have and each Bi is eventually removed from the stack in order so that there are states q , q , . . . , where q and v v v . . . v v such that 1 2 qn -1 qn qn = = 1 2 n -1 n \end{verbatim} ```
727. ph-9cb7a2ce19d58cd933a6 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) By the induction hypothesis, 〈 qi -1 , Bi , qi 〉 ⇒ ∗ v i . But since the production is type (3), so that 〈 p , B , q 〉 ⇒ ∗ w . /square Theorem 4.6 Alanguage is context-free if and only if it is accepted by a PDA. ## Exercises - (1) Construct a pushdown automaton which reads the same language as the grammar /Gamma1 = ( N , /Sigma1 , S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by - (2) Construct a pushdown automaton which reads the same language as generated by the grammar /Gamma1 =… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{By the induction hypothesis, 〈 qi -1 , Bi , qi 〉 ⇒ ∗ v i . But since the production is type (3), so that 〈 p , B , q 〉 ⇒ ∗ w . /square Theorem 4.6 Alanguage is context-free if and only if it is accepted by a PDA. \#\# Exercises - (1) Construct a pushdo…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=98 \begin{verbatim} By the induction hypothesis, 〈 qi -1 , Bi , qi 〉 ⇒ ∗ v i . But since the production is type (3), so that 〈 p , B , q 〉 ⇒ ∗ w . /square Theorem 4.6 Alanguage is context-free if and only if it is accepted by a PDA. ## Exercises - (1) Construct a pushdown automaton which reads the same language as the grammar /Gamma1 = ( N , /Sigma1 , S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by - (2) Construct a pushdown automaton which reads the same language as generated by the grammar /Gamma1 =… \end{verbatim} ``` </details>
728. ph-116edada57f50d52a961 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) By the induction hypothesis, 〈 qi -1 , Bi , qi 〉 ⇒ ∗ v i . But since the production is type (3), so that 〈 p , B , q 〉 ⇒ ∗ w . /square Theorem 4.6 Alanguage is context-free if and only if it is accepted by a PDA. ## Exercises - (1) Construct a pushdown automaton which reads the same language as the grammar /Gamma1 = ( N , /Sigma1 , S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by - (2) Construct a pushdown automaton which reads the same language as generated by the grammar /Gamma1 =… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{By the induction hypothesis, 〈 qi -1 , Bi , qi 〉 ⇒ ∗ v i . But since the production is type (3), so that 〈 p , B , q 〉 ⇒ ∗ w . /square Theorem 4.6 Alanguage is context-free if and only if it is accepted by a PDA. \#\# Exercises - (1) Construct a pushdo…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=473 \begin{verbatim} By the induction hypothesis, 〈 qi -1 , Bi , qi 〉 ⇒ ∗ v i . But since the production is type (3), so that 〈 p , B , q 〉 ⇒ ∗ w . /square Theorem 4.6 Alanguage is context-free if and only if it is accepted by a PDA. ## Exercises - (1) Construct a pushdown automaton which reads the same language as the grammar /Gamma1 = ( N , /Sigma1 , S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by - (2) Construct a pushdown automaton which reads the same language as generated by the grammar /Gamma1 =… \end{verbatim} ``` </details>
729. ph-acee6f7fc39d553cacc4 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) ccepted by a PDA. ## Exercises - (1) Construct a pushdown automaton which reads the same language as the grammar /Gamma1 = ( N , /Sigma1 , S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by - (2) Construct a pushdown automaton which reads the same language as generated by the grammar /Gamma1 = ( N , /Sigma1 , S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by - (3) Construct a pushdown automaton which reads the same language as gener… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ccepted by a PDA. \#\# Exercises - (1) Construct a pushdown automaton which reads the same language as the grammar /Gamma1 = ( N , /Sigma1 , S , P ) defined by N = \{ S , A , B \} , /Sigma1 = \{ a , b , c \} , and the set of productions P given by - (2) Co…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=728 \begin{verbatim} ccepted by a PDA. ## Exercises - (1) Construct a pushdown automaton which reads the same language as the grammar /Gamma1 = ( N , /Sigma1 , S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by - (2) Construct a pushdown automaton which reads the same language as generated by the grammar /Gamma1 = ( N , /Sigma1 , S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by - (3) Construct a pushdown automaton which reads the same language as gener… \end{verbatim} ``` </details>
730. ph-5489e61e99d7a37d9cfd — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) -not-decoded --> - (2) Construct a pushdown automaton which reads the same language as generated by the grammar /Gamma1 = ( N , /Sigma1 , S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by - (3) Construct a pushdown automaton which reads the same language as generated by the grammar /Gamma1 = ( N , /Sigma1 , S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c , } , and the set of productions P given by - (4) Construct a pushdown automaton which reads the same language as gene… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{-not-decoded --> - (2) Construct a pushdown automaton which reads the same language as generated by the grammar /Gamma1 = ( N , /Sigma1 , S , P ) defined by N = \{ S , A , B \} , /Sigma1 = \{ a , b , c \} , and the set of productions P given by - (3) Con…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=985 \begin{verbatim} -not-decoded --> - (2) Construct a pushdown automaton which reads the same language as generated by the grammar /Gamma1 = ( N , /Sigma1 , S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by - (3) Construct a pushdown automaton which reads the same language as generated by the grammar /Gamma1 = ( N , /Sigma1 , S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c , } , and the set of productions P given by - (4) Construct a pushdown automaton which reads the same language as gene… \end{verbatim} ``` </details>
731. ph-d926e088bad4e6295079 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) -not-decoded --> - (3) Construct a pushdown automaton which reads the same language as generated by the grammar /Gamma1 = ( N , /Sigma1 , S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c , } , and the set of productions P given by - (4) Construct a pushdown automaton which reads the same language as generated by the grammar /Gamma1 = ( N , /Sigma1 , S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by - (5) Construct a pushdown automaton which reads the same language as gene… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{-not-decoded --> - (3) Construct a pushdown automaton which reads the same language as generated by the grammar /Gamma1 = ( N , /Sigma1 , S , P ) defined by N = \{ S , A , B \} , /Sigma1 = \{ a , b , c , \} , and the set of productions P given by - (4) C…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=1240 \begin{verbatim} -not-decoded --> - (3) Construct a pushdown automaton which reads the same language as generated by the grammar /Gamma1 = ( N , /Sigma1 , S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c , } , and the set of productions P given by - (4) Construct a pushdown automaton which reads the same language as generated by the grammar /Gamma1 = ( N , /Sigma1 , S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by - (5) Construct a pushdown automaton which reads the same language as gene… \end{verbatim} ``` </details>
732. ph-583606d3d50e2a297752 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) ot-decoded --> - (4) Construct a pushdown automaton which reads the same language as generated by the grammar /Gamma1 = ( N , /Sigma1 , S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by - (5) Construct a pushdown automaton which reads the same language as generated by the grammar /Gamma1 = ( N , /Sigma1 , S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c , d } , and the set of productions P given by - (6) Construct a grammar which generates the language read by the pushdow… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ot-decoded --> - (4) Construct a pushdown automaton which reads the same language as generated by the grammar /Gamma1 = ( N , /Sigma1 , S , P ) defined by N = \{ S , A , B \} , /Sigma1 = \{ a , b , c \} , and the set of productions P given by - (5) Const…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=1499 \begin{verbatim} ot-decoded --> - (4) Construct a pushdown automaton which reads the same language as generated by the grammar /Gamma1 = ( N , /Sigma1 , S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by - (5) Construct a pushdown automaton which reads the same language as generated by the grammar /Gamma1 = ( N , /Sigma1 , S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c , d } , and the set of productions P given by - (6) Construct a grammar which generates the language read by the pushdow… \end{verbatim} ``` </details>
733. ph-79d00ec01aa0a1e595bf — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) bach normal form. - (11) Construct a pushdown automaton that reads the same language as the grammar /Gamma1 = ( N , /Sigma1 , S , P ) defined by N = { S , B } ∪ /Sigma1 , /Sigma1 = { a , b , c } , and the set of productions P given by ![Image](./AutomataTheory_chapter_2.2_artifacts/image_000002_e9b0569031c5d3dd136abfc521bc0580ef1b2fd1ec6dc42aac4a9642c31c9a9e.png) ![Image](./AutomataTheory_chapter_2.2_artifacts/image_000003_49abd552d81bd1620b2260dcd62a38991bfd7a0dc38b7a7819b097408548b7fe.png) - (12) Construct a pushdown automaton that reads the same language as the… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{bach normal form. - (11) Construct a pushdown automaton that reads the same language as the grammar /Gamma1 = ( N , /Sigma1 , S , P ) defined by N = \{ S , B \} ∪ /Sigma1 , /Sigma1 = \{ a , b , c \} , and the set of productions P given by ![Image](./AutomataTheory\_chapter\_2.2\_artifa…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=2761 \begin{verbatim} bach normal form. - (11) Construct a pushdown automaton that reads the same language as the grammar /Gamma1 = ( N , /Sigma1 , S , P ) defined by N = { S , B } ∪ /Sigma1 , /Sigma1 = { a , b , c } , and the set of productions P given by ![Image](./AutomataTheory_chapter_2.2_artifacts/image_000002_e9b0569031c5d3dd136abfc521bc0580ef1b2fd1ec6dc42aac4a9642c31c9a9e.png) ![Image](./AutomataTheory_chapter_2.2_artifacts/image_000003_49abd552d81bd1620b2260dcd62a38991bfd7a0dc38b7a7819b097408548b7fe.png) - (12) Construct a pushdown automaton that reads the same language as the… \end{verbatim} ``` </details>
734. ph-67c34c4264d40f2b594d — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) e](./AutomataTheory_chapter_2.2_artifacts/image_000002_e9b0569031c5d3dd136abfc521bc0580ef1b2fd1ec6dc42aac4a9642c31c9a9e.png) ![Image](./AutomataTheory_chapter_2.2_artifacts/image_000003_49abd552d81bd1620b2260dcd62a38991bfd7a0dc38b7a7819b097408548b7fe.png) - (12) Construct a pushdown automaton that reads the same language as the grammar /Gamma1 = ( N , /Sigma1 , S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by - (13) Construct a pushdown automaton that reads the same language as the … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{e](./AutomataTheory\_chapter\_2.2\_artifacts/image\_000002\_e9b0569031c5d3dd136abfc521bc0580ef1b2fd1ec6dc42aac4a9642c31c9a9e.png) ![Image](./AutomataTheory\_chapter\_2.2\_artifacts/image\_000003\_49abd552d81bd1620b2260dcd62a38991bfd7a0dc38b7a7819b097408548b7fe.png) - (12) Construct a pushdown automaton that reads the same language as the grammar /Gamma1 = ( N , /Sigma1 , S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by - (13) Construct a pushdown automaton that reads the same language as the … \end{verbatim} ```
735. ph-18f7f586cb45eb2621dc — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) 19b097408548b7fe.png) - (12) Construct a pushdown automaton that reads the same language as the grammar /Gamma1 = ( N , /Sigma1 , S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by - (13) Construct a pushdown automaton that reads the same language as the grammar /Gamma1 = ( N , ϒ, S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by - (14) Construct a pushdown automaton that reads the same language as the g… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{19b097408548b7fe.png) - (12) Construct a pushdown automaton that reads the same language as the grammar /Gamma1 = ( N , /Sigma1 , S , P ) defined by N = \{ S , A , B \} , /Sigma1 = \{ a , b , c \} , and the set of productions P given by - (12) Construct a pushdown automaton that reads the same language as the grammar /Gamma1 = ( N , /Sigma1 , S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by - (13) Construct a pushdown automaton that reads the same language as the grammar /Gamma1 = ( N , ϒ, S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by - (14) Construct a pushdown automaton that reads the same language as the g… \end{verbatim} ```
736. ph-db14ad18c7f34853d608 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) the set of productions P given by - (13) Construct a pushdown automaton that reads the same language as the grammar /Gamma1 = ( N , ϒ, S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by - (14) Construct a pushdown automaton that reads the same language as the grammar /Gamma1 = ( N , ϒ, S , P ) defined by N = { S , A , B } , ϒ = { a , b , c } , and the set of productions P given by - (15) Construct a pushdown automaton that reads the same language as the gr… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{the set of productions P given by - (13) Construct a pushdown automaton that reads the same language as the grammar /Gamma1 = ( N , ϒ, S , P ) defined by N = \{ S , A , B \} , /Sigma1 = \{ a , b , c \} , and the set of productions P given by - (13) Construct a pushdown automaton that reads the same language as the grammar /Gamma1 = ( N , ϒ, S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by - (14) Construct a pushdown automaton that reads the same language as the grammar /Gamma1 = ( N , ϒ, S , P ) defined by N = { S , A , B } , ϒ = { a , b , c } , and the set of productions P given by - (15) Construct a pushdown automaton that reads the same language as the gr… \end{verbatim} ```
737. ph-6c99f36997fe3905d51c — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) nd the set of productions P given by - (14) Construct a pushdown automaton that reads the same language as the grammar /Gamma1 = ( N , ϒ, S , P ) defined by N = { S , A , B } , ϒ = { a , b , c } , and the set of productions P given by - (15) Construct a pushdown automaton that reads the same language as the grammar /Gamma1 = ( N , ϒ, S , P ) defined by N = { S , A , B } , ϒ = { a , b , c , d } , and the set of productions P given by ## 4.4 The Pumping Lemma and decidability Just as we were able to show that… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{nd the set of productions P given by - (14) Construct a pushdown automaton that reads the same language as the grammar /Gamma1 = ( N , ϒ, S , P ) defined by N = \{ S , A , B \} , ϒ = \{ a , b , c \} , and the set of productions P given by - (14) Construct a pushdown automaton that reads the same language as the grammar /Gamma1 = ( N , ϒ, S , P ) defined by N = { S , A , B } , ϒ = { a , b , c } , and the set of productions P given by - (15) Construct a pushdown automaton that reads the same language as the grammar /Gamma1 = ( N , ϒ, S , P ) defined by N = { S , A , B } , ϒ = { a , b , c , d } , and the set of productions P given by ## 4.4 The Pumping Lemma and decidability Just as we were able to show that… \end{verbatim} ```
738. ph-d90ddc1080183750f3da — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) . Let M = 2 p . Assume there is a word w in L with length greater than or equal to M . Then by the previous theorem, the derivation tree has height greater than p . Therefore there is a path S →··· → a where a is a letter in the derivation tree with length greater than p and a is a letter of w . Since there are only p productions, some nonterminal occurs more than once on the left-hand side of a production. Let C be the first nonterminal to occur the second time. Therefore we have a derivation where α ⇒ ∗ x , β ⇒ ∗ y , C ⇒ ∗ uC v and C ⇒ w . But using these deriva… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{. Let M = 2 p . Assume there is a word w in L with length greater than or equal to M . Then by the previous theorem, the derivation tree has height greater than p . Therefore there is a path S →··· → a where a is a letter in the derivation tree with length greater than p and a i…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=6030 \begin{verbatim} . Let M = 2 p . Assume there is a word w in L with length greater than or equal to M . Then by the previous theorem, the derivation tree has height greater than p . Therefore there is a path S →··· → a where a is a letter in the derivation tree with length greater than p and a is a letter of w . Since there are only p productions, some nonterminal occurs more than once on the left-hand side of a production. Let C be the first nonterminal to occur the second time. Therefore we have a derivation where α ⇒ ∗ x , β ⇒ ∗ y , C ⇒ ∗ uC v and C ⇒ w . But using these deriva… \end{verbatim} ``` </details>
739. ph-dd99c66c5359b74ecb7f — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) 1 S 1 · · · S 1 . Using leftmost derivations we can derive S ⇒ ∗ S 1 S 1 S 1 · · · S 1 ⇒ ∗ w 1 S 1 S 1 · · · S 1 ⇒ ∗ w 1 w 2 S 1 · · · S 1 ⇒ ∗ w 1 w 2 · · · w n in /Gamma1 . Hence L ∗ 1 = L , the language generated by /Gamma1 . /square Theorem 4.9 The set of context-free languages is not closed under the operations of intersection and complement. Proof The sets { a n b n c m : m , n ≥ 0 } and { a n b m c m : m , n ≥ 0 } are contextfree. The first is generated by the grammar with productions The second is generated by the grammar with productions The second is generated by the grammar with productions The second is generated by the grammar with productions However, the intersection is the language L = { a m b m a m : m ≥ 0 } , whic… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{⇒ ∗ w 1 S 1 S 1 · · · S 1 ⇒ ∗ w 1 w 2 S 1 · · · S 1 ⇒ ∗ w 1 w 2 · · · w n in /Gamma1 . Hence L ∗ 1 = L , the language generated by /Gamma1 . /square Theorem 4.9 The set of context-free languages is not closed under the operations of intersection and complement. Proof The sets \{ …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=11060 \begin{verbatim} ⇒ ∗ w 1 S 1 S 1 · · · S 1 ⇒ ∗ w 1 w 2 S 1 · · · S 1 ⇒ ∗ w 1 w 2 · · · w n in /Gamma1 . Hence L ∗ 1 = L , the language generated by /Gamma1 . /square Theorem 4.9 The set of context-free languages is not closed under the operations of intersection and complement. Proof The sets { a n b n c m : m , n ≥ 0 } and { a n b m c m : m , n ≥ 0 } are contextfree. The first is generated by the grammar with productions The second is generated by the grammar with productions However, the intersection is the language L = { a m b m a m : m ≥ 0 } , whic… \end{verbatim} ``` </details>
741. ph-fe361970f172f952710a — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) generated by a context-free grammar is finite or infinite. Proof Since it is possible to determine whether a word is in the language of a context-free grammar, simply try all words with length between 2 p and 2 p + 1 to see if one of them is in the context-free grammar. If one is, the grammar is infinite. If not the grammar is finite. /square ## Exercises - (1) Let grammar /Gamma1 = ( N , ϒ, S , P ) be defined by N = { S , A , B } , ϒ = { a , b , c } , and the set of productions P given by Let L be the language generated by /Gamma1 . Find the grammar that generate… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{generated by a context-free grammar is finite or infinite. Proof Since it is possible to determine whether a word is in the language of a context-free grammar, simply try all words with length between 2 p and 2 p + 1 to see if one of them is in the context-free grammar. If one i…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=17112 \begin{verbatim} generated by a context-free grammar is finite or infinite. Proof Since it is possible to determine whether a word is in the language of a context-free grammar, simply try all words with length between 2 p and 2 p + 1 to see if one of them is in the context-free grammar. If one is, the grammar is infinite. If not the grammar is finite. /square ## Exercises - (1) Let grammar /Gamma1 = ( N , ϒ, S , P ) be defined by N = { S , A , B } , ϒ = { a , b , c } , and the set of productions P given by Let L be the language generated by /Gamma1 . Find the grammar that generate… \end{verbatim} ``` </details>
742. ph-2076587d736e0e683c11 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) inite. If not the grammar is finite. /square ## Exercises - (1) Let grammar /Gamma1 = ( N , ϒ, S , P ) be defined by N = { S , A , B } , ϒ = { a , b , c } , and the set of productions P given by Let L be the language generated by /Gamma1 . Find the grammar that generates L ∗ . - (2) Let L 1 be the language generated by the grammar /Gamma1 1 = ( N , /Sigma1 , S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by and L 2 be the language generated by the grammar /Gamma1 2 = ( N , /Sigma1 , … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{inite. If not the grammar is finite. /square \#\# Exercises - (1) Let grammar /Gamma1 = ( N , ϒ, S , P ) be defined by N = \{ S , A , B \} , ϒ = \{ a , b , c \} , and the set of productions P given by Let L be the language generated by /Gamma1 . Find the g…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=17414 \begin{verbatim} inite. If not the grammar is finite. /square ## Exercises - (1) Let grammar /Gamma1 = ( N , ϒ, S , P ) be defined by N = { S , A , B } , ϒ = { a , b , c } , and the set of productions P given by Let L be the language generated by /Gamma1 . Find the grammar that generates L ∗ . - (2) Let L 1 be the language generated by the grammar /Gamma1 1 = ( N , /Sigma1 , S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by and L 2 be the language generated by the grammar /Gamma1 2 = ( N , /Sigma1 , … \end{verbatim} ``` </details>
743. ph-62f8b264fac4521873a9 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) not-decoded --> Let L be the language generated by /Gamma1 . Find the grammar that generates L ∗ . - (2) Let L 1 be the language generated by the grammar /Gamma1 1 = ( N , /Sigma1 , S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by and L 2 be the language generated by the grammar /Gamma1 2 = ( N , /Sigma1 , S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by Find the grammar that generates L 1 L 2 . - (3) Let L 1 be the language gene… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{not-decoded --> Let L be the language generated by /Gamma1 . Find the grammar that generates L ∗ . - (2) Let L 1 be the language generated by the grammar /Gamma1 1 = ( N , /Sigma1 , S , P ) defined by N = \{ S , A , B \} , /Sigma1 = \{ a , b , c \} , and the set of productions P giv…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=17625 \begin{verbatim} not-decoded --> Let L be the language generated by /Gamma1 . Find the grammar that generates L ∗ . - (2) Let L 1 be the language generated by the grammar /Gamma1 1 = ( N , /Sigma1 , S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by and L 2 be the language generated by the grammar /Gamma1 2 = ( N , /Sigma1 , S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by Find the grammar that generates L 1 L 2 . - (3) Let L 1 be the language gene… \end{verbatim} ``` </details>
744. ph-af525a1440b1d9bc37b1 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) the set of productions P given by and L 2 be the language generated by the grammar /Gamma1 2 = ( N , /Sigma1 , S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by Find the grammar that generates L 1 L 2 . - (3) Let L 1 be the language generated by the grammar /Gamma1 1 = ( N , ϒ, S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by and L 2 be the language generated by the grammar /Gamma1 2 = ( N , ϒ, S , P )… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{the set of productions P given by and L 2 be the language generated by the grammar /Gamma1 2 = ( N , /Sigma1 , S , P ) defined by N = \{ S , A , B \} , /Sigma1 = \{ a , b , c \} , and the set of productions P given by Find th…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=17878 \begin{verbatim} the set of productions P given by and L 2 be the language generated by the grammar /Gamma1 2 = ( N , /Sigma1 , S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by Find the grammar that generates L 1 L 2 . - (3) Let L 1 be the language generated by the grammar /Gamma1 1 = ( N , ϒ, S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by and L 2 be the language generated by the grammar /Gamma1 2 = ( N , ϒ, S , P )… \end{verbatim} ``` </details>
745. ph-bd24ee93ef96b9abb006 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) b , c } , and the set of productions P given by Find the grammar that generates L 1 L 2 . - (3) Let L 1 be the language generated by the grammar /Gamma1 1 = ( N , ϒ, S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by and L 2 be the language generated by the grammar /Gamma1 2 = ( N , ϒ, S , P ) defined by N = { S , A , B } , ϒ = { a , b , c } , and the set of productions P given by Find the grammar that generates L 1 ∪ L 2 . Determine whether the following … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{b , c \} , and the set of productions P given by Find the grammar that generates L 1 L 2 . - (3) Let L 1 be the language generated by the grammar /Gamma1 1 = ( N , ϒ, S , P ) defined by N = \{ S , A , B \} , /Sigma1 = \{ a , b , c \} , and the set of prod…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=18076 \begin{verbatim} b , c } , and the set of productions P given by Find the grammar that generates L 1 L 2 . - (3) Let L 1 be the language generated by the grammar /Gamma1 1 = ( N , ϒ, S , P ) defined by N = { S , A , B } , /Sigma1 = { a , b , c } , and the set of productions P given by and L 2 be the language generated by the grammar /Gamma1 2 = ( N , ϒ, S , P ) defined by N = { S , A , B } , ϒ = { a , b , c } , and the set of productions P given by Find the grammar that generates L 1 ∪ L 2 . Determine whether the following … \end{verbatim} ``` </details>
746. ph-b54eee415b5a79f322f9 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) e symbols. As input, readingablankissimplyreadingtheabsenceofanyofthetapesymbols.Printing a blank is considered to be erasing the symbol currently in that square. We use # for blank. The Turing machine shown below is in state s 1 and is reading letter a . ![Image](./AutomataTheory_chapter_2.2_artifacts/image_000004_93b8824c7887d5a3cd67e9a38e9de031df722093638a54d1b81993447d1cf49e.png) More formally we have the following definition. Definition 5.1 A deterministic Turing machine is a quintuple where Q is the set of states, /Gamma1 is a finite set of tape symbols, whi… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{e symbols. As input, readingablankissimplyreadingtheabsenceofanyofthetapesymbols.Printing a blank is considered to be erasing the symbol currently in that square. We use \# for blank. The Turing machine shown below is in state s 1 and is reading letter a . ![Image](./AutomataTheo…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=21939 \begin{verbatim} e symbols. As input, readingablankissimplyreadingtheabsenceofanyofthetapesymbols.Printing a blank is considered to be erasing the symbol currently in that square. We use # for blank. The Turing machine shown below is in state s 1 and is reading letter a . ![Image](./AutomataTheory_chapter_2.2_artifacts/image_000004_93b8824c7887d5a3cd67e9a38e9de031df722093638a54d1b81993447d1cf49e.png) More formally we have the following definition. Definition 5.1 A deterministic Turing machine is a quintuple where Q is the set of states, /Gamma1 is a finite set of tape symbols, whi… \end{verbatim} ``` </details>
747. ph-8dee7fbfa475b717229f — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) set of tape symbols, which includes the alphabet and #, s 0 is the starting state, h is the halt state, and δ is a function from Q × /Gamma1 to Q × /Gamma1 × N where N consists of L which indicates a movement on the tape one position to the left, R which indicates a movement on the tape one position to the right, and # which indicates that no movement takes place. Just like any computer, a Turing machine has a program or set of rules which tell the machine what to do. An example of a rule is This rules says that if the machine is in st… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{set of tape symbols, which includes the alphabet and \#, s 0 is the starting state, h is the halt state, and δ is a function from Q × /Gamma1 to Q × /Gamma1 × N where N consists of L which indicates a movement on the tape one position to the left, R which indicates a movement on …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=22518 \begin{verbatim} set of tape symbols, which includes the alphabet and #, s 0 is the starting state, h is the halt state, and δ is a function from Q × /Gamma1 to Q × /Gamma1 × N where N consists of L which indicates a movement on the tape one position to the left, R which indicates a movement on the tape one position to the right, and # which indicates that no movement takes place. Just like any computer, a Turing machine has a program or set of rules which tell the machine what to do. An example of a rule is This rules says that if the machine is in st… \end{verbatim} ``` </details>
748. ph-80a08798d7b76b5e8d0a — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) cludes the alphabet and #, s 0 is the starting state, h is the halt state, and δ is a function from Q × /Gamma1 to Q × /Gamma1 × N where N consists of L which indicates a movement on the tape one position to the left, R which indicates a movement on the tape one position to the right, and # which indicates that no movement takes place. Just like any computer, a Turing machine has a program or set of rules which tell the machine what to do. An example of a rule is This rules says that if the machine is in state s 1 and reads the letter … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{cludes the alphabet and \#, s 0 is the starting state, h is the halt state, and δ is a function from Q × /Gamma1 to Q × /Gamma1 × N where N consists of L which indicates a movement on the tape one position to the left, R which indicates a movement on the tape one position to the …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=22548 \begin{verbatim} cludes the alphabet and #, s 0 is the starting state, h is the halt state, and δ is a function from Q × /Gamma1 to Q × /Gamma1 × N where N consists of L which indicates a movement on the tape one position to the left, R which indicates a movement on the tape one position to the right, and # which indicates that no movement takes place. Just like any computer, a Turing machine has a program or set of rules which tell the machine what to do. An example of a rule is This rules says that if the machine is in state s 1 and reads the letter … \end{verbatim} ``` </details>
749. ph-0ad070cf5cd3fb25923e — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) indicates a movement on the tape one position to the right, and # which indicates that no movement takes place. Just like any computer, a Turing machine has a program or set of rules which tell the machine what to do. An example of a rule is This rules says that if the machine is in state s 1 and reads the letter a , it is to change to state s 2 , print the letter b in place of the letter a and move one square to the left. The rule which we shall denote as says that if the machine is in state s 1 and reads … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{indicates a movement on the tape one position to the right, and \# which indicates that no movement takes place. Just like any computer, a Turing machine has a program or set of rules which tell the machine what to do. An example of a rule is This rules says that if the machine is in state s 1 and reads the letter a , it is to change to state s 2 , print the letter b in place of the letter a and move one square to the left. The rule which we shall denote as says that if the machine is in state s 1 and reads … \end{verbatim} ```
750. ph-3860ce76350fd98beb46 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) e what to do. An example of a rule is This rules says that if the machine is in state s 1 and reads the letter a , it is to change to state s 2 , print the letter b in place of the letter a and move one square to the left. The rule which we shall denote as says that if the machine is in state s 1 and reads the letter a , it changes to state s 2 , erases the a and moves one square to the right. The rule says that if the machine is in state s 1 and reads a blank then it halts, and… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{e what to do. An example of a rule is This rules says that if the machine is in state s 1 and reads the letter a , it is to change to state s 2 , print the letter b in place of the letter a and move one square to the left…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=22979 \begin{verbatim} e what to do. An example of a rule is This rules says that if the machine is in state s 1 and reads the letter a , it is to change to state s 2 , print the letter b in place of the letter a and move one square to the left. The rule which we shall denote as says that if the machine is in state s 1 and reads the letter a , it changes to state s 2 , erases the a and moves one square to the right. The rule says that if the machine is in state s 1 and reads a blank then it halts, and… \end{verbatim} ``` </details>
751. ph-acd3af867a7c2366e16f — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) he tape to the left or to the right. Some definitions allow a machine either to print a letter or to move the head, but not both. Thus it requires two separate rules to print a letter and move the position on the tape. Weshall begin with a program that simply moves the position of the machine on the tape from the beginning to the end of a string. The alphabet is /Sigma1 = { a , b } and symbols /Gamma1 = { a , b , # } . We shall have the set of states Q = { s 0 , s 1 , h } and the set of rules This program leaves everything alone. It simply reads each letter and th… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{he tape to the left or to the right. Some definitions allow a machine either to print a letter or to move the head, but not both. Thus it requires two separate rules to print a letter and move the position on the tape. Weshall begin with a program that simply moves the position …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=24435 \begin{verbatim} he tape to the left or to the right. Some definitions allow a machine either to print a letter or to move the head, but not both. Thus it requires two separate rules to print a letter and move the position on the tape. Weshall begin with a program that simply moves the position of the machine on the tape from the beginning to the end of a string. The alphabet is /Sigma1 = { a , b } and symbols /Gamma1 = { a , b , # } . We shall have the set of states Q = { s 0 , s 1 , h } and the set of rules This program leaves everything alone. It simply reads each letter and th… \end{verbatim} ``` </details>
752. ph-3f8f961a8f5c04871416 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) monstrate this program, it would be rather tiresome to continually draw the Turing machine so rather than draw ![Image](./AutomataTheory_chapter_2.2_artifacts/image_000005_64e9f170c5ccc95c50b0d86fa97c8052043867611bbceb8dbf7c12f355037ae4.png) which shows the position of the machine at the second square of the tape and in state s 1, while the first and third squares of the tape contain an a , the second, fourth and fifth squares contain a b and the other squares are blank; we replace this with where the line below the b denotes the location of the head, and the 1 ab… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{monstrate this program, it would be rather tiresome to continually draw the Turing machine so rather than draw ![Image](./AutomataTheory\_chapter\_2.2\_artifacts/image\_000005\_64e9f170c5ccc95c50b0d86fa97c8052043867611bbceb8dbf7c12f355037ae4.png) which shows the position of the machi…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=25491 \begin{verbatim} monstrate this program, it would be rather tiresome to continually draw the Turing machine so rather than draw ![Image](./AutomataTheory_chapter_2.2_artifacts/image_000005_64e9f170c5ccc95c50b0d86fa97c8052043867611bbceb8dbf7c12f355037ae4.png) which shows the position of the machine at the second square of the tape and in state s 1, while the first and third squares of the tape contain an a , the second, fourth and fifth squares contain a b and the other squares are blank; we replace this with where the line below the b denotes the location of the head, and the 1 ab… \end{verbatim} ``` </details>
753. ph-5c2f38712ce01b99a1d5 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) the machine at the second square of the tape and in state s 1, while the first and third squares of the tape contain an a , the second, fourth and fifth squares contain a b and the other squares are blank; we replace this with where the line below the b denotes the location of the head, and the 1 above the b denotes the current state of the machine. We shall call this the configuration of the Turing machine. As we begin our program the machine has configuration moving the head to the right and changing from… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{the machine at the second square of the tape and in state s 1, while the first and third squares of the tape contain an a , the second, fourth and fifth squares contain a b and the other squares are blank; we replace this with where the line below th…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=25762 \begin{verbatim} the machine at the second square of the tape and in state s 1, while the first and third squares of the tape contain an a , the second, fourth and fifth squares contain a b and the other squares are blank; we replace this with where the line below the b denotes the location of the head, and the 1 above the b denotes the current state of the machine. We shall call this the configuration of the Turing machine. As we begin our program the machine has configuration moving the head to the right and changing from… \end{verbatim} ``` </details>
754. ph-0f74c0a0ba3f49f2de6e — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) are of the tape and in state s 1, while the first and third squares of the tape contain an a , the second, fourth and fifth squares contain a b and the other squares are blank; we replace this with where the line below the b denotes the location of the head, and the 1 above the b denotes the current state of the machine. We shall call this the configuration of the Turing machine. As we begin our program the machine has configuration moving the head to the right and changing from state s 0 to state s 1 and o… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{are of the tape and in state s 1, while the first and third squares of the tape contain an a , the second, fourth and fifth squares contain a b and the other squares are blank; we replace this with where the line below the b denotes the location of t…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=25792 \begin{verbatim} are of the tape and in state s 1, while the first and third squares of the tape contain an a , the second, fourth and fifth squares contain a b and the other squares are blank; we replace this with where the line below the b denotes the location of the head, and the 1 above the b denotes the current state of the machine. We shall call this the configuration of the Turing machine. As we begin our program the machine has configuration moving the head to the right and changing from state s 0 to state s 1 and o… \end{verbatim} ``` </details>
755. ph-6f2f3f06c739c39cb72e — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) a b and the other squares are blank; we replace this with where the line below the b denotes the location of the head, and the 1 above the b denotes the current state of the machine. We shall call this the configuration of the Turing machine. As we begin our program the machine has configuration moving the head to the right and changing from state s 0 to state s 1 and our machine then has configuration moving the head to the right again and our machi… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{a b and the other squares are blank; we replace this with where the line below the b denotes the location of the head, and the 1 above the b denotes the current state of the machine. We shall call this the configuration of the Turing machine. As we b…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=25932 \begin{verbatim} a b and the other squares are blank; we replace this with where the line below the b denotes the location of the head, and the 1 above the b denotes the current state of the machine. We shall call this the configuration of the Turing machine. As we begin our program the machine has configuration moving the head to the right and changing from state s 0 to state s 1 and our machine then has configuration moving the head to the right again and our machi… \end{verbatim} ``` </details>
756. ph-5be84361539b0925be2c — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) blank; we replace this with where the line below the b denotes the location of the head, and the 1 above the b denotes the current state of the machine. We shall call this the configuration of the Turing machine. As we begin our program the machine has configuration moving the head to the right and changing from state s 0 to state s 1 and our machine then has configuration moving the head to the right again and our machine then has configuration where the line below the b denotes the location of the head, and the 1 above the b denotes the current state of the machine. We shall call this the configuration of the Turing machine. As we begin our program the machine h…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=25962 \begin{verbatim} blank; we replace this with where the line below the b denotes the location of the head, and the 1 above the b denotes the current state of the machine. We shall call this the configuration of the Turing machine. As we begin our program the machine has configuration moving the head to the right and changing from state s 0 to state s 1 and our machine then has configuration moving the head to the right again and our machine then has configuration moving the head to the right and changing from state s 0 to state s 1 and our machine then has configuration moving the head to the right again and our machine then has configuration moving the head to the right again and our machin… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{n of the head, and the 1 above the b denotes the current state of the machine. We shall call this the configuration of the Turing machine. As we begin our program the machine has configuration moving the head to the right…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=26067 \begin{verbatim} n of the head, and the 1 above the b denotes the current state of the machine. We shall call this the configuration of the Turing machine. As we begin our program the machine has configuration moving the head to the right and changing from state s 0 to state s 1 and our machine then has configuration moving the head to the right again and our machine then has configuration moving the head to the right again and our machin… \end{verbatim} ``` </details>
758. ph-5b7fa2034bdfd0ee75e7 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) the b denotes the current state of the machine. We shall call this the configuration of the Turing machine. As we begin our program the machine has configuration moving the head to the right and changing from state s 0 to state s 1 and our machine then has configuration moving the head to the right again and our machine then has configuration moving the head to the right again and our machine then has configuration moving the head to the right and changing from state s 0 to…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=26097 \begin{verbatim} the b denotes the current state of the machine. We shall call this the configuration of the Turing machine. As we begin our program the machine has configuration moving the head to the right and changing from state s 0 to state s 1 and our machine then has configuration moving the head to the right again and our machine then has configuration moving the head to the right again and our machine then has configuration moving the head to the right and changing from state s 0 to state s 1 and our machine then has configuration moving the head to the right again and our machine then has configuration moving the head to the right again and our machine then has configuration moving the head to the right again and our machine t… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ne. As we begin our program the machine has configuration moving the head to the right and changing from state s 0 to state s 1 and our machine then has configuration moving the head to the right and changing from state s 0 to state s 1 and our machine then has configuration moving the head to the right again and our machine then has configuration moving the head to the right again and our machine then has configuration moving the head to the right again and our machine t… \end{verbatim} ``` </details>
760. ph-5714b061e5f1af3a54e2 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) he machine has configuration moving the head to the right and changing from state s 0 to state s 1 and our machine then has configuration moving the head to the right again and our machine then has configuration moving the head to the right again and our machine then has configuration moving the head to the right again and our machine then has configuration moving the head to the right and changing from state s 0 to state s 1 and our machine then has configuration moving the head to the ri…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=26232 \begin{verbatim} he machine has configuration moving the head to the right and changing from state s 0 to state s 1 and our machine then has configuration moving the head to the right again and our machine then has configuration moving the head to the right again and our machine then has configuration moving the head to the right again and our machine then has configuration moving the head to the right again and our machine then has configuration moving the head to the right again and our machine then has configuration moving the head to the right again and our machine then has configuration We then apply rule We then apply rule We then apply rule We again apply rule We a… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{to the right and changing from state s 0 to state s 1 and our machine then has configuration moving the head to the right again and our machine then has configuration moving the head to the right again and our machine then has configuration moving the head to the right again and our machine then has configuration moving the head to the right again and our machine then has configuration We then apply rule We then apply rule We then apply rule We again apply rule We a… \end{verbatim} ``` </details>
762. ph-b4e0cc493df39c7deccf — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) --> moving the head to the right again and our machine then has configuration moving the head to the right again and our machine then has configuration moving the head to the right again and our machine then has configuration We then apply rule We then apply rule We then apply rule We again apply rule We apply the same rule again and have We then use rule and the machine shuts down. We mention… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{--> moving the head to the right again and our machine then has configuration moving the head to the right again and our machine then has configuration moving the …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=26487 \begin{verbatim} --> moving the head to the right again and our machine then has configuration moving the head to the right again and our machine then has configuration moving the head to the right again and our machine then has configuration We then apply rule We then apply rule We then apply rule We again apply rule We apply the same rule again and have We then use rule and the machine shuts down. We mention… \end{verbatim} ``` </details>
763. ph-68f1d8e22df88430f317 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) machine then has configuration moving the head to the right again and our machine then has configuration moving the head to the right again and our machine then has configuration We then apply rule We then apply rule We then apply rule We again apply rule We apply the same rule again and have We then use rule and the machine shuts down. We mentioned previously that if the position on the tape … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{machine then has configuration moving the head to the right again and our machine then has configuration moving the head to the right again and our machine then ha…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=26535 \begin{verbatim} machine then has configuration moving the head to the right again and our machine then has configuration moving the head to the right again and our machine then has configuration We then apply rule We then apply rule We then apply rule We again apply rule We apply the same rule again and have We then use rule and the machine shuts down. We mentioned previously that if the position on the tape … \end{verbatim} ``` </details>
764. ph-d8cff6e7d63f9a4c76f0 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) d the machine shuts down. We mentioned previously that if the position on the tape is on the leftmost position on the tape and gets an instruction to move left, we say that the machine crashes , and the machine ceases functioning. We shall now construct a rather unusual program. This program causes the machine to crash. We shall again let the input alphabet /Gamma1 be the set { a , b , # } . We shall also assume we have states Q = { s 0 , s 1 , . . . , s j , . . . } . It shall have the rules If we have a larger alphabet, we simply add more rules, so that regardles… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{d the machine shuts down. We mentioned previously that if the position on the tape is on the leftmost position on the tape and gets an instruction to move left, we say that the machine crashes , and the machine ceases functioning. We shall now construct a rather unusual program.…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=27067 \begin{verbatim} d the machine shuts down. We mentioned previously that if the position on the tape is on the leftmost position on the tape and gets an instruction to move left, we say that the machine crashes , and the machine ceases functioning. We shall now construct a rather unusual program. This program causes the machine to crash. We shall again let the input alphabet /Gamma1 be the set { a , b , # } . We shall also assume we have states Q = { s 0 , s 1 , . . . , s j , . . . } . It shall have the rules If we have a larger alphabet, we simply add more rules, so that regardles… \end{verbatim} ``` </details>
765. ph-e14ced4cb5f9cb3f4bbc — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) efine a function. If the rules do not define a function then there is a state s and an input letter a for which there is no rule. When this happens, we say that the system hangs , since it cannot go on. We shall again meet this problem with nondeterministic automata. Suppose we would like the set of rules to define a function, but we still want the program to stop when it is in state s and reads a . The system cannot hang since the function is defined for every input. We can however add a rule which puts the system into the suicide state and causes it to crash usi… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{efine a function. If the rules do not define a function then there is a state s and an input letter a for which there is no rule. When this happens, we say that the system hangs , since it cannot go on. We shall again meet this problem with nondeterministic automata. Suppose we …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=28540 \begin{verbatim} efine a function. If the rules do not define a function then there is a state s and an input letter a for which there is no rule. When this happens, we say that the system hangs , since it cannot go on. We shall again meet this problem with nondeterministic automata. Suppose we would like the set of rules to define a function, but we still want the program to stop when it is in state s and reads a . The system cannot hang since the function is defined for every input. We can however add a rule which puts the system into the suicide state and causes it to crash usi… \end{verbatim} ``` </details>
766. ph-b326bda95bcb0772a3e2 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) c automata. Suppose we would like the set of rules to define a function, but we still want the program to stop when it is in state s and reads a . The system cannot hang since the function is defined for every input. We can however add a rule which puts the system into the suicide state and causes it to crash using go-crash. Thus the system crashes instead of hanging and we have expanded our rules so that we have a function. In this discussion, we will state only relevant rules with the understanding that we could produce a function us… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{c automata. Suppose we would like the set of rules to define a function, but we still want the program to stop when it is in state s and reads a . The system cannot hang since the function is defined for every input. We can however add a rule which p…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=28796 \begin{verbatim} c automata. Suppose we would like the set of rules to define a function, but we still want the program to stop when it is in state s and reads a . The system cannot hang since the function is defined for every input. We can however add a rule which puts the system into the suicide state and causes it to crash using go-crash. Thus the system crashes instead of hanging and we have expanded our rules so that we have a function. In this discussion, we will state only relevant rules with the understanding that we could produce a function us… \end{verbatim} ``` </details>
767. ph-45a76332ab88d88c779d — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) for the Turing machine. We begin by showing some of its properties as a text editor. Our first step is not exactly a giant one. We show how to move the position on the tape to the right n steps. Again we assume both the input and output alphabet are the set { a , b } . If we have a larger alphabet, we simply add appropriate rules for each new letter. The set of states Q = { s 1 , . . . , s j , . . . , sn , sn + 1 } . We shall call this new subroutine go-right ( n ). It has the following rules: It is easily seen that if we begin in state s 1, each application of a … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{for the Turing machine. We begin by showing some of its properties as a text editor. Our first step is not exactly a giant one. We show how to move the position on the tape to the right n steps. Again we assume both the input and output alphabet are the set \{ a , b \} . If we hav…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=29507 \begin{verbatim} for the Turing machine. We begin by showing some of its properties as a text editor. Our first step is not exactly a giant one. We show how to move the position on the tape to the right n steps. Again we assume both the input and output alphabet are the set { a , b } . If we have a larger alphabet, we simply add appropriate rules for each new letter. The set of states Q = { s 1 , . . . , s j , . . . , sn , sn + 1 } . We shall call this new subroutine go-right ( n ). It has the following rules: It is easily seen that if we begin in state s 1, each application of a … \end{verbatim} ``` </details>
768. ph-bc8124443577145b62dd — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) position on the tape one step to the right and increases the state. After n steps the head has been moved to the right by n squares and we are in state sn + 1 . It is hoped that, with little effort, the reader can create a subroutine for moving to the left by n squares. Suppose that after moving left or right by n squares, or without moving at all we want to change the letter in the current square occupied from a to b . Assuming that we are in state si at the time then we simply use the rule Moving along, suppose that /Gamma1 = /Sigma1… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{position on the tape one step to the right and increases the state. After n steps the head has been moved to the right by n squares and we are in state sn + 1 . It is hoped that, with little effort, the reader can create a subroutine for moving to the left by n squares. Suppose …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=30158 \begin{verbatim} position on the tape one step to the right and increases the state. After n steps the head has been moved to the right by n squares and we are in state sn + 1 . It is hoped that, with little effort, the reader can create a subroutine for moving to the left by n squares. Suppose that after moving left or right by n squares, or without moving at all we want to change the letter in the current square occupied from a to b . Assuming that we are in state si at the time then we simply use the rule Moving along, suppose that /Gamma1 = /Sigma1… \end{verbatim} ``` </details>
769. ph-38845705e6ad1f3378d9 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) the state. After n steps the head has been moved to the right by n squares and we are in state sn + 1 . It is hoped that, with little effort, the reader can create a subroutine for moving to the left by n squares. Suppose that after moving left or right by n squares, or without moving at all we want to change the letter in the current square occupied from a to b . Assuming that we are in state si at the time then we simply use the rule Moving along, suppose that /Gamma1 = /Sigma1 ∪ { # } and we want to replace Moving along, suppose that /Gamma1 = /Sigma1 ∪ { # } and we want to replace Moving along, suppose that /Gamma1 = /Sigma1 ∪ { # } and we want to replace We first use go-right ( i ) to move to the prop… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{e are in state sn + 1 . It is hoped that, with little effort, the reader can create a subroutine for moving to the left by n squares. Suppose that after moving left or right by n squares, or without moving at all we want to change the letter in the current square occupied from a…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=30296 \begin{verbatim} e are in state sn + 1 . It is hoped that, with little effort, the reader can create a subroutine for moving to the left by n squares. Suppose that after moving left or right by n squares, or without moving at all we want to change the letter in the current square occupied from a to b . Assuming that we are in state si at the time then we simply use the rule Moving along, suppose that /Gamma1 = /Sigma1 ∪ { # } and we want to replace We first use go-right ( i ) to move to the prop… \end{verbatim} ``` </details>
771. ph-44776946d95370d8245a — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) hoped that, with little effort, the reader can create a subroutine for moving to the left by n squares. Suppose that after moving left or right by n squares, or without moving at all we want to change the letter in the current square occupied from a to b . Assuming that we are in state si at the time then we simply use the rule Moving along, suppose that /Gamma1 = /Sigma1 ∪ { # } and we want to replace We first use go-right ( i ) to move to the proper position so the head is on … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{hoped that, with little effort, the reader can create a subroutine for moving to the left by n squares. Suppose that after moving left or right by n squares, or without moving at all we want to change the letter in the current square occupied from a to b . Assuming that we are i…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=30326 \begin{verbatim} hoped that, with little effort, the reader can create a subroutine for moving to the left by n squares. Suppose that after moving left or right by n squares, or without moving at all we want to change the letter in the current square occupied from a to b . Assuming that we are in state si at the time then we simply use the rule Moving along, suppose that /Gamma1 = /Sigma1 ∪ { # } and we want to replace We first use go-right ( i ) to move to the proper position so the head is on … \end{verbatim} ``` </details>
772. ph-ee829c3ecc50e87b04ca — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) t moving at all we want to change the letter in the current square occupied from a to b . Assuming that we are in state si at the time then we simply use the rule Moving along, suppose that /Gamma1 = /Sigma1 ∪ { # } and we want to replace We first use go-right ( i ) to move to the proper position so the head is on ai + 1 . Assume we are in state s ′ , We then use the rules to replace the letters and use go-left ( j ) to return to the original spot. w… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{t moving at all we want to change the letter in the current square occupied from a to b . Assuming that we are in state si at the time then we simply use the rule Moving along, suppose that /Gamma1 = /Sigma1 ∪ \{ \# \} and w…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=30494 \begin{verbatim} t moving at all we want to change the letter in the current square occupied from a to b . Assuming that we are in state si at the time then we simply use the rule Moving along, suppose that /Gamma1 = /Sigma1 ∪ { # } and we want to replace We first use go-right ( i ) to move to the proper position so the head is on ai + 1 . Assume we are in state s ′ , We then use the rules to replace the letters and use go-left ( j ) to return to the original spot. w… \end{verbatim} ``` </details>
773. ph-dd0d752b944aaf4e0d0a — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) > We first use go-right ( i ) to move to the proper position so the head is on ai + 1 . Assume we are in state s ′ , We then use the rules to replace the letters and use go-left ( j ) to return to the original spot. with The next text edit feature which we shall illustrate is to insert a letter in a string. We shall find this feature very handy in the near future. We shall call this subroutine insert(c) . Say that we have a string and we want to replace it with so that the strin… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{> We first use go-right ( i ) to move to the proper position so the head is on ai + 1 . Assume we are in state s ′ , We then use the rules to replace the letters and use go-left ( j ) to return to the original spot. with …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=30823 \begin{verbatim} > We first use go-right ( i ) to move to the proper position so the head is on ai + 1 . Assume we are in state s ′ , We then use the rules to replace the letters and use go-left ( j ) to return to the original spot. with The next text edit feature which we shall illustrate is to insert a letter in a string. We shall find this feature very handy in the near future. We shall call this subroutine insert(c) . Say that we have a string and we want to replace it with so that the strin… \end{verbatim} ``` </details>
774. ph-bf7a3a74dffc3b6c1e89 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) o move to the proper position so the head is on ai + 1 . Assume we are in state s ′ , We then use the rules to replace the letters and use go-left ( j ) to return to the original spot. with The next text edit feature which we shall illustrate is to insert a letter in a string. We shall find this feature very handy in the near future. We shall call this subroutine insert(c) . Say that we have a string and we want to replace it with so that the string ai + 1 · · · an -1 an must be moved one square to the righ… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{o move to the proper position so the head is on ai + 1 . Assume we are in state s ′ , We then use the rules to replace the letters and use go-left ( j ) to return to the original spot. with The next text edit feature which we shall illustrate is to i…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=30885 \begin{verbatim} o move to the proper position so the head is on ai + 1 . Assume we are in state s ′ , We then use the rules to replace the letters and use go-left ( j ) to return to the original spot. with The next text edit feature which we shall illustrate is to insert a letter in a string. We shall find this feature very handy in the near future. We shall call this subroutine insert(c) . Say that we have a string and we want to replace it with so that the string ai + 1 · · · an -1 an must be moved one square to the righ… \end{verbatim} ``` </details>
775. ph-cf6b08c79a7a939b1d13 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) ponding to the letter destroyed and in this way 'remember' this letter so it can be printed in the next square. Remember in state sai + j , we print ai + j regardless of the letter read. Finally, when we reach a blank square, we print an and then use go-left ( n ) to return to the beginning of the string. Also it is possible that c occurs elsewhere in the string; however, we shall assume that ai + 1 is not already c . Thus our rules for actually printing c and moving over the other letters are and we end up in state sy . For example assume we have the word abbac a… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ponding to the letter destroyed and in this way 'remember' this letter so it can be printed in the next square. Remember in state sai + j , we print ai + j regardless of the letter read. Finally, when we reach a blank square, we print an and then use go-left ( n ) to return to t…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=32674 \begin{verbatim} ponding to the letter destroyed and in this way 'remember' this letter so it can be printed in the next square. Remember in state sai + j , we print ai + j regardless of the letter read. Finally, when we reach a blank square, we print an and then use go-left ( n ) to return to the beginning of the string. Also it is possible that c occurs elsewhere in the string; however, we shall assume that ai + 1 is not already c . Thus our rules for actually printing c and moving over the other letters are and we end up in state sy . For example assume we have the word abbac a… \end{verbatim} ``` </details>
776. ph-551f7514f6eaf4fb691a — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) ally, when we reach a blank square, we print an and then use go-left ( n ) to return to the beginning of the string. Also it is possible that c occurs elsewhere in the string; however, we shall assume that ai + 1 is not already c . Thus our rules for actually printing c and moving over the other letters are and we end up in state sy . For example assume we have the word abbac and want to insert c so that we have abcbbc . Using go-right (2), we have configuration ## Applying rule we have configuration and we end up in state sy . For example assume we have the word abbac and want to insert c so that we have abcbbc . Using go-right (2), we have configuration ## Applying rule we have configuration and we end up in state sy . For example assume we have the word abbac and want to insert c so that we have abcbbc . Using go-right (2), we have configuration ## Applying rule we have configuration In the future we will condense this statement to… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{n ) to return to the beginning of the string. Also it is possible that c occurs elsewhere in the string; however, we shall assume that ai + 1 is not already c . Thus our rules for actually printing c and moving over the other letters are and we end u…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=32935 \begin{verbatim} n ) to return to the beginning of the string. Also it is possible that c occurs elsewhere in the string; however, we shall assume that ai + 1 is not already c . Thus our rules for actually printing c and moving over the other letters are and we end up in state sy . For example assume we have the word abbac and want to insert c so that we have abcbbc . Using go-right (2), we have configuration ## Applying rule we have configuration In the future we will condense this statement to… \end{verbatim} ``` </details>
778. ph-0592b773701afa43cd26 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) of the string. Also it is possible that c occurs elsewhere in the string; however, we shall assume that ai + 1 is not already c . Thus our rules for actually printing c and moving over the other letters are and we end up in state sy . For example assume we have the word abbac and want to insert c so that we have abcbbc . Using go-right (2), we have configuration ## Applying rule we have configuration In the future we will condense this statement to W… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{of the string. Also it is possible that c occurs elsewhere in the string; however, we shall assume that ai + 1 is not already c . Thus our rules for actually printing c and moving over the other letters are and we end up in state sy . For example ass…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=32965 \begin{verbatim} of the string. Also it is possible that c occurs elsewhere in the string; however, we shall assume that ai + 1 is not already c . Thus our rules for actually printing c and moving over the other letters are and we end up in state sy . For example assume we have the word abbac and want to insert c so that we have abcbbc . Using go-right (2), we have configuration ## Applying rule we have configuration In the future we will condense this statement to W… \end{verbatim} ``` </details>
779. ph-cd770eb0704139032395 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) er, we shall assume that ai + 1 is not already c . Thus our rules for actually printing c and moving over the other letters are and we end up in state sy . For example assume we have the word abbac and want to insert c so that we have abcbbc . Using go-right (2), we have configuration ## Applying rule we have configuration In the future we will condense this statement to We then have the following rules and configurations … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{er, we shall assume that ai + 1 is not already c . Thus our rules for actually printing c and moving over the other letters are and we end up in state sy . For example assume we have the word abbac and want to insert c so that we have abcbbc . Using …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=33045 \begin{verbatim} er, we shall assume that ai + 1 is not already c . Thus our rules for actually printing c and moving over the other letters are and we end up in state sy . For example assume we have the word abbac and want to insert c so that we have abcbbc . Using go-right (2), we have configuration ## Applying rule we have configuration In the future we will condense this statement to We then have the following rules and configurations … \end{verbatim} ``` </details>
780. ph-505214dfe7afd1af7198 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) ting c and moving over the other letters are and we end up in state sy . For example assume we have the word abbac and want to insert c so that we have abcbbc . Using go-right (2), we have configuration ## Applying rule we have configuration In the future we will condense this statement to We then have the following rules and configurations and we now use go-left (5) to return to our original … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ting c and moving over the other letters are and we end up in state sy . For example assume we have the word abbac and want to insert c so that we have abcbbc . Using go-right (2), we have configuration \#\# Applying rule w…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=33128 \begin{verbatim} ting c and moving over the other letters are and we end up in state sy . For example assume we have the word abbac and want to insert c so that we have abcbbc . Using go-right (2), we have configuration ## Applying rule we have configuration In the future we will condense this statement to We then have the following rules and configurations and we now use go-left (5) to return to our original … \end{verbatim} ``` </details>
781. ph-bc082db2ab3e356b2f80 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) er letters are and we end up in state sy . For example assume we have the word abbac and want to insert c so that we have abcbbc . Using go-right (2), we have configuration ## Applying rule we have configuration In the future we will condense this statement to We then have the following rules and configurations and we now use go-left (5) to return to our original position. Suppose we began at … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{er letters are and we end up in state sy . For example assume we have the word abbac and want to insert c so that we have abcbbc . Using go-right (2), we have configuration \#\# Applying rule we have configuration and we end up in state sy . For example assume we have the word abbac and want to insert c so that we have abcbbc . Using go-right (2), we have configuration ## Applying rule we have configuration In the future we will condense this statement to We then have the following rules and configurations and we now use go-left (5) to return to our original position. Suppose we began at … \end{verbatim} ```
782. ph-b021630daf6509d81065 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) nd replace the marker with that letter. Then go right again to the letter which has been duplicated and replace it with a marker. Continue this process until reaching the end of the string. Let /Delta1 be the special marker. Assume that we have used go-right ( i ) to reach the letter c to be deleted. Again assume that we begin in state sx and want to end in state sy . We shall also let the /Sigma1 = { a , b , c } and /Gamma1 = { a , b , c } ∪ { # , } . We then have the following set of rules: Note that the marker is not actually needed. It is used to make the rule… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{nd replace the marker with that letter. Then go right again to the letter which has been duplicated and replace it with a marker. Continue this process until reaching the end of the string. Let /Delta1 be the special marker. Assume that we have used go-right ( i ) to reach the l…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=35246 \begin{verbatim} nd replace the marker with that letter. Then go right again to the letter which has been duplicated and replace it with a marker. Continue this process until reaching the end of the string. Let /Delta1 be the special marker. Assume that we have used go-right ( i ) to reach the letter c to be deleted. Again assume that we begin in state sx and want to end in state sy . We shall also let the /Sigma1 = { a , b , c } and /Gamma1 = { a , b , c } ∪ { # , } . We then have the following set of rules: Note that the marker is not actually needed. It is used to make the rule… \end{verbatim} ``` </details>
783. ph-af73f20059ce2dbf151d — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) o be deleted. Again assume that we begin in state sx and want to end in state sy . We shall also let the /Sigma1 = { a , b , c } and /Gamma1 = { a , b , c } ∪ { # , } . We then have the following set of rules: Note that the marker is not actually needed. It is used to make the rules easier to read. For example, suppose we have the string abcbac and wish to remove the c in the third space. We use go-right (2) to get to the desired space and have the configuration We then have the following rules and configurations: Note that the marker is not actually nee…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=35535 \begin{verbatim} o be deleted. Again assume that we begin in state sx and want to end in state sy . We shall also let the /Sigma1 = { a , b , c } and /Gamma1 = { a , b , c } ∪ { # , } . We then have the following set of rules: Note that the marker is not actually needed. It is used to make the rules easier to read. For example, suppose we have the string abcbac and wish to remove the c in the third space. We use go-right (2) to get to the desired space and have the configuration We then have the following rules and configurations: Note that the marker is not actually needed. It is used to make the rules easier to read. For example, suppose we have the string abcbac and wish to remove the c in the third space. We use go-right (2) to get to the desired space and have the configuration We then have the following rules and configurations: Finally applying rule we have configuration <!-… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{e shall also let the /Sigma1 = \{ a , b , c \} and /Gamma1 = \{ a , b , c \} ∪ \{ \# , \} . We then have the following set of rules: Note that the marker is not actually needed. It is used to make the rules easier to read. For example, suppose we have the s…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=35619 \begin{verbatim} e shall also let the /Sigma1 = { a , b , c } and /Gamma1 = { a , b , c } ∪ { # , } . We then have the following set of rules: Note that the marker is not actually needed. It is used to make the rules easier to read. For example, suppose we have the string abcbac and wish to remove the c in the third space. We use go-right (2) to get to the desired space and have the configuration We then have the following rules and configurations: Finally applying rule we have configuration <!-… \end{verbatim} ``` </details>
785. ph-52fe242a9265e0544482 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) { # , } . We then have the following set of rules: Note that the marker is not actually needed. It is used to make the rules easier to read. For example, suppose we have the string abcbac and wish to remove the c in the third space. We use go-right (2) to get to the desired space and have the configuration We then have the following rules and configurations: Finally applying rule we have configuration and using go-left (5), we return to the beginning… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{\{ \# , \} . We then have the following set of rules: Note that the marker is not actually needed. It is used to make the rules easier to read. For example, suppose we have the string abcbac and wish to remove the c in the third space. We use go-right (…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=35694 \begin{verbatim} { # , } . We then have the following set of rules: Note that the marker is not actually needed. It is used to make the rules easier to read. For example, suppose we have the string abcbac and wish to remove the c in the third space. We use go-right (2) to get to the desired space and have the configuration We then have the following rules and configurations: Finally applying rule we have configuration and using go-left (5), we return to the beginning… \end{verbatim} ``` </details>
786. ph-d94cc4d1c0c96cbb30e5 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) lowing set of rules: Note that the marker is not actually needed. It is used to make the rules easier to read. For example, suppose we have the string abcbac and wish to remove the c in the third space. We use go-right (2) to get to the desired space and have the configuration We then have the following rules and configurations: Finally applying rule we have configuration and using go-left (5), we return to the beginning of the string. Finally we sho… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{lowing set of rules: Note that the marker is not actually needed. It is used to make the rules easier to read. For example, suppose we have the string abcbac and wish to remove the c in the third space. We use go-right (2) to get to the desired space…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=35724 \begin{verbatim} lowing set of rules: Note that the marker is not actually needed. It is used to make the rules easier to read. For example, suppose we have the string abcbac and wish to remove the c in the third space. We use go-right (2) to get to the desired space and have the configuration We then have the following rules and configurations: Finally applying rule we have configuration and using go-left (5), we return to the beginning of the string. Finally we sho… \end{verbatim} ``` </details>
787. ph-5faf5126ad09d38369ee — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) tring is replaced by λ a if the letter is a and by λ b if the letter is b . We then go to the end of the string and place a corresponding σ a , or σ b . We then return to the first symbol and replace it with the original letter, go to the second letter and repeat the process. We continue until we have a string followed by corresponding σ a s, and σ b s. We then replace each σ a with an a and σ b with a b . Assumethatwestartinstate sx andendinstate sy . We then have the following set of rules: For example, we shall duplicate the word bab . The initial configuration… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{tring is replaced by λ a if the letter is a and by λ b if the letter is b . We then go to the end of the string and place a corresponding σ a , or σ b . We then return to the first symbol and replace it with the original letter, go to the second letter and repeat the process. We…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=36661 \begin{verbatim} tring is replaced by λ a if the letter is a and by λ b if the letter is b . We then go to the end of the string and place a corresponding σ a , or σ b . We then return to the first symbol and replace it with the original letter, go to the second letter and repeat the process. We continue until we have a string followed by corresponding σ a s, and σ b s. We then replace each σ a with an a and σ b with a b . Assumethatwestartinstate sx andendinstate sy . We then have the following set of rules: For example, we shall duplicate the word bab . The initial configuration… \end{verbatim} ``` </details>
788. ph-746197eae6798873ef49 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) ring and place a corresponding σ a , or σ b . We then return to the first symbol and replace it with the original letter, go to the second letter and repeat the process. We continue until we have a string followed by corresponding σ a s, and σ b s. We then replace each σ a with an a and σ b with a b . Assumethatwestartinstate sx andendinstate sy . We then have the following set of rules: For example, we shall duplicate the word bab . The initial configuration is We then have the following rules and configurations: For example, we shall duplicate the word bab . The initial configuration is We then have the following rules and configurations: For example, we shall duplicate the word bab . The initial configuration is We then have the following rules and configurations: ⇒ ( sx ′ , a , sx ′ , a , L ) /turnstileleft x ′ λ b a b σ b ⇒ ( sx ′ , λ b ,… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{replace it with the original letter, go to the second letter and repeat the process. We continue until we have a string followed by corresponding σ a s, and σ b s. We then replace each σ a with an a and σ b with a b . Assumethatwestartinstate sx andendinstate sy . We then have t…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=36852 \begin{verbatim} replace it with the original letter, go to the second letter and repeat the process. We continue until we have a string followed by corresponding σ a s, and σ b s. We then replace each σ a with an a and σ b with a b . Assumethatwestartinstate sx andendinstate sy . We then have the following set of rules: For example, we shall duplicate the word bab . The initial configuration is We then have the following rules and configurations: ⇒ ( sx ′ , a , sx ′ , a , L ) /turnstileleft x ′ λ b a b σ b ⇒ ( sx ′ , λ b ,… \end{verbatim} ``` </details>
790. ph-8907df43ce0db03a5087 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) g that a Turing machine can recognize a regular language. We already know that an automaton recognizes a regular language, so what we shall basically do is program it to imitate an automaton. Assume that we have a word in the Turing machine which we want the machine to read so that it can determine whether it wants to accept it. An automaton reads a word beginning with the first letter and reads from left to right until it has reached the last letter. We need our Turing machine to do the same. and we are ready to begin. Wehave another way of representing a Turing … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{g that a Turing machine can recognize a regular language. We already know that an automaton recognizes a regular language, so what we shall basically do is program it to imitate an automaton. Assume that we have a word in the Turing machine which we want the machine to read so t…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=38999 \begin{verbatim} g that a Turing machine can recognize a regular language. We already know that an automaton recognizes a regular language, so what we shall basically do is program it to imitate an automaton. Assume that we have a word in the Turing machine which we want the machine to read so that it can determine whether it wants to accept it. An automaton reads a word beginning with the first letter and reads from left to right until it has reached the last letter. We need our Turing machine to do the same. and we are ready to begin. Wehave another way of representing a Turing … \end{verbatim} ``` </details>
791. ph-c4280ad9a77aef55eb58 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) tomaton. Assume that we have a word in the Turing machine which we want the machine to read so that it can determine whether it wants to accept it. An automaton reads a word beginning with the first letter and reads from left to right until it has reached the last letter. We need our Turing machine to do the same. and we are ready to begin. Wehave another way of representing a Turing machine which makes it look more like an automaton. We shall represent the rule so that the program go-end which has rules <!… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{tomaton. Assume that we have a word in the Turing machine which we want the machine to read so that it can determine whether it wants to accept it. An automaton reads a word beginning with the first letter and reads from left to right until it has reached the last letter. We nee…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=39182 \begin{verbatim} tomaton. Assume that we have a word in the Turing machine which we want the machine to read so that it can determine whether it wants to accept it. An automaton reads a word beginning with the first letter and reads from left to right until it has reached the last letter. We need our Turing machine to do the same. and we are ready to begin. Wehave another way of representing a Turing machine which makes it look more like an automaton. We shall represent the rule so that the program go-end which has rules <!… \end{verbatim} ``` </details>
792. ph-07569323fe976bb4e4af — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) word in the Turing machine which we want the machine to read so that it can determine whether it wants to accept it. An automaton reads a word beginning with the first letter and reads from left to right until it has reached the last letter. We need our Turing machine to do the same. and we are ready to begin. Wehave another way of representing a Turing machine which makes it look more like an automaton. We shall represent the rule so that the program go-end which has rules by t… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{word in the Turing machine which we want the machine to read so that it can determine whether it wants to accept it. An automaton reads a word beginning with the first letter and reads from left to right until it has reached the last letter. We need our Turing machine to do the …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=39212 \begin{verbatim} word in the Turing machine which we want the machine to read so that it can determine whether it wants to accept it. An automaton reads a word beginning with the first letter and reads from left to right until it has reached the last letter. We need our Turing machine to do the same. and we are ready to begin. Wehave another way of representing a Turing machine which makes it look more like an automaton. We shall represent the rule so that the program go-end which has rules by t… \end{verbatim} ``` </details>
793. ph-2ac184605d2e2484bcee — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) an determine whether it wants to accept it. An automaton reads a word beginning with the first letter and reads from left to right until it has reached the last letter. We need our Turing machine to do the same. and we are ready to begin. Wehave another way of representing a Turing machine which makes it look more like an automaton. We shall represent the rule so that the program go-end which has rules by the symbol may b… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{an determine whether it wants to accept it. An automaton reads a word beginning with the first letter and reads from left to right until it has reached the last letter. We need our Turing machine to do the same. and we are ready to begin. Wehave anot…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=39286 \begin{verbatim} an determine whether it wants to accept it. An automaton reads a word beginning with the first letter and reads from left to right until it has reached the last letter. We need our Turing machine to do the same. and we are ready to begin. Wehave another way of representing a Turing machine which makes it look more like an automaton. We shall represent the rule so that the program go-end which has rules by the symbol may b… \end{verbatim} ``` </details>
794. ph-59bf3627e55456584fce — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) n automaton reads a word beginning with the first letter and reads from left to right until it has reached the last letter. We need our Turing machine to do the same. and we are ready to begin. Wehave another way of representing a Turing machine which makes it look more like an automaton. We shall represent the rule so that the program go-end which has rules by the symbol may be represented by or the symbol shown as ![Ima… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{n automaton reads a word beginning with the first letter and reads from left to right until it has reached the last letter. We need our Turing machine to do the same. and we are ready to begin. Wehave another way of representing a Turing machine whic…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=39331 \begin{verbatim} n automaton reads a word beginning with the first letter and reads from left to right until it has reached the last letter. We need our Turing machine to do the same. and we are ready to begin. Wehave another way of representing a Turing machine which makes it look more like an automaton. We shall represent the rule so that the program go-end which has rules by the symbol may be represented by or the symbol shown as ![Ima… \end{verbatim} ``` </details>
795. ph-446c2f5479d38a92eb09 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) ning with the first letter and reads from left to right until it has reached the last letter. We need our Turing machine to do the same. and we are ready to begin. Wehave another way of representing a Turing machine which makes it look more like an automaton. We shall represent the rule so that the program go-end which has rules by the symbol may be represented by or the symbol shown as ![Image](./AutomataTheory_chapter_2… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ning with the first letter and reads from left to right until it has reached the last letter. We need our Turing machine to do the same. and we are ready to begin. Wehave another way of representing a Turing machine which makes it look more like an a…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=39361 \begin{verbatim} ning with the first letter and reads from left to right until it has reached the last letter. We need our Turing machine to do the same. and we are ready to begin. Wehave another way of representing a Turing machine which makes it look more like an automaton. We shall represent the rule so that the program go-end which has rules by the symbol may be represented by or the symbol shown as ![Image](./AutomataTheory_chapter_2… \end{verbatim} ``` </details>
796. ph-98f8ac5441116fbe07a8 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) could print a # in each square as it is read or we could simply print back the letter that is read. We shall choose to do the latter. Each time a letter is read, we wish the machine to move one square to the left, so that the next letter is read. We are now ready to imitate an automaton. If the symbol a ![Image](./AutomataTheory_chapter_2.2_artifacts/image_000007_98dcb5ef041f06761b94e94218522149a14a68488147ae18880c0c34c5aef771.png) occurs in an automaton, we shall imitate it with the rule ![Image](./AutomataTheory_chapter_2.2_artifacts/image_000008_cb4113b9fe290fd… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{could print a \# in each square as it is read or we could simply print back the letter that is read. We shall choose to do the latter. Each time a letter is read, we wish the machine to move one square to the left, so that the next letter is read. We are now ready to imitate an a…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=40198 \begin{verbatim} could print a # in each square as it is read or we could simply print back the letter that is read. We shall choose to do the latter. Each time a letter is read, we wish the machine to move one square to the left, so that the next letter is read. We are now ready to imitate an automaton. If the symbol a ![Image](./AutomataTheory_chapter_2.2_artifacts/image_000007_98dcb5ef041f06761b94e94218522149a14a68488147ae18880c0c34c5aef771.png) occurs in an automaton, we shall imitate it with the rule ![Image](./AutomataTheory_chapter_2.2_artifacts/image_000008_cb4113b9fe290fd… \end{verbatim} ``` </details>
797. ph-ad5eaa729e597f616a9f — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) image_000007_98dcb5ef041f06761b94e94218522149a14a68488147ae18880c0c34c5aef771.png) occurs in an automaton, we shall imitate it with the rule ![Image](./AutomataTheory_chapter_2.2_artifacts/image_000008_cb4113b9fe290fd03719777786524fe1a737760b2c167b6ca62948e65e1df94a.png) It may be recalled that a word is accepted by an automaton if, after the word is read, the automaton is in an acceptance state. For every acceptance state s of the automaton, we will add a rule ![Image](./AutomataTheory_chapter_2.2_artifacts/image_000009_65c2a96dd26b65… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{image\_000007\_98dcb5ef041f06761b94e94218522149a14a68488147ae18880c0c34c5aef771.png) occurs in an automaton, we shall imitate it with the rule ![Image](./AutomataTheory\_chapter\_2.2\_artifacts/image\_000008\_cb4113b9fe290fd03719777786524fe1a737760b2c167b6c…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=40555 \begin{verbatim} image_000007_98dcb5ef041f06761b94e94218522149a14a68488147ae18880c0c34c5aef771.png) occurs in an automaton, we shall imitate it with the rule ![Image](./AutomataTheory_chapter_2.2_artifacts/image_000008_cb4113b9fe290fd03719777786524fe1a737760b2c167b6ca62948e65e1df94a.png) It may be recalled that a word is accepted by an automaton if, after the word is read, the automaton is in an acceptance state. For every acceptance state s of the automaton, we will add a rule ![Image](./AutomataTheory_chapter_2.2_artifacts/image_000009_65c2a96dd26b65… \end{verbatim} ``` </details>
798. ph-48c0039c1ec8e95d1e46 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) tive integer } , which is not context-free. We begin by designing a program for a Turing machine that will recognize the language { a n b n : n is a positive integer } . We basically want the Turing machine to read an a , then read a b , return to read an a , and continue until all of the a s and b s have been read, if there is an equal number of them. We begin by reading an a in the first square. We want to know that we have counted this a , so we shall change it to A . We do this with rule We now want to go right until we reach a b , which we shall change to a B… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{tive integer \} , which is not context-free. We begin by designing a program for a Turing machine that will recognize the language \{ a n b n : n is a positive integer \} . We basically want the Turing machine to read an a , then read a b , return to read an a , and continue until …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=42512 \begin{verbatim} tive integer } , which is not context-free. We begin by designing a program for a Turing machine that will recognize the language { a n b n : n is a positive integer } . We basically want the Turing machine to read an a , then read a b , return to read an a , and continue until all of the a s and b s have been read, if there is an equal number of them. We begin by reading an a in the first square. We want to know that we have counted this a , so we shall change it to A . We do this with rule We now want to go right until we reach a b , which we shall change to a B… \end{verbatim} ``` </details>
799. ph-f84d72ce4843410f8514 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) a s and b s have been read, if there is an equal number of them. We begin by reading an a in the first square. We want to know that we have counted this a , so we shall change it to A . We do this with rule We now want to go right until we reach a b , which we shall change to a B . To get to b , we need to pass over each a without changing it and also after the first time we may have to pass over B s without changing them to reach a b . We do this with the rules When we reach a b , we want to change it to a B and go back left. We do th… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{a s and b s have been read, if there is an equal number of them. We begin by reading an a in the first square. We want to know that we have counted this a , so we shall change it to A . We do this with rule We now want to go right until we reach a b …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=42803 \begin{verbatim} a s and b s have been read, if there is an equal number of them. We begin by reading an a in the first square. We want to know that we have counted this a , so we shall change it to A . We do this with rule We now want to go right until we reach a b , which we shall change to a B . To get to b , we need to pass over each a without changing it and also after the first time we may have to pass over B s without changing them to reach a b . We do this with the rules When we reach a b , we want to change it to a B and go back left. We do th… \end{verbatim} ``` </details>
800. ph-a257e55d192399f88295 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) know that we have counted this a , so we shall change it to A . We do this with rule We now want to go right until we reach a b , which we shall change to a B . To get to b , we need to pass over each a without changing it and also after the first time we may have to pass over B s without changing them to reach a b . We do this with the rules When we reach a b , we want to change it to a B and go back left. We do that with the rule We now need to go back to find the second a . To do this we go left until we… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{know that we have counted this a , so we shall change it to A . We do this with rule We now want to go right until we reach a b , which we shall change to a B . To get to b , we need to pass over each a without changing it and also after the first ti…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=42925 \begin{verbatim} know that we have counted this a , so we shall change it to A . We do this with rule We now want to go right until we reach a b , which we shall change to a B . To get to b , we need to pass over each a without changing it and also after the first time we may have to pass over B s without changing them to reach a b . We do this with the rules When we reach a b , we want to change it to a B and go back left. We do that with the rule We now need to go back to find the second a . To do this we go left until we… \end{verbatim} ``` </details>
801. ph-93af844aa9c05cc46466 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) me we may have to pass over B s without changing them to reach a b . We do this with the rules When we reach a b , we want to change it to a B and go back left. We do that with the rule We now need to go back to find the second a . To do this we go left until we reach an A . This will tell us that the next letter to the right should be the next a . To go back, we need to pass over B s, and a s to get to A . We do this with the rules When we reach A , we want to go one square to the right to read another a ,… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{me we may have to pass over B s without changing them to reach a b . We do this with the rules When we reach a b , we want to change it to a B and go back left. We do that with the rule We now need to go back to find the …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=43207 \begin{verbatim} me we may have to pass over B s without changing them to reach a b . We do this with the rules When we reach a b , we want to change it to a B and go back left. We do that with the rule We now need to go back to find the second a . To do this we go left until we reach an A . This will tell us that the next letter to the right should be the next a . To go back, we need to pass over B s, and a s to get to A . We do this with the rules When we reach A , we want to go one square to the right to read another a ,… \end{verbatim} ``` </details>
802. ph-ebb58671d4b8ee946848 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) want to change it to a B and go back left. We do that with the rule We now need to go back to find the second a . To do this we go left until we reach an A . This will tell us that the next letter to the right should be the next a . To go back, we need to pass over B s, and a s to get to A . We do this with the rules When we reach A , we want to go one square to the right to read another a , if there is one. We do this with the rule This puts us back into the cycle of reading another a and another b . If we… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{want to change it to a B and go back left. We do that with the rule We now need to go back to find the second a . To do this we go left until we reach an A . This will tell us that the next letter to the right should be the next a . To go back, we ne…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=43356 \begin{verbatim} want to change it to a B and go back left. We do that with the rule We now need to go back to find the second a . To do this we go left until we reach an A . This will tell us that the next letter to the right should be the next a . To go back, we need to pass over B s, and a s to get to A . We do this with the rules When we reach A , we want to go one square to the right to read another a , if there is one. We do this with the rule This puts us back into the cycle of reading another a and another b . If we… \end{verbatim} ``` </details>
803. ph-2f0a0e918b9d209834d7 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) ere is one. We do this with the rule This puts us back into the cycle of reading another a and another b . If we run out of b s before we run out of a s the system will be in state s 1 and eventually try to read a blank so it will hang. If we have read the last a , then when we reach A and go right one square, we will read a B . At this point we need to check to see if there is another b . First we change state if we are in s 0 and read a B . We do this with rule In state s 3, read nothing but B s and a blank. Thus we have the rules <!… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ere is one. We do this with the rule This puts us back into the cycle of reading another a and another b . If we run out of b s before we run out of a s the system will be in state s 1 and eventually try to read a blank so it will hang. If we have re…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=43818 \begin{verbatim} ere is one. We do this with the rule This puts us back into the cycle of reading another a and another b . If we run out of b s before we run out of a s the system will be in state s 1 and eventually try to read a blank so it will hang. If we have read the last a , then when we reach A and go right one square, we will read a B . At this point we need to check to see if there is another b . First we change state if we are in s 0 and read a B . We do this with rule In state s 3, read nothing but B s and a blank. Thus we have the rules <!… \end{verbatim} ``` </details>
804. ph-eeb2dd7fa471b3e78f43 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) f reading another a and another b . If we run out of b s before we run out of a s the system will be in state s 1 and eventually try to read a blank so it will hang. If we have read the last a , then when we reach A and go right one square, we will read a B . At this point we need to check to see if there is another b . First we change state if we are in s 0 and read a B . We do this with rule In state s 3, read nothing but B s and a blank. Thus we have the rules ## This may also be shown as the labeled graph ![Image](./AutomataTheory_… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{f reading another a and another b . If we run out of b s before we run out of a s the system will be in state s 1 and eventually try to read a blank so it will hang. If we have read the last a , then when we reach A and go right one square, we will read a B . At this point we ne…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=43920 \begin{verbatim} f reading another a and another b . If we run out of b s before we run out of a s the system will be in state s 1 and eventually try to read a blank so it will hang. If we have read the last a , then when we reach A and go right one square, we will read a B . At this point we need to check to see if there is another b . First we change state if we are in s 0 and read a B . We do this with rule In state s 3, read nothing but B s and a blank. Thus we have the rules ## This may also be shown as the labeled graph ![Image](./AutomataTheory_… \end{verbatim} ``` </details>
805. ph-aa30ddc6e905c1927987 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) ed to check to see if there is another b . First we change state if we are in s 0 and read a B . We do this with rule In state s 3, read nothing but B s and a blank. Thus we have the rules ## This may also be shown as the labeled graph ![Image](./AutomataTheory_chapter_2.2_artifacts/image_000012_127ef03fe2afa9feb2186c5df8414e0ff7003f9d4eaa3c13fa91382090c32e81.png) For example consider the string aabb . The initial configuration is We then have the following rules and configurations: In state s 3, read nothing but B s and a blank. Thus we have the rules \#\# This may also be shown as the…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=44199 \begin{verbatim} ed to check to see if there is another b . First we change state if we are in s 0 and read a B . We do this with rule In state s 3, read nothing but B s and a blank. Thus we have the rules ## This may also be shown as the labeled graph ![Image](./AutomataTheory_chapter_2.2_artifacts/image_000012_127ef03fe2afa9feb2186c5df8414e0ff7003f9d4eaa3c13fa91382090c32e81.png) For example consider the string aabb . The initial configuration is We then have the following rules and configurations: In state s 3, read nothing but B s and a blank. Thus we have the rules ## This may also be shown as the labeled graph ![Image](./AutomataTheory_chapter_2.2_artifacts/image_000012_127ef03fe2afa9feb2186c5df8414e0ff7003f9d4eaa3c13fa91382090c32e81.png) For example consider the string aabb . The initial configuration is We then have the following rules and configurations: Next we design a program for a Turing machine that… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{d read a B . We do this with rule In state s 3, read nothing but B s and a blank. Thus we have the rules \#\# This may also be shown as the labeled graph ![Image](./AutomataTheory\_chapter\_2.2\_artifacts/image\_000012\_127ef03f…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=44283 \begin{verbatim} d read a B . We do this with rule In state s 3, read nothing but B s and a blank. Thus we have the rules ## This may also be shown as the labeled graph ![Image](./AutomataTheory_chapter_2.2_artifacts/image_000012_127ef03fe2afa9feb2186c5df8414e0ff7003f9d4eaa3c13fa91382090c32e81.png) For example consider the string aabb . The initial configuration is We then have the following rules and configurations: Next we design a program for a Turing machine that… \end{verbatim} ``` </details>
807. ph-2c55669c054570e88275 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) ule In state s 3, read nothing but B s and a blank. Thus we have the rules ## This may also be shown as the labeled graph ![Image](./AutomataTheory_chapter_2.2_artifacts/image_000012_127ef03fe2afa9feb2186c5df8414e0ff7003f9d4eaa3c13fa91382090c32e81.png) For example consider the string aabb . The initial configuration is We then have the following rules and configurations: Next we design a program for a Turing machine that will recognize the language {… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ule In state s 3, read nothing but B s and a blank. Thus we have the rules \#\# This may also be shown as the labeled graph ![Image](./AutomataTheory\_chapter\_2.2\_artifacts/image\_000012\_127ef03fe2afa9feb2186c5df8414e0ff7003f…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=44313 \begin{verbatim} ule In state s 3, read nothing but B s and a blank. Thus we have the rules ## This may also be shown as the labeled graph ![Image](./AutomataTheory_chapter_2.2_artifacts/image_000012_127ef03fe2afa9feb2186c5df8414e0ff7003f9d4eaa3c13fa91382090c32e81.png) For example consider the string aabb . The initial configuration is We then have the following rules and configurations: Next we design a program for a Turing machine that will recognize the language {… \end{verbatim} ``` </details>
808. ph-1b9fc6fc39f36752395e — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) 3fa91382090c32e81.png) For example consider the string aabb . The initial configuration is We then have the following rules and configurations: Next we design a program for a Turing machine that will recognize the language { a n b n : n is a positive integer } . We begin by reading an a in the first square. We want to know that we have counted this a , so we shall change it to A . We do this with rule We now want to go right until we reach a b , which we shall change to a B . To… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{3fa91382090c32e81.png) For example consider the string aabb . The initial configuration is We then have the following rules and configurations: Next we design a program for a Turing machine th…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=44606 \begin{verbatim} 3fa91382090c32e81.png) For example consider the string aabb . The initial configuration is We then have the following rules and configurations: Next we design a program for a Turing machine that will recognize the language { a n b n : n is a positive integer } . We begin by reading an a in the first square. We want to know that we have counted this a , so we shall change it to A . We do this with rule We now want to go right until we reach a b , which we shall change to a B . To… \end{verbatim} ``` </details>
809. ph-f3969c267cc530400399 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) l recognize the language { a n b n : n is a positive integer } . We begin by reading an a in the first square. We want to know that we have counted this a , so we shall change it to A . We do this with rule We now want to go right until we reach a b , which we shall change to a B . To get to b , we need to pass over each a without changing it and also after the first time we may have to pass over B s without changing them to reach a b . We do this with the rules When we reach a b , we want to change it to a B and start back to look for… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{l recognize the language \{ a n b n : n is a positive integer \} . We begin by reading an a in the first square. We want to know that we have counted this a , so we shall change it to A . We do this with rule We now want to go right until we reach a b …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=44897 \begin{verbatim} l recognize the language { a n b n : n is a positive integer } . We begin by reading an a in the first square. We want to know that we have counted this a , so we shall change it to A . We do this with rule We now want to go right until we reach a b , which we shall change to a B . To get to b , we need to pass over each a without changing it and also after the first time we may have to pass over B s without changing them to reach a b . We do this with the rules When we reach a b , we want to change it to a B and start back to look for… \end{verbatim} ``` </details>
810. ph-6be2875e16f13f45d609 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) unted this a , so we shall change it to A . We do this with rule We now want to go right until we reach a b , which we shall change to a B . To get to b , we need to pass over each a without changing it and also after the first time we may have to pass over B s without changing them to reach a b . We do this with the rules When we reach a b , we want to change it to a B and start back to look for another a . We do this with the rule To go back, we need to pass over B s and a s to get to A . We do this with … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{unted this a , so we shall change it to A . We do this with rule We now want to go right until we reach a b , which we shall change to a B . To get to b , we need to pass over each a without changing it and also after the first time we may have to pa…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=45040 \begin{verbatim} unted this a , so we shall change it to A . We do this with rule We now want to go right until we reach a b , which we shall change to a B . To get to b , we need to pass over each a without changing it and also after the first time we may have to pass over B s without changing them to reach a b . We do this with the rules When we reach a b , we want to change it to a B and start back to look for another a . We do this with the rule To go back, we need to pass over B s and a s to get to A . We do this with … \end{verbatim} ``` </details>
811. ph-accbf8e48ab3f2a9c021 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) ght until we reach a b , which we shall change to a B . To get to b , we need to pass over each a without changing it and also after the first time we may have to pass over B s without changing them to reach a b . We do this with the rules When we reach a b , we want to change it to a B and start back to look for another a . We do this with the rule To go back, we need to pass over B s and a s to get to A . We do this with the rules When we reach A , we want to go one square to the right to read another a ,… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ght until we reach a b , which we shall change to a B . To get to b , we need to pass over each a without changing it and also after the first time we may have to pass over B s without changing them to reach a b . We do this with the rules When we re…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=45156 \begin{verbatim} ght until we reach a b , which we shall change to a B . To get to b , we need to pass over each a without changing it and also after the first time we may have to pass over B s without changing them to reach a b . We do this with the rules When we reach a b , we want to change it to a B and start back to look for another a . We do this with the rule To go back, we need to pass over B s and a s to get to A . We do this with the rules When we reach A , we want to go one square to the right to read another a ,… \end{verbatim} ``` </details>
812. ph-95ee0c4605931321ed03 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) e may have to pass over B s without changing them to reach a b . We do this with the rules When we reach a b , we want to change it to a B and start back to look for another a . We do this with the rule To go back, we need to pass over B s and a s to get to A . We do this with the rules When we reach A , we want to go one square to the right to read another a , if there is one. We do this with the rule This puts us back into the cycle of reading another a and b . If we run out o… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{e may have to pass over B s without changing them to reach a b . We do this with the rules When we reach a b , we want to change it to a B and start back to look for another a . We do this with the rule To go back, we nee…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=45305 \begin{verbatim} e may have to pass over B s without changing them to reach a b . We do this with the rules When we reach a b , we want to change it to a B and start back to look for another a . We do this with the rule To go back, we need to pass over B s and a s to get to A . We do this with the rules When we reach A , we want to go one square to the right to read another a , if there is one. We do this with the rule This puts us back into the cycle of reading another a and b . If we run out o… \end{verbatim} ``` </details>
813. ph-fd8c8acbb52b9d4c8841 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) ch A , we want to go one square to the right to read another a , if there is one. We do this with the rule This puts us back into the cycle of reading another a and b . If we run out of b s before we run out of a s the system will hang. If we have read the last a , then when we reach A and go right one square, we will read a B . At this point we need to check to see if there is another b . First we change state if we are in s 0 and read a B . We do this with rule In state s 3, we expect to read nothing but B s, b , and a blank. Thus we… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ch A , we want to go one square to the right to read another a , if there is one. We do this with the rule This puts us back into the cycle of reading another a and b . If we run out of b s before we run out of a s the system will hang. If we have re…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=45697 \begin{verbatim} ch A , we want to go one square to the right to read another a , if there is one. We do this with the rule This puts us back into the cycle of reading another a and b . If we run out of b s before we run out of a s the system will hang. If we have read the last a , then when we reach A and go right one square, we will read a B . At this point we need to check to see if there is another b . First we change state if we are in s 0 and read a B . We do this with rule In state s 3, we expect to read nothing but B s, b , and a blank. Thus we… \end{verbatim} ``` </details>
814. ph-65b71a727f3c15d357ac — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) -not-decoded --> This puts us back into the cycle of reading another a and b . If we run out of b s before we run out of a s the system will hang. If we have read the last a , then when we reach A and go right one square, we will read a B . At this point we need to check to see if there is another b . First we change state if we are in s 0 and read a B . We do this with rule In state s 3, we expect to read nothing but B s, b , and a blank. Thus we have the rules We now design a program for a Turing machine that will recognize the langu… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{-not-decoded --> This puts us back into the cycle of reading another a and b . If we run out of b s before we run out of a s the system will hang. If we have read the last a , then when we reach A and go right one square, we will read a B . At this point we need to check to see …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=45817 \begin{verbatim} -not-decoded --> This puts us back into the cycle of reading another a and b . If we run out of b s before we run out of a s the system will hang. If we have read the last a , then when we reach A and go right one square, we will read a B . At this point we need to check to see if there is another b . First we change state if we are in s 0 and read a B . We do this with rule In state s 3, we expect to read nothing but B s, b , and a blank. Thus we have the rules We now design a program for a Turing machine that will recognize the langu… \end{verbatim} ``` </details>
815. ph-ace3c62fee95ffac1331 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) t-decoded --> We now design a program for a Turing machine that will recognize the language { a n b n c n : n is a positive integer } . In a manner similar to the previous example we want the Turing machine to read an a , then read a b , then read a c , and continue until all of the a s, b s, and c s have been read, if there are an equal number of them. We begin by reading an a in the first square. We want to know that we have counted this a , so we shall change it to A . We do this with rule We now want to go right until we reach a b , which we shall change to a … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{t-decoded --> We now design a program for a Turing machine that will recognize the language \{ a n b n c n : n is a positive integer \} . In a manner similar to the previous example we want the Turing machine to read an a , then read a b , then read a c , and continue until all of…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=46332 \begin{verbatim} t-decoded --> We now design a program for a Turing machine that will recognize the language { a n b n c n : n is a positive integer } . In a manner similar to the previous example we want the Turing machine to read an a , then read a b , then read a c , and continue until all of the a s, b s, and c s have been read, if there are an equal number of them. We begin by reading an a in the first square. We want to know that we have counted this a , so we shall change it to A . We do this with rule We now want to go right until we reach a b , which we shall change to a … \end{verbatim} ``` </details>
816. ph-13d8055cc33155fa10f0 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) s, and c s have been read, if there are an equal number of them. We begin by reading an a in the first square. We want to know that we have counted this a , so we shall change it to A . We do this with rule We now want to go right until we reach a b , which we shall change to a B . To get to b , we need to pass over each a without changing it and also after the first time we may have to pass over B s without changing them to reach a b . We do this with the rules When we reach a b , we want to change it to a B and continue onward. We do… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{s, and c s have been read, if there are an equal number of them. We begin by reading an a in the first square. We want to know that we have counted this a , so we shall change it to A . We do this with rule We now want to go right until we reach a b …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=46623 \begin{verbatim} s, and c s have been read, if there are an equal number of them. We begin by reading an a in the first square. We want to know that we have counted this a , so we shall change it to A . We do this with rule We now want to go right until we reach a b , which we shall change to a B . To get to b , we need to pass over each a without changing it and also after the first time we may have to pass over B s without changing them to reach a b . We do this with the rules When we reach a b , we want to change it to a B and continue onward. We do… \end{verbatim} ``` </details>
817. ph-9f9159f9da3c8c52832b — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) ow that we have counted this a , so we shall change it to A . We do this with rule We now want to go right until we reach a b , which we shall change to a B . To get to b , we need to pass over each a without changing it and also after the first time we may have to pass over B s without changing them to reach a b . We do this with the rules When we reach a b , we want to change it to a B and continue onward. We do that with the rule We now need to continue until we find a c . We will need to pass over b s a… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ow that we have counted this a , so we shall change it to A . We do this with rule We now want to go right until we reach a b , which we shall change to a B . To get to b , we need to pass over each a without changing it and also after the first time…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=46748 \begin{verbatim} ow that we have counted this a , so we shall change it to A . We do this with rule We now want to go right until we reach a b , which we shall change to a B . To get to b , we need to pass over each a without changing it and also after the first time we may have to pass over B s without changing them to reach a b . We do this with the rules When we reach a b , we want to change it to a B and continue onward. We do that with the rule We now need to continue until we find a c . We will need to pass over b s a… \end{verbatim} ``` </details>
818. ph-d890e57459f81aa59820 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) til we reach a b , which we shall change to a B . To get to b , we need to pass over each a without changing it and also after the first time we may have to pass over B s without changing them to reach a b . We do this with the rules When we reach a b , we want to change it to a B and continue onward. We do that with the rule We now need to continue until we find a c . We will need to pass over b s and C s. We do this with the rules Wenext want to read c , replace it with a C , and start back to look for an… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{til we reach a b , which we shall change to a B . To get to b , we need to pass over each a without changing it and also after the first time we may have to pass over B s without changing them to reach a b . We do this with the rules When we reach a …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=46888 \begin{verbatim} til we reach a b , which we shall change to a B . To get to b , we need to pass over each a without changing it and also after the first time we may have to pass over B s without changing them to reach a b . We do this with the rules When we reach a b , we want to change it to a B and continue onward. We do that with the rule We now need to continue until we find a c . We will need to pass over b s and C s. We do this with the rules Wenext want to read c , replace it with a C , and start back to look for an… \end{verbatim} ``` </details>
819. ph-f2af0dd538f6b335b272 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) we may have to pass over B s without changing them to reach a b . We do this with the rules When we reach a b , we want to change it to a B and continue onward. We do that with the rule We now need to continue until we find a c . We will need to pass over b s and C s. We do this with the rules Wenext want to read c , replace it with a C , and start back to look for another a . We do this with the rule To go back, we need to pass over C s, b s, B s, and a s to get to A . We do th… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{we may have to pass over B s without changing them to reach a b . We do this with the rules When we reach a b , we want to change it to a B and continue onward. We do that with the rule We now need to continue until we fi…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=47029 \begin{verbatim} we may have to pass over B s without changing them to reach a b . We do this with the rules When we reach a b , we want to change it to a B and continue onward. We do that with the rule We now need to continue until we find a c . We will need to pass over b s and C s. We do this with the rules Wenext want to read c , replace it with a C , and start back to look for another a . We do this with the rule To go back, we need to pass over C s, b s, B s, and a s to get to A . We do th… \end{verbatim} ``` </details>
820. ph-cc52700b4e43370ae24b — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) n we reach a b , we want to change it to a B and continue onward. We do that with the rule We now need to continue until we find a c . We will need to pass over b s and C s. We do this with the rules Wenext want to read c , replace it with a C , and start back to look for another a . We do this with the rule To go back, we need to pass over C s, b s, B s, and a s to get to A . We do this with the rules When we reach A , we want to go one square to the right to read another a , i… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{n we reach a b , we want to change it to a B and continue onward. We do that with the rule We now need to continue until we find a c . We will need to pass over b s and C s. We do this with the rules Wenext want to read c…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=47156 \begin{verbatim} n we reach a b , we want to change it to a B and continue onward. We do that with the rule We now need to continue until we find a c . We will need to pass over b s and C s. We do this with the rules Wenext want to read c , replace it with a C , and start back to look for another a . We do this with the rule To go back, we need to pass over C s, b s, B s, and a s to get to A . We do this with the rules When we reach A , we want to go one square to the right to read another a , i… \end{verbatim} ``` </details>
821. ph-7a8c13227c278263377f — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) il we find a c . We will need to pass over b s and C s. We do this with the rules Wenext want to read c , replace it with a C , and start back to look for another a . We do this with the rule To go back, we need to pass over C s, b s, B s, and a s to get to A . We do this with the rules When we reach A , we want to go one square to the right to read another a , if there is one. We do this with the rule This puts us back into the cycle of reading another a , b , and c . If we run… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{il we find a c . We will need to pass over b s and C s. We do this with the rules Wenext want to read c , replace it with a C , and start back to look for another a . We do this with the rule To go back, we need to pass o…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=47305 \begin{verbatim} il we find a c . We will need to pass over b s and C s. We do this with the rules Wenext want to read c , replace it with a C , and start back to look for another a . We do this with the rule To go back, we need to pass over C s, b s, B s, and a s to get to A . We do this with the rules When we reach A , we want to go one square to the right to read another a , if there is one. We do this with the rule This puts us back into the cycle of reading another a , b , and c . If we run… \end{verbatim} ``` </details>
822. ph-872c65633b0af7e41b97 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md ### Plain (markdown context) t to go one square to the right to read another a , if there is one. We do this with the rule This puts us back into the cycle of reading another a , b , and c . If we run out of b s or c s before we run out of a s the system will hang. If we have read the last a , then when we reach A and go right one square, we will read a B . At this point we need to check to see if there is another b . First we change state if we are in s 0 and read a B . We do this with rule ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{t to go one square to the right to read another a , if there is one. We do this with the rule This puts us back into the cycle of reading another a , b , and c . If we run out of b s or c s before we run out of a s the system will hang. If we have re…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.2.md:offset=47710 \begin{verbatim} t to go one square to the right to read another a , if there is one. We do this with the rule This puts us back into the cycle of reading another a , b , and c . If we run out of b s or c s before we run out of a s the system will hang. If we have read the last a , then when we reach A and go right one square, we will read a B . At this point we need to check to see if there is another b . First we change state if we are in s 0 and read a B . We do this with rule \end{verbatim} ``` </details>
823. ph-2231480de02be460429b — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md ### Plain (markdown context) In state s 4, we expect to read nothing but B s, C s, and a blank. Thus we have the rules This may also be shown as the labeled directed graph ![Image](./AutomataTheory_chapter_2.3_artifacts/image_000000_67e323c0d0c3806682cd955a3528a306a11f5ffd40534ed4d616f6a74ef4c6c6.png) For example consider the string aabbcc . The initial configuration is We then have the following rules and configurations: WenowshowhowtoperformtwoarithmeticoperationsonaTuringmach… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{In state s 4, we expect to read nothing but B s, C s, and a blank. Thus we have the rules This may also be shown as the labeled directed graph ![Image](./AutomataTheory\_chapter\_2.3\_artifacts/image\_000000\_67e323c0d0c3806682cd955a3528a306a11f5ffd40534e…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md:offset=91 \begin{verbatim} In state s 4, we expect to read nothing but B s, C s, and a blank. Thus we have the rules This may also be shown as the labeled directed graph ![Image](./AutomataTheory_chapter_2.3_artifacts/image_000000_67e323c0d0c3806682cd955a3528a306a11f5ffd40534ed4d616f6a74ef4c6c6.png) For example consider the string aabbcc . The initial configuration is We then have the following rules and configurations: WenowshowhowtoperformtwoarithmeticoperationsonaTuringmach… \end{verbatim} ``` </details>
824. ph-40bc32ec9697b5af21da — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md ### Plain (markdown context) In state s 4, we expect to read nothing but B s, C s, and a blank. Thus we have the rules This may also be shown as the labeled directed graph ![Image](./AutomataTheory_chapter_2.3_artifacts/image_000000_67e323c0d0c3806682cd955a3528a306a11f5ffd40534ed4d616f6a74ef4c6c6.png) For example consider the string aabbcc . The initial configuration is We then have the following rules and configurations: WenowshowhowtoperformtwoarithmeticoperationsonaTuringmach… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{In state s 4, we expect to read nothing but B s, C s, and a blank. Thus we have the rules This may also be shown as the labeled directed graph ![Image](./AutomataTheory\_chapter\_2.3\_artifacts/image\_000000\_67e323c0d0c3806682cd955a3528a306a11f5ffd40534e…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md:offset=378 \begin{verbatim} In state s 4, we expect to read nothing but B s, C s, and a blank. Thus we have the rules This may also be shown as the labeled directed graph ![Image](./AutomataTheory_chapter_2.3_artifacts/image_000000_67e323c0d0c3806682cd955a3528a306a11f5ffd40534ed4d616f6a74ef4c6c6.png) For example consider the string aabbcc . The initial configuration is We then have the following rules and configurations: WenowshowhowtoperformtwoarithmeticoperationsonaTuringmach… \end{verbatim} ``` </details>
825. ph-bb98b7e3eac82d66d7d1 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md ### Plain (markdown context) In state s 4, we expect to read nothing but B s, C s, and a blank. Thus we have the rules This may also be shown as the labeled directed graph ![Image](./AutomataTheory_chapter_2.3_artifacts/image_000000_67e323c0d0c3806682cd955a3528a306a11f5ffd40534ed4d616f6a74ef4c6c6.png) For example consider the string aabbcc . The initial configuration is We then have the following rules and configurations: WenowshowhowtoperformtwoarithmeticoperationsonaTuringmach… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{In state s 4, we expect to read nothing but B s, C s, and a blank. Thus we have the rules This may also be shown as the labeled directed graph ![Image](./AutomataTheory\_chapter\_2.3\_artifacts/image\_000000\_67e323c0d0c3806682cd955a3528a306a11f5ffd40534e…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md:offset=408 \begin{verbatim} In state s 4, we expect to read nothing but B s, C s, and a blank. Thus we have the rules This may also be shown as the labeled directed graph ![Image](./AutomataTheory_chapter_2.3_artifacts/image_000000_67e323c0d0c3806682cd955a3528a306a11f5ffd40534ed4d616f6a74ef4c6c6.png) For example consider the string aabbcc . The initial configuration is We then have the following rules and configurations: WenowshowhowtoperformtwoarithmeticoperationsonaTuringmach… \end{verbatim} ``` </details>
826. ph-136ba784d43de3144796 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md ### Plain (markdown context) In state s 4, we expect to read nothing but B s, C s, and a blank. Thus we have the rules This may also be shown as the labeled directed graph ![Image](./AutomataTheory_chapter_2.3_artifacts/image_000000_67e323c0d0c3806682cd955a3528a306a11f5ffd40534ed4d616f6a74ef4c6c6.png) For example consider the string aabbcc . The initial configuration is We then have the following rules and configurations: WenowshowhowtoperformtwoarithmeticoperationsonaTuringmach… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{In state s 4, we expect to read nothing but B s, C s, and a blank. Thus we have the rules This may also be shown as the labeled directed graph ![Image](./AutomataTheory\_chapter\_2.3\_artifacts/image\_000000\_67e323c0d0c3806682cd955a3528a306a11f5ffd40534e…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md:offset=492 \begin{verbatim} In state s 4, we expect to read nothing but B s, C s, and a blank. Thus we have the rules This may also be shown as the labeled directed graph ![Image](./AutomataTheory_chapter_2.3_artifacts/image_000000_67e323c0d0c3806682cd955a3528a306a11f5ffd40534ed4d616f6a74ef4c6c6.png) For example consider the string aabbcc . The initial configuration is We then have the following rules and configurations: WenowshowhowtoperformtwoarithmeticoperationsonaTuringmach… \end{verbatim} ``` </details>
827. ph-f1c075540bd56475e0d9 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md ### Plain (markdown context) t to read nothing but B s, C s, and a blank. Thus we have the rules This may also be shown as the labeled directed graph ![Image](./AutomataTheory_chapter_2.3_artifacts/image_000000_67e323c0d0c3806682cd955a3528a306a11f5ffd40534ed4d616f6a74ef4c6c6.png) For example consider the string aabbcc . The initial configuration is We then have the following rules and configurations: WenowshowhowtoperformtwoarithmeticoperationsonaTuringmachine. The first of thes… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{t to read nothing but B s, C s, and a blank. Thus we have the rules This may also be shown as the labeled directed graph ![Image](./AutomataTheory\_chapter\_2.3\_artifacts/image\_000000\_67e323c0d0c3806682cd955a3528a306a11f5ffd40534ed4d616f6a74ef4c6c6.png…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md:offset=522 \begin{verbatim} t to read nothing but B s, C s, and a blank. Thus we have the rules This may also be shown as the labeled directed graph ![Image](./AutomataTheory_chapter_2.3_artifacts/image_000000_67e323c0d0c3806682cd955a3528a306a11f5ffd40534ed4d616f6a74ef4c6c6.png) For example consider the string aabbcc . The initial configuration is We then have the following rules and configurations: WenowshowhowtoperformtwoarithmeticoperationsonaTuringmachine. The first of thes… \end{verbatim} ``` </details>
828. ph-d17906341d72fbccb9de — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md ### Plain (markdown context) θ m 2 , . . . , θ mp , which together with the state and input would give us the rules to use. If we apply all possible relevant sequences, we can produce all possible computations . Hence if a word is accepted by the Turing machine T , it will be accepted in one of these computations. The next problem is the production of the sequences of integers. We shall label these sequences N 0 , N 1 , N 2 , . . . , Ni , . . . We begin with N 0 = 0 and simply count in base n + 1. Thus the sequences are The sequence following is and the sequence following is The sequence following is and the sequence following is The sequence following is and the sequence following is (1 , 3 , 4 , n , n , n ) The sequence following is and the sequence following is (1 , 3 , 4 , n , n , n ) The sequence following is and the sequence following is (1 , 3 , 4 , n , n , n ) The subroutine in whic… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{le relevant sequences, we can produce all possible computations . Hence if a word is accepted by the Turing machine T , it will be accepted in one of these computations. The next problem is the production of the sequences of integers. We shall label these sequences N 0 , N 1 , N…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md:offset=9538 \begin{verbatim} le relevant sequences, we can produce all possible computations . Hence if a word is accepted by the Turing machine T , it will be accepted in one of these computations. The next problem is the production of the sequences of integers. We shall label these sequences N 0 , N 1 , N 2 , . . . , Ni , . . . We begin with N 0 = 0 and simply count in base n + 1. Thus the sequences are The sequence following is and the sequence following is (1 , 3 , 4 , n , n , n ) The subroutine in whic… \end{verbatim} ``` </details>
831. ph-afc0092aaa9fa52484a2 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md ### Plain (markdown context) tations . Hence if a word is accepted by the Turing machine T , it will be accepted in one of these computations. The next problem is the production of the sequences of integers. We shall label these sequences N 0 , N 1 , N 2 , . . . , Ni , . . . We begin with N 0 = 0 and simply count in base n + 1. Thus the sequences are The sequence following is and the sequence following is (1 , 3 , 4 , n , n , n ) The subroutine in which a Turing machine changes the number Nk to Nk + 1 is st… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{tations . Hence if a word is accepted by the Turing machine T , it will be accepted in one of these computations. The next problem is the production of the sequences of integers. We shall label these sequences N 0 , N 1 , N 2 , . . . , Ni , . . . We begin with N 0 = 0 and simply…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md:offset=9594 \begin{verbatim} tations . Hence if a word is accepted by the Turing machine T , it will be accepted in one of these computations. The next problem is the production of the sequences of integers. We shall label these sequences N 0 , N 1 , N 2 , . . . , Ni , . . . We begin with N 0 = 0 and simply count in base n + 1. Thus the sequences are The sequence following is and the sequence following is (1 , 3 , 4 , n , n , n ) The subroutine in which a Turing machine changes the number Nk to Nk + 1 is st… \end{verbatim} ``` </details>
832. ph-66aa21db040ba9df9148 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md ### Plain (markdown context) nce we mark it with a ′ , so that we proceed each time to the first unmarked number in the sequence, select it, mark it, and then return to the marked letter with the information needed to select the proper path for the Turing machine to take, given the state and the letter being read. For example suppose the Turing machine is in state s , reads letter a , and finds that j is the number selected, it then proceeds with θ j ( s , a ) to supply the rule to use. As an illustration suppose we have We change b to bs 4 so we have We then move… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{nce we mark it with a ′ , so that we proceed each time to the first unmarked number in the sequence, select it, mark it, and then return to the marked letter with the information needed to select the proper path for the Turing machine to take, given the state and the letter bein…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md:offset=10890 \begin{verbatim} nce we mark it with a ′ , so that we proceed each time to the first unmarked number in the sequence, select it, mark it, and then return to the marked letter with the information needed to select the proper path for the Turing machine to take, given the state and the letter being read. For example suppose the Turing machine is in state s , reads letter a , and finds that j is the number selected, it then proceeds with θ j ( s , a ) to supply the rule to use. As an illustration suppose we have We change b to bs 4 so we have We then move… \end{verbatim} ``` </details>
833. ph-b82704bc148cd7b4c180 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md ### Plain (markdown context) first unmarked number in the sequence, select it, mark it, and then return to the marked letter with the information needed to select the proper path for the Turing machine to take, given the state and the letter being read. For example suppose the Turing machine is in state s , reads letter a , and finds that j is the number selected, it then proceeds with θ j ( s , a ) to supply the rule to use. As an illustration suppose we have We change b to bs 4 so we have We then move to 1, the first unmarked integer and mark it, so we have We change b to bs 4 so we have We then move to 1, the first unmarked integer and mark it, so we have We change b to bs 4 so we have We then move to 1, the first unmarked integer and mark it, so we have where the subscript of the state is the number selected. We then return to b… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{the information needed to select the proper path for the Turing machine to take, given the state and the letter being read. For example suppose the Turing machine is in state s , reads letter a , and finds that j is the number selected, it then proceeds with θ j ( s , a ) to sup…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md:offset=11053 \begin{verbatim} the information needed to select the proper path for the Turing machine to take, given the state and the letter being read. For example suppose the Turing machine is in state s , reads letter a , and finds that j is the number selected, it then proceeds with θ j ( s , a ) to supply the rule to use. As an illustration suppose we have We change b to bs 4 so we have We then move to 1, the first unmarked integer and mark it, so we have where the subscript of the state is the number selected. We then return to b… \end{verbatim} ``` </details>
835. ph-22715ccf687c2c6d64b3 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md ### Plain (markdown context) inds that j is the number selected, it then proceeds with θ j ( s , a ) to supply the rule to use. As an illustration suppose we have We change b to bs 4 so we have We then move to 1, the first unmarked integer and mark it, so we have where the subscript of the state is the number selected. We then return to bs 4 where we have θ ∗ ( q 1 , bs 4 ) = θ 1 ( s 4 , b ). The instructions could be as follows Informally we state the procedure for testing a word for acceptance by a Turing… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{inds that j is the number selected, it then proceeds with θ j ( s , a ) to supply the rule to use. As an illustration suppose we have We change b to bs 4 so we have We then move to 1, the first unmarked integer and mark i…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md:offset=11254 \begin{verbatim} inds that j is the number selected, it then proceeds with θ j ( s , a ) to supply the rule to use. As an illustration suppose we have We change b to bs 4 so we have We then move to 1, the first unmarked integer and mark it, so we have where the subscript of the state is the number selected. We then return to bs 4 where we have θ ∗ ( q 1 , bs 4 ) = θ 1 ( s 4 , b ). The instructions could be as follows Informally we state the procedure for testing a word for acceptance by a Turing… \end{verbatim} ``` </details>
836. ph-00e5e2c1bd8e36f6c841 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md ### Plain (markdown context) > We then move to 1, the first unmarked integer and mark it, so we have where the subscript of the state is the number selected. We then return to bs 4 where we have θ ∗ ( q 1 , bs 4 ) = θ 1 ( s 4 , b ). The instructions could be as follows Informally we state the procedure for testing a word for acceptance by a Turing machine as follows: First, given the word, duplicate the word and follow it by the first sequence so that we have Perform the process above for testing the second copy of the word w following… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{> We then move to 1, the first unmarked integer and mark it, so we have where the subscript of the state is the number selected. We then return to bs 4 where we have θ ∗ ( q 1 , bs 4 ) = θ 1 ( s 4 , b ). The instructions could be as follows where the subscript of the state is the number selected. We then return to bs 4 where we have θ ∗ ( q 1 , bs 4 ) = θ 1 ( s 4 , b ). The instructions could be as follows Informally we state the procedure for testing a word for acceptance by a Turing machine as follows: First, given the word, duplicate the word and follow it by the first sequence so that we have Perform the process above for testing the second copy of the word w following… \end{verbatim} ```
837. ph-81f90a553d7188c2d9cf — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md ### Plain (markdown context) y a Turing machine M ′′ . Hence L is Turing decidable and, by Theorem 5.4, its complement L ′ is also Turing decidable. /square Before proceeding further we need to show that every Turing machine with alphabet A = { a , b } can be uniquely described by a string of a s and b s. It is obvious that a Turing machine is uniquely determined by the set of rules for the machine. We shall show this for the set of states S = { s 1 , s 2 , s 3 , . . . , sn } . It may be recalled that a rule has the form where the first and third components are states, the second and fourth c… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{y a Turing machine M ′′ . Hence L is Turing decidable and, by Theorem 5.4, its complement L ′ is also Turing decidable. /square Before proceeding further we need to show that every Turing machine with alphabet A = \{ a , b \} can be uniquely described by a string of a s and b s. I…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md:offset=17484 \begin{verbatim} y a Turing machine M ′′ . Hence L is Turing decidable and, by Theorem 5.4, its complement L ′ is also Turing decidable. /square Before proceeding further we need to show that every Turing machine with alphabet A = { a , b } can be uniquely described by a string of a s and b s. It is obvious that a Turing machine is uniquely determined by the set of rules for the machine. We shall show this for the set of states S = { s 1 , s 2 , s 3 , . . . , sn } . It may be recalled that a rule has the form where the first and third components are states, the second and fourth c… \end{verbatim} ``` </details>
838. ph-6ec5ce3656b0ca29f8ab — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md ### Plain (markdown context) hm that would satisfy the halting problem is the algorithm which describes MM 2 and it does not exist. Theorem 5.8 Given a Turing machine T and an input string w , there is no algorithm which will determine whether the Turing machine T , given the input string w , will reach the halt state. ## Exercises - (1) Show that a finite set is Turing decidable. - (2) Find the string representing the rule ( s 5 , /Delta1 , s 2 , a , R ) . - (3) Find c ( M ) where M is the machine defined by the rules - (4) Find c ( M ) where M is the machine defined by the rules - (4) Find c ( M ) where M is the machine defined by the rules - (4) Find c ( M ) where M is the machine defined by the rules - (5) Find the rule that corresponds to the string aaabababbaa . - (6) Find … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{t exist. Theorem 5.8 Given a Turing machine T and an input string w , there is no algorithm which will determine whether the Turing machine T , given the input string w , will reach the halt state. \#\# Exercises - (1) Show that a finite set is Turing decidable. - (2) Find the str…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md:offset=24021 \begin{verbatim} t exist. Theorem 5.8 Given a Turing machine T and an input string w , there is no algorithm which will determine whether the Turing machine T , given the input string w , will reach the halt state. ## Exercises - (1) Show that a finite set is Turing decidable. - (2) Find the string representing the rule ( s 5 , /Delta1 , s 2 , a , R ) . - (3) Find c ( M ) where M is the machine defined by the rules - (4) Find c ( M ) where M is the machine defined by the rules - (5) Find the rule that corresponds to the string aaabababbaa . - (6) Find … \end{verbatim} ``` </details>
840. ph-58cf2144c9f532660bec — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md ### Plain (markdown context) abaaaabbbbabbbaabbbbaabbbabbbaababaaababbbaaabbb . - (11) Find the Turing machine and input that correspond to the string abaaaabbbbabbbaabaabaabbbabbbaaabaaabbbaaababaa ababababaabb . - (12) Devise a method of coding that allows the use of A and B as well as a and b by allowing strings of length 3 to represent input and output symbols. - (13) Use the coding in the previous problem to find the string corresponding to ( s 1 , a , s 3 , A , R ). - (14) Find the string that represents the machine together with input ababaab . - (15) Find the string that represents th… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{abaaaabbbbabbbaabbbbaabbbabbbaababaaababbbaaabbb . - (11) Find the Turing machine and input that correspond to the string abaaaabbbbabbbaabaabaabbbabbbaaabaaabbbaaababaa ababababaabb . - (12) Devise a method of coding that allows the use of A and B as well as a and b by allowing…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md:offset=25106 \begin{verbatim} abaaaabbbbabbbaabbbbaabbbabbbaababaaababbbaaabbb . - (11) Find the Turing machine and input that correspond to the string abaaaabbbbabbbaabaabaabbbabbbaaabaaabbbaaababaa ababababaabb . - (12) Devise a method of coding that allows the use of A and B as well as a and b by allowing strings of length 3 to represent input and output symbols. - (13) Use the coding in the previous problem to find the string corresponding to ( s 1 , a , s 3 , A , R ). - (14) Find the string that represents the machine together with input ababaab . - (15) Find the string that represents th… \end{verbatim} ``` </details>
841. ph-6602481c178d6e6af502 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md ### Plain (markdown context) e string abaaaabbbbabbbaabaabaabbbabbbaaabaaabbbaaababaa ababababaabb . - (12) Devise a method of coding that allows the use of A and B as well as a and b by allowing strings of length 3 to represent input and output symbols. - (13) Use the coding in the previous problem to find the string corresponding to ( s 1 , a , s 3 , A , R ). - (14) Find the string that represents the machine together with input ababaab . - (15) Find the string that represents the machine together with input babbab . - (16) Let L be a language. Prove that one an… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{e string abaaaabbbbabbbaabaabaabbbabbbaaabaaabbbaaababaa ababababaabb . - (12) Devise a method of coding that allows the use of A and B as well as a and b by allowing strings of length 3 to represent input and output symbols. - (13) Use the coding in the previous problem to find…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md:offset=25219 \begin{verbatim} e string abaaaabbbbabbbaabaabaabbbabbbaaabaaabbbaaababaa ababababaabb . - (12) Devise a method of coding that allows the use of A and B as well as a and b by allowing strings of length 3 to represent input and output symbols. - (13) Use the coding in the previous problem to find the string corresponding to ( s 1 , a , s 3 , A , R ). - (14) Find the string that represents the machine together with input ababaab . - (15) Find the string that represents the machine together with input babbab . - (16) Let L be a language. Prove that one an… \end{verbatim} ``` </details>
842. ph-008c6782ca2fe3bc6733 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md ### Plain (markdown context) de, ( /star $ , $) in P 2 . It is obvious that only ( L ( u 0 ) , L ( u 0 ) /star ) can begin a match in P 2 , since it is the only pair where we do not have one word in the pair beginning with a star while the other does not. It is also obvious that the only pair that can end a pair in P 2 , is ( /star $ , $), since it is the only word where the last symbols match, that is we do not have one ending in a star while the other does not. It is also obvious that if there exist a sequence of pairs in P 1, such that w = u 0 ui 1 ui 2 · · · uim = v 0 v i 1 v i 2 · · · v … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{de, ( /star $ , $) in P 2 . It is obvious that only ( L ( u 0 ) , L ( u 0 ) /star ) can begin a match in P 2 , since it is the only pair where we do not have one word in the pair beginning with a star while the other does not. It is also obvious that the only pair that can end a…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md:offset=29216 \begin{verbatim} de, ( /star $ , $) in P 2 . It is obvious that only ( L ( u 0 ) , L ( u 0 ) /star ) can begin a match in P 2 , since it is the only pair where we do not have one word in the pair beginning with a star while the other does not. It is also obvious that the only pair that can end a pair in P 2 , is ( /star $ , $), since it is the only word where the last symbols match, that is we do not have one ending in a star while the other does not. It is also obvious that if there exist a sequence of pairs in P 1, such that w = u 0 ui 1 ui 2 · · · uim = v 0 v i 1 v i 2 · · · v … \end{verbatim} ``` </details>
843. ph-39917851cd925993974a — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md ### Plain (markdown context) only pair where we do not have one word in the pair beginning with a star while the other does not. It is also obvious that the only pair that can end a pair in P 2 , is ( /star $ , $), since it is the only word where the last symbols match, that is we do not have one ending in a star while the other does not. It is also obvious that if there exist a sequence of pairs in P 1, such that w = u 0 ui 1 ui 2 · · · uim = v 0 v i 1 v i 2 · · · v im . Then the sequence produces a match in P 2. The words and in P 1, such that w = u 0 ui 1 ui 2 · · · uim = v 0 v i 1 v i 2 · · · v im . Then the sequence produces a match in P 2. The words and in P 1, such that w = u 0 ui 1 ui 2 · · · uim = v 0 v i 1 v i 2 · · · v im . Then the sequence produces a match in P 2. The words and in P 1, such that w = u 0 ui 1 ui 2 · · · uim = v 0 v i 1 v i 2 · · · v im . Then the sequence produces a match in P 2. The words and in P 1, such that w = u 0 ui 1 ui 2 · · · uim = v 0 v i 1 v i 2 · · · v im . Then the sequence produces a match in P 2. The words and in P 2 differ from the words u 0 ui 1 ui 2 · · · … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{It is also obvious that the only pair that can end a pair in P 2 , is ( /star $ , $), since it is the only word where the last symbols match, that is we do not have one ending in a star while the other does not. It is also obvious that if there exist a sequence of pairs in P 1, such that w = u 0 ui 1 ui 2 · · · uim = v 0 v i 1 v i 2 · · · v im . Then the sequence produces a match in P 2. The words and in P 2 differ from the words u 0 ui 1 ui 2 · · · … \end{verbatim} ```
846. ph-a59b5110118de3413855 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md ### Plain (markdown context) ly pair that can end a pair in P 2 , is ( /star $ , $), since it is the only word where the last symbols match, that is we do not have one ending in a star while the other does not. It is also obvious that if there exist a sequence of pairs in P 1, such that w = u 0 ui 1 ui 2 · · · uim = v 0 v i 1 v i 2 · · · v im . Then the sequence produces a match in P 2. The words and in P 2 differ from the words u 0 ui 1 ui 2 · · · uim and v 0 v i 1 v i 2 · · · … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ly pair that can end a pair in P 2 , is ( /star $ , $), since it is the only word where the last symbols match, that is we do not have one ending in a star while the other does not. It is also obvious that if there exist a sequence of pairs in P 1, s…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md:offset=29473 \begin{verbatim} ly pair that can end a pair in P 2 , is ( /star $ , $), since it is the only word where the last symbols match, that is we do not have one ending in a star while the other does not. It is also obvious that if there exist a sequence of pairs in P 1, such that w = u 0 ui 1 ui 2 · · · uim = v 0 v i 1 v i 2 · · · v im . Then the sequence produces a match in P 2. The words and in P 2 differ from the words u 0 ui 1 ui 2 · · · uim and v 0 v i 1 v i 2 · · · … \end{verbatim} ``` </details>
847. ph-f37af08f5667bbb5b906 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md ### Plain (markdown context) -decoded --> in P 2 differ from the words u 0 ui 1 ui 2 · · · uim and v 0 v i 1 v i 2 · · · v im respectively in P 1 in the fact that that they have stars between the letters and end in $. Hence, since a match in the modified Post's correspondence system has a corresponding match in Post's correspondence system, if Post's Correspondence Problem is decidable, then the modified Post's Correspondence Problem is decidable. /square ## Example 5.2 Using the previous modified Post's correspondence with match abcded f , we have with match ∗ a … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{-decoded --> in P 2 differ from the words u 0 ui 1 ui 2 · · · uim and v 0 v i 1 v i 2 · · · v im respectively in P 1 in the fact that that they have stars between the letters and end in $. Hence, since a match in the modified Post's correspondence system has a corresponding matc…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md:offset=29989 \begin{verbatim} -decoded --> in P 2 differ from the words u 0 ui 1 ui 2 · · · uim and v 0 v i 1 v i 2 · · · v im respectively in P 1 in the fact that that they have stars between the letters and end in $. Hence, since a match in the modified Post's correspondence system has a corresponding match in Post's correspondence system, if Post's Correspondence Problem is decidable, then the modified Post's Correspondence Problem is decidable. /square ## Example 5.2 Using the previous modified Post's correspondence with match abcded f , we have with match ∗ a … \end{verbatim} ``` </details>
848. ph-a72195120f554ca9f04b — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md ### Plain (markdown context) · uim and v 0 v i 1 v i 2 · · · v im respectively in P 1 in the fact that that they have stars between the letters and end in $. Hence, since a match in the modified Post's correspondence system has a corresponding match in Post's correspondence system, if Post's Correspondence Problem is decidable, then the modified Post's Correspondence Problem is decidable. /square ## Example 5.2 Using the previous modified Post's correspondence with match abcded f , we have with match ∗ a ∗ b ∗ c ∗ d ∗ e ∗ d ∗ f /star $. Theorem 5.9 Post's Correspo… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{· uim and v 0 v i 1 v i 2 · · · v im respectively in P 1 in the fact that that they have stars between the letters and end in $. Hence, since a match in the modified Post's correspondence system has a corresponding match in Post's correspondence system, if Post's Correspondence …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md:offset=30050 \begin{verbatim} · uim and v 0 v i 1 v i 2 · · · v im respectively in P 1 in the fact that that they have stars between the letters and end in $. Hence, since a match in the modified Post's correspondence system has a corresponding match in Post's correspondence system, if Post's Correspondence Problem is decidable, then the modified Post's Correspondence Problem is decidable. /square ## Example 5.2 Using the previous modified Post's correspondence with match abcded f , we have with match ∗ a ∗ b ∗ c ∗ d ∗ e ∗ d ∗ f /star $. Theorem 5.9 Post's Correspo… \end{verbatim} ``` </details>
849. ph-ceaa222a3b3bbd124684 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md ### Plain (markdown context) Correspondence Problem is undecidable by showing that the modified Post's Correspondence Problem is undecidable. We do this by showing that if the modified Post's Correspondence Problem is decidable, then L 0 (see previous section) is acceptable, which means that it is decidable if a Turing machine accepts a given word. Assuming the sequence for a given Turing machine and word, we construct a modified Post's correspondence system that has a match if and only if M accepts w . Intuitively assume describes the process used by the Turing machine to read w , where each… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{Correspondence Problem is undecidable by showing that the modified Post's Correspondence Problem is undecidable. We do this by showing that if the modified Post's Correspondence Problem is decidable, then L 0 (see previous section) is acceptable, which means that it is decidable…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md:offset=30713 \begin{verbatim} Correspondence Problem is undecidable by showing that the modified Post's Correspondence Problem is undecidable. We do this by showing that if the modified Post's Correspondence Problem is decidable, then L 0 (see previous section) is acceptable, which means that it is decidable if a Turing machine accepts a given word. Assuming the sequence for a given Turing machine and word, we construct a modified Post's correspondence system that has a match if and only if M accepts w . Intuitively assume describes the process used by the Turing machine to read w , where each… \end{verbatim} ``` </details>
850. ph-ebc74e9dabd09cf89cdd — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md ### Plain (markdown context) 90b61b04fb37d87f08efbf2c12024a7dd442.png) Since this is a modified Post's correspondence system, we can require that we begin with this pair. For each X in /Gamma1 we have ![Image](./AutomataTheory_chapter_2.3_artifacts/image_000004_03250beda82ca304c7e4564bdee577c2eec62f0611e205b5a01177a95696b5d5.png) Wenext use the following pairs to guide us in selecting the next string in our match: For each state s , which is not a final state and each state s ′ , and symbols X , Y , and Z in /Gamma1 , ![Image](./AutomataTheory_chapter_2.3_artifacts/image_000005_541c7f305d6e24… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{90b61b04fb37d87f08efbf2c12024a7dd442.png) Since this is a modified Post's correspondence system, we can require that we begin with this pair. For each X in /Gamma1 we have ![Image](./AutomataTheory\_chapter\_2.3\_artifacts/image\_000004\_03250beda82ca304c7e4564bdee577c2eec62f0611e205…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md:offset=32474 \begin{verbatim} 90b61b04fb37d87f08efbf2c12024a7dd442.png) Since this is a modified Post's correspondence system, we can require that we begin with this pair. For each X in /Gamma1 we have ![Image](./AutomataTheory_chapter_2.3_artifacts/image_000004_03250beda82ca304c7e4564bdee577c2eec62f0611e205b5a01177a95696b5d5.png) Wenext use the following pairs to guide us in selecting the next string in our match: For each state s , which is not a final state and each state s ′ , and symbols X , Y , and Z in /Gamma1 , ![Image](./AutomataTheory_chapter_2.3_artifacts/image_000005_541c7f305d6e24… \end{verbatim} ``` </details>
851. ph-0a92af6ec9a18a4595b7 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md ### Plain (markdown context) .png) Wenext use the following pairs to guide us in selecting the next string in our match: For each state s , which is not a final state and each state s ′ , and symbols X , Y , and Z in /Gamma1 , ![Image](./AutomataTheory_chapter_2.3_artifacts/image_000005_541c7f305d6e24ef2385b7ddb0c359c22b5789564aafc4752fd95dd99ad7d81c.png) We shall call these the pairs generated by δ. In trying to get our match this set guides us to the next string. For example if we have in our match and one of the pairs above is ( si 0 , 1 s j ), we will want the… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{.png) Wenext use the following pairs to guide us in selecting the next string in our match: For each state s , which is not a final state and each state s ′ , and symbols X , Y , and Z in /Gamma1 , ![Image](./AutomataTheory\_chapter\_2.3\_artifacts/imag…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md:offset=32773 \begin{verbatim} .png) Wenext use the following pairs to guide us in selecting the next string in our match: For each state s , which is not a final state and each state s ′ , and symbols X , Y , and Z in /Gamma1 , ![Image](./AutomataTheory_chapter_2.3_artifacts/image_000005_541c7f305d6e24ef2385b7ddb0c359c22b5789564aafc4752fd95dd99ad7d81c.png) We shall call these the pairs generated by δ. In trying to get our match this set guides us to the next string. For example if we have in our match and one of the pairs above is ( si 0 , 1 s j ), we will want the… \end{verbatim} ``` </details>
852. ph-d0dbf1ac8988dc39629f — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md ### Plain (markdown context) 541c7f305d6e24ef2385b7ddb0c359c22b5789564aafc4752fd95dd99ad7d81c.png) We shall call these the pairs generated by δ. In trying to get our match this set guides us to the next string. For example if we have in our match and one of the pairs above is ( si 0 , 1 s j ), we will want the next string to be #111 s j 11#. Note however that the two 1s at the beginning and end of the string are not affected by the pair above. Hence we need pairs (# , #) and (1 , 1) to get More precisely we would use 1 1 , 1 1 , si 0 1 , 1 1 , 1 1 , # # . Hence we… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{541c7f305d6e24ef2385b7ddb0c359c22b5789564aafc4752fd95dd99ad7d81c.png) We shall call these the pairs generated by δ. In trying to get our match this set guides us to the next string. For example if we have in our match and one of the pairs above is ( …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md:offset=33065 \begin{verbatim} 541c7f305d6e24ef2385b7ddb0c359c22b5789564aafc4752fd95dd99ad7d81c.png) We shall call these the pairs generated by δ. In trying to get our match this set guides us to the next string. For example if we have in our match and one of the pairs above is ( si 0 , 1 s j ), we will want the next string to be #111 s j 11#. Note however that the two 1s at the beginning and end of the string are not affected by the pair above. Hence we need pairs (# , #) and (1 , 1) to get More precisely we would use 1 1 , 1 1 , si 0 1 , 1 1 , 1 1 , # # . Hence we… \end{verbatim} ``` </details>
853. ph-7fa3b0d388ad1018f968 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md ### Plain (markdown context) In trying to get our match this set guides us to the next string. For example if we have in our match and one of the pairs above is ( si 0 , 1 s j ), we will want the next string to be #111 s j 11#. Note however that the two 1s at the beginning and end of the string are not affected by the pair above. Hence we need pairs (# , #) and (1 , 1) to get More precisely we would use 1 1 , 1 1 , si 0 1 , 1 1 , 1 1 , # # . Hence we need pairs Obviously if we never get to an acceptance sta… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{In trying to get our match this set guides us to the next string. For example if we have in our match and one of the pairs above is ( si 0 , 1 s j ), we will want the next string to be \#111 s j 11\#. Note however that the two 1s at the beginning and e…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md:offset=33183 \begin{verbatim} In trying to get our match this set guides us to the next string. For example if we have in our match and one of the pairs above is ( si 0 , 1 s j ), we will want the next string to be #111 s j 11#. Note however that the two 1s at the beginning and end of the string are not affected by the pair above. Hence we need pairs (# , #) and (1 , 1) to get More precisely we would use 1 1 , 1 1 , si 0 1 , 1 1 , 1 1 , # # . Hence we need pairs Obviously if we never get to an acceptance sta… \end{verbatim} ``` </details>
854. ph-c3e54de1ced18fcf493f — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md ### Plain (markdown context) s set guides us to the next string. For example if we have in our match and one of the pairs above is ( si 0 , 1 s j ), we will want the next string to be #111 s j 11#. Note however that the two 1s at the beginning and end of the string are not affected by the pair above. Hence we need pairs (# , #) and (1 , 1) to get More precisely we would use 1 1 , 1 1 , si 0 1 , 1 1 , 1 1 , # # . Hence we need pairs Obviously if we never get to an acceptance state (and hence a final state) w… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{s set guides us to the next string. For example if we have in our match and one of the pairs above is ( si 0 , 1 s j ), we will want the next string to be \#111 s j 11\#. Note however that the two 1s at the beginning and end of the string are not affec…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md:offset=33213 \begin{verbatim} s set guides us to the next string. For example if we have in our match and one of the pairs above is ( si 0 , 1 s j ), we will want the next string to be #111 s j 11#. Note however that the two 1s at the beginning and end of the string are not affected by the pair above. Hence we need pairs (# , #) and (1 , 1) to get More precisely we would use 1 1 , 1 1 , si 0 1 , 1 1 , 1 1 , # # . Hence we need pairs Obviously if we never get to an acceptance state (and hence a final state) w… \end{verbatim} ``` </details>
855. ph-573dd2559897acfb0187 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md ### Plain (markdown context) Hence we need pairs (# , #) and (1 , 1) to get More precisely we would use 1 1 , 1 1 , si 0 1 , 1 1 , 1 1 , # # . Hence we need pairs Obviously if we never get to an acceptance state (and hence a final state) we will never have a match since there will always be an overlap at the bottom. We thus need rules to get a match if we reach a halt state h . We use the following pairs to get rid of the overlap. The last term gets rid of the overlap sm when all of the other symbols have b… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{Hence we need pairs (\# , \#) and (1 , 1) to get More precisely we would use 1 1 , 1 1 , si 0 1 , 1 1 , 1 1 , \# \# . Hence we need pairs Obviously if we never get to an acceptance state (and henc…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md:offset=33517 \begin{verbatim} Hence we need pairs (# , #) and (1 , 1) to get More precisely we would use 1 1 , 1 1 , si 0 1 , 1 1 , 1 1 , # # . Hence we need pairs Obviously if we never get to an acceptance state (and hence a final state) we will never have a match since there will always be an overlap at the bottom. We thus need rules to get a match if we reach a halt state h . We use the following pairs to get rid of the overlap. The last term gets rid of the overlap sm when all of the other symbols have b… \end{verbatim} ``` </details>
856. ph-51e1e15ea6f58aa92231 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md ### Plain (markdown context) . Hence we need pairs Obviously if we never get to an acceptance state (and hence a final state) we will never have a match since there will always be an overlap at the bottom. We thus need rules to get a match if we reach a halt state h . We use the following pairs to get rid of the overlap. The last term gets rid of the overlap sm when all of the other symbols have been eliminated. Thus if we reached as follows Obviously if we never get to an acceptance state (and hence a final state) we will never have a match since there will always be an overlap at the bottom. We thus need rules to get a match if we reac…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md:offset=33660 \begin{verbatim} . Hence we need pairs Obviously if we never get to an acceptance state (and hence a final state) we will never have a match since there will always be an overlap at the bottom. We thus need rules to get a match if we reach a halt state h . We use the following pairs to get rid of the overlap. The last term gets rid of the overlap sm when all of the other symbols have been eliminated. Thus if we reached as follows Obviously if we never get to an acceptance state (and hence a final state) we will never have a match since there will always be an overlap at the bottom. We thus need rules to get a match if we reach a halt state h . We use the following pairs to get rid of the overlap. The last term gets rid of the overlap sm when all of the other symbols have been eliminated. Thus if we reached as follows Formally… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{rmula-not-decoded --> Obviously if we never get to an acceptance state (and hence a final state) we will never have a match since there will always be an overlap at the bottom. We thus need rules to get a match if we reach a halt state h . We use the…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md:offset=33690 \begin{verbatim} rmula-not-decoded --> Obviously if we never get to an acceptance state (and hence a final state) we will never have a match since there will always be an overlap at the bottom. We thus need rules to get a match if we reach a halt state h . We use the following pairs to get rid of the overlap. The last term gets rid of the overlap sm when all of the other symbols have been eliminated. Thus if we reached as follows Formally… \end{verbatim} ``` </details>
858. ph-42412fc6dc638703c3b7 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md ### Plain (markdown context) coded --> Obviously if we never get to an acceptance state (and hence a final state) we will never have a match since there will always be an overlap at the bottom. We thus need rules to get a match if we reach a halt state h . We use the following pairs to get rid of the overlap. The last term gets rid of the overlap sm when all of the other symbols have been eliminated. Thus if we reached as follows Formally we give a proof of the theorem. If we ha… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{coded --> Obviously if we never get to an acceptance state (and hence a final state) we will never have a match since there will always be an overlap at the bottom. We thus need rules to get a match if we reach a halt state h . We use the following pairs to get rid of the overla…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md:offset=33732 \begin{verbatim} coded --> Obviously if we never get to an acceptance state (and hence a final state) we will never have a match since there will always be an overlap at the bottom. We thus need rules to get a match if we reach a halt state h . We use the following pairs to get rid of the overlap. The last term gets rid of the overlap sm when all of the other symbols have been eliminated. Thus if we reached as follows Formally we give a proof of the theorem. If we ha… \end{verbatim} ``` </details>
859. ph-de7f57d1d1ab45ca5769 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md ### Plain (markdown context) er get to an acceptance state (and hence a final state) we will never have a match since there will always be an overlap at the bottom. We thus need rules to get a match if we reach a halt state h . We use the following pairs to get rid of the overlap. The last term gets rid of the overlap sm when all of the other symbols have been eliminated. Thus if we reached as follows Formally we give a proof of the theorem. If we have a valid set of sequences d… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{er get to an acceptance state (and hence a final state) we will never have a match since there will always be an overlap at the bottom. We thus need rules to get a match if we reach a halt state h . We use the following pairs to get rid of the overlap. The last term gets rid of the overlap sm when all of the other symbols have been eliminated. Thus if we reached as follows Formally we give a proof of the theorem. If we have a valid set of sequences d… \end{verbatim} ```
860. ph-40b512b335ea6a7394ec — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md ### Plain (markdown context) rid of the overlap. The last term gets rid of the overlap sm when all of the other symbols have been eliminated. Thus if we reached as follows Formally we give a proof of the theorem. If we have a valid set of sequences describing the acceptance of w by M , using induction on the number of computations we show that there is a partial solution For n = 0, we have ![Image](./AutomataTheory_chapter_2.3_artifacts/image_000006_… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{rid of the overlap. The last term gets rid of the overlap sm when all of the other symbols have been eliminated. Thus if we reached as follows The last term gets rid of the overlap sm when all of the other symbols have been eliminated. Thus if we reached as follows Formally we give a proof of the theorem. If we have a valid set of sequences describing the acceptance of w by M , using induction on the number of computations we show that there is a partial solution For n = 0, we have ![Image](./AutomataTheory_chapter_2.3_artifacts/image_000006_… \end{verbatim} ```
861. ph-a6eb8ee3749ca5a59af4 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md ### Plain (markdown context) ed --> Formally we give a proof of the theorem. If we have a valid set of sequences describing the acceptance of w by M , using induction on the number of computations we show that there is a partial solution For n = 0, we have ![Image](./AutomataTheory_chapter_2.3_artifacts/image_000006_5b727be33d51dd2f49e9af007eaafec411a761064191798997f44cbbd4f68413.png) Assuming the statement is true for k , and sk is not the halt state we have The next pairs are chosen so the string at the top forms # α k sk β k # using… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ed --> Formally we give a proof of the theorem. If we have a valid set of sequences describing the acceptance of w by M , using induction on the number of computations we show that there is a partial solution For n = 0, w…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md:offset=34254 \begin{verbatim} ed --> Formally we give a proof of the theorem. If we have a valid set of sequences describing the acceptance of w by M , using induction on the number of computations we show that there is a partial solution For n = 0, we have ![Image](./AutomataTheory_chapter_2.3_artifacts/image_000006_5b727be33d51dd2f49e9af007eaafec411a761064191798997f44cbbd4f68413.png) Assuming the statement is true for k , and sk is not the halt state we have The next pairs are chosen so the string at the top forms # α k sk β k # using… \end{verbatim} ``` </details>
862. ph-305a33943cb0746b754b — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md ### Plain (markdown context) t there is a partial solution For n = 0, we have ![Image](./AutomataTheory_chapter_2.3_artifacts/image_000006_5b727be33d51dd2f49e9af007eaafec411a761064191798997f44cbbd4f68413.png) Assuming the statement is true for k , and sk is not the halt state we have The next pairs are chosen so the string at the top forms # α k sk β k # using the rules above. There is at most one pair in the pairs generated by δ that works. We can thus form and we have extended a new partial solution. Since rules generated by δ apply … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{t there is a partial solution For n = 0, we have ![Image](./AutomataTheory\_chapter\_2.3\_artifacts/image\_000006\_5b727be33d51dd2f49e9af007eaafec411a761064191798997f44cbbd4f68413.png) Assuming the statement is true for k , and sk is not the halt state we…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md:offset=34464 \begin{verbatim} t there is a partial solution For n = 0, we have ![Image](./AutomataTheory_chapter_2.3_artifacts/image_000006_5b727be33d51dd2f49e9af007eaafec411a761064191798997f44cbbd4f68413.png) Assuming the statement is true for k , and sk is not the halt state we have The next pairs are chosen so the string at the top forms # α k sk β k # using the rules above. There is at most one pair in the pairs generated by δ that works. We can thus form and we have extended a new partial solution. Since rules generated by δ apply … \end{verbatim} ``` </details>
863. ph-720ef664c4272ac7c5ff — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md ### Plain (markdown context) is not a rule and there can be no match. If, for some k , β k is a halt state, then as mentioned above, there are rules to make the upper and lower lists agree. As already mentioned, if we do not reach the halt state, we cannot have a match. If we do reach the halt state, we can produce a match. Hence if the modified Post's Correspondence Problem is decidable, L 0 is decidable. Therefore the modified Post's Correspondence Problem is undecidable. /square ## Example 5.3 Let the Turing Machine and word 0110 where with corresponding pairs In addition we have pairs <!-… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{is not a rule and there can be no match. If, for some k , β k is a halt state, then as mentioned above, there are rules to make the upper and lower lists agree. As already mentioned, if we do not reach the halt state, we cannot have a match. If we do reach the halt state, we can…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md:offset=35300 \begin{verbatim} is not a rule and there can be no match. If, for some k , β k is a halt state, then as mentioned above, there are rules to make the upper and lower lists agree. As already mentioned, if we do not reach the halt state, we cannot have a match. If we do reach the halt state, we can produce a match. Hence if the modified Post's Correspondence Problem is decidable, L 0 is decidable. Therefore the modified Post's Correspondence Problem is undecidable. /square ## Example 5.3 Let the Turing Machine and word 0110 where with corresponding pairs In addition we have pairs <!-… \end{verbatim} ``` </details>
864. ph-317136c3cb260086e15e — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md ### Plain (markdown context) , there are rules to make the upper and lower lists agree. As already mentioned, if we do not reach the halt state, we cannot have a match. If we do reach the halt state, we can produce a match. Hence if the modified Post's Correspondence Problem is decidable, L 0 is decidable. Therefore the modified Post's Correspondence Problem is undecidable. /square ## Example 5.3 Let the Turing Machine and word 0110 where with corresponding pairs In addition we have pairs Our first pair is … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{, there are rules to make the upper and lower lists agree. As already mentioned, if we do not reach the halt state, we cannot have a match. If we do reach the halt state, we can produce a match. Hence if the modified Post's Correspondence Problem is decidable, L 0 is decidable. …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md:offset=35403 \begin{verbatim} , there are rules to make the upper and lower lists agree. As already mentioned, if we do not reach the halt state, we cannot have a match. If we do reach the halt state, we can produce a match. Hence if the modified Post's Correspondence Problem is decidable, L 0 is decidable. Therefore the modified Post's Correspondence Problem is undecidable. /square ## Example 5.3 Let the Turing Machine and word 0110 where with corresponding pairs In addition we have pairs Our first pair is … \end{verbatim} ``` </details>
865. ph-979f4528948031d803ff — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md ### Plain (markdown context) upper and lower lists agree. As already mentioned, if we do not reach the halt state, we cannot have a match. If we do reach the halt state, we can produce a match. Hence if the modified Post's Correspondence Problem is decidable, L 0 is decidable. Therefore the modified Post's Correspondence Problem is undecidable. /square ## Example 5.3 Let the Turing Machine and word 0110 where with corresponding pairs In addition we have pairs Our first pair is (# , # s 0 w ) which produces … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{upper and lower lists agree. As already mentioned, if we do not reach the halt state, we cannot have a match. If we do reach the halt state, we can produce a match. Hence if the modified Post's Correspondence Problem is decidable, L 0 is decidable. Therefore the modified Post's …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md:offset=35433 \begin{verbatim} upper and lower lists agree. As already mentioned, if we do not reach the halt state, we cannot have a match. If we do reach the halt state, we can produce a match. Hence if the modified Post's Correspondence Problem is decidable, L 0 is decidable. Therefore the modified Post's Correspondence Problem is undecidable. /square ## Example 5.3 Let the Turing Machine and word 0110 where with corresponding pairs In addition we have pairs Our first pair is (# , # s 0 w ) which produces … \end{verbatim} ``` </details>
866. ph-a533a22b2f0dae14eb57 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md ### Plain (markdown context) As already mentioned, if we do not reach the halt state, we cannot have a match. If we do reach the halt state, we can produce a match. Hence if the modified Post's Correspondence Problem is decidable, L 0 is decidable. Therefore the modified Post's Correspondence Problem is undecidable. /square ## Example 5.3 Let the Turing Machine and word 0110 where with corresponding pairs In addition we have pairs Our first pair is (# , # s 0 w ) which produces # # s 0010#. and word 0110 where with corresponding pairs In addition we have pairs Our first pair is (# , # s 0 w ) which produces # # s 0010#. and word 0110 where with corresponding pairs In addition we have pairs Our first pair is (# , # s 0 w ) which produces # # s 0010#. We then use (1 , 1) twice, (0 , 0), and (# , #) to get and word …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md:offset=35557 \begin{verbatim} h the halt state, we can produce a match. Hence if the modified Post's Correspondence Problem is decidable, L 0 is decidable. Therefore the modified Post's Correspondence Problem is undecidable. /square ## Example 5.3 Let the Turing Machine and word 0110 where with corresponding pairs In addition we have pairs Our first pair is (# , # s 0 w ) which produces # # s 0010#. We then use (1 , 1) twice, (0 , 0), and (# , #) to get and word 0110 where with corresponding pairs In addition we have pairs Our first pair is (# , # s 0 w ) which produces # # s 0010#. We then use (1 , 1) twice, (0 , 0), and (# , #) to get We next use ( /star, /star ) , ( s 11 , 1 s 1 ) , (1 , 1),(0 , 0), and (# , #) , to… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{Problem is decidable, L 0 is decidable. Therefore the modified Post's Correspondence Problem is undecidable. /square \#\# Example 5.3 Let the Turing Machine and word 0110 where with corresponding pairs In addition we have pairs and word 0110 where with corresponding pairs In addition we have pairs Our first pair is (# , # s 0 w ) which produces # # s 0010#. We then use (1 , 1) twice, (0 , 0), and (# , #) to get We next use ( /star, /star ) , ( s 11 , 1 s 1 ) , (1 , 1),(0 , 0), and (# , #) , to… \end{verbatim} ``` </details>
869. ph-f90685e04da0e35dfac4 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md ### Plain (markdown context) Our first pair is (# , # s 0 w ) which produces # # s 0010#. We then use (1 , 1) twice, (0 , 0), and (# , #) to get We next use ( /star, /star ) , ( s 11 , 1 s 1 ) , (1 , 1),(0 , 0), and (# , #) , to get # s # 00110# s 00110# /star /star s s 1110# 1110# /star 1 s 110#, again using ( /star, /star ) , (1 , 1) , ( s 11 , 1 s 1 ) , (0 , 0), and (# , #) we get Now using ( /star, /star ) , (1 , 1) , (1 s 10 , s 210), and (# , #) we get Our first pair is (\# , \# s 0 w ) which produces \# \# s 0010\#. We then use (1 , 1) twice, (0 , 0), and (\# , \#) to get We next use ( /star, /star ) , ( s 11 , 1 s 1 )…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md:offset=35932 \begin{verbatim} Our first pair is (# , # s 0 w ) which produces # # s 0010#. We then use (1 , 1) twice, (0 , 0), and (# , #) to get We next use ( /star, /star ) , ( s 11 , 1 s 1 ) , (1 , 1),(0 , 0), and (# , #) , to get # s # 00110# s 00110# /star /star s s 1110# 1110# /star 1 s 110#, again using ( /star, /star ) , (1 , 1) , ( s 11 , 1 s 1 ) , (0 , 0), and (# , #) we get Now using ( /star, /star ) , (1 , 1) , (1 s 10 , s 210), and (# , #) we get We then use (1 , 1) twice, (0 , 0), and (# , #) to get We next use ( /star, /star ) , ( s 11 , 1 s 1 ) , (1 , 1),(0 , 0), and (# , #) , to get # s # 00110# s 00110# /star /star s s 1110# 1110# /star 1 s 110#, again using ( /star, /star ) , (1 , 1) , ( s 11 , 1 s 1 ) , (0 , 0), and (# , #) we get Now using ( /star, /star ) , (1 , 1) , (1 s 10 , s 210), and (# , #) we get Using ( /star, /star ) , ( /star s 11 , s 2 /star 1) , (1 , 1) , (0 , 0), and (# , #) we get <!… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{s \# \# s 0010\#. We then use (1 , 1) twice, (0 , 0), and (\# , \#) to get We next use ( /star, /star ) , ( s 11 , 1 s 1 ) , (1 , 1),(0 , 0), and (\# , \#) , to get \# s \# 00110\# s 00110\# /star /star s s 1110\# 1110\# /star 1 s 110…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md:offset=36039 \begin{verbatim} s # # s 0010#. We then use (1 , 1) twice, (0 , 0), and (# , #) to get We next use ( /star, /star ) , ( s 11 , 1 s 1 ) , (1 , 1),(0 , 0), and (# , #) , to get # s # 00110# s 00110# /star /star s s 1110# 1110# /star 1 s 110#, again using ( /star, /star ) , (1 , 1) , ( s 11 , 1 s 1 ) , (0 , 0), and (# , #) we get Now using ( /star, /star ) , (1 , 1) , (1 s 10 , s 210), and (# , #) we get Using ( /star, /star ) , ( /star s 11 , s 2 /star 1) , (1 , 1) , (0 , 0), and (# , #) we get <!… \end{verbatim} ``` </details>
871. ph-714b8d1344c40d02622c — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md ### Plain (markdown context) oded --> We next use ( /star, /star ) , ( s 11 , 1 s 1 ) , (1 , 1),(0 , 0), and (# , #) , to get # s # 00110# s 00110# /star /star s s 1110# 1110# /star 1 s 110#, again using ( /star, /star ) , (1 , 1) , ( s 11 , 1 s 1 ) , (0 , 0), and (# , #) we get Now using ( /star, /star ) , (1 , 1) , (1 s 10 , s 210), and (# , #) we get Using ( /star, /star ) , ( /star s 11 , s 2 /star 1) , (1 , 1) , (0 , 0), and (# , #) we get Now using ( s 2 /star, h 0), (1 , 1) twice, (0 , 0), and (# , #) , we get We next use ( /star, /star ) , ( s 11 , 1 s 1 ) , (1 , 1),(0 , 0), and (\# , \#) , to get \# s \# 00110\# s 00110\# /star /star s s 1110\# 1110\# /star 1 s 110\#, again using ( /star, /star ) , (1 , 1) , ( s 11 , 1 s 1 ) , (0 , 0), and (\# , \#) we get …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md:offset=36163 \begin{verbatim} oded --> We next use ( /star, /star ) , ( s 11 , 1 s 1 ) , (1 , 1),(0 , 0), and (# , #) , to get # s # 00110# s 00110# /star /star s s 1110# 1110# /star 1 s 110#, again using ( /star, /star ) , (1 , 1) , ( s 11 , 1 s 1 ) , (0 , 0), and (# , #) we get Now using ( /star, /star ) , (1 , 1) , (1 s 10 , s 210), and (# , #) we get Using ( /star, /star ) , ( /star s 11 , s 2 /star 1) , (1 , 1) , (0 , 0), and (# , #) we get Now using ( s 2 /star, h 0), (1 , 1) twice, (0 , 0), and (# , #) , we get Now using ( /star, /star ) , (1 , 1) , (1 s 10 , s 210), and (# , #) we get Using ( /star, /star ) , ( /star s 11 , s 2 /star 1) , (1 , 1) , (0 , 0), and (# , #) we get Now using ( s 2 /star, h 0), (1 , 1) twice, (0 , 0), and (# , #) , we get Finally, using the pairs containing h , together with (1 , 1) , (0 , 0), and (# , #) , we … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{\# 00110\# s 00110\# /star /star s s 1110\# 1110\# /star 1 s 110\#, again using ( /star, /star ) , (1 , 1) , ( s 11 , 1 s 1 ) , (0 , 0), and (\# , \#) we get Now using ( /star, /star ) , (1 , 1) , (1 s 10 , s 210), and (\# , \#) we get Now using ( /star, /star ) , (1 , 1) , (1 s 10 , s 210), and (# , #) we get Using ( /star, /star ) , ( /star s 11 , s 2 /star 1) , (1 , 1) , (0 , 0), and (# , #) we get Now using ( s 2 /star, h 0), (1 , 1) twice, (0 , 0), and (# , #) , we get Finally, using the pairs containing h , together with (1 , 1) , (0 , 0), and (# , #) , we … \end{verbatim} ``` </details>
873. ph-e93b20621c3db69d3056 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md ### Plain (markdown context) 1 s 1 ) , (0 , 0), and (# , #) we get Now using ( /star, /star ) , (1 , 1) , (1 s 10 , s 210), and (# , #) we get Using ( /star, /star ) , ( /star s 11 , s 2 /star 1) , (1 , 1) , (0 , 0), and (# , #) we get Now using ( s 2 /star, h 0), (1 , 1) twice, (0 , 0), and (# , #) , we get Finally, using the pairs containing h , together with (1 , 1) , (0 , 0), and (# , #) , we get Wecan now use the fact that Post's Correspondence Problem is undecidable to sol… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{1 s 1 ) , (0 , 0), and (\# , \#) we get Now using ( /star, /star ) , (1 , 1) , (1 s 10 , s 210), and (\# , \#) we get Using ( /star, /star ) , ( /star s 11 , s 2 /star 1) , (1 , 1) , (0 , 0), and (\# , \#) we get Now using ( /star, /star ) , (1 , 1) , (1 s 10 , s 210), and (# , #) we get Using ( /star, /star ) , ( /star s 11 , s 2 /star 1) , (1 , 1) , (0 , 0), and (# , #) we get Now using ( s 2 /star, h 0), (1 , 1) twice, (0 , 0), and (# , #) , we get Finally, using the pairs containing h , together with (1 , 1) , (0 , 0), and (# , #) , we get Wecan now use the fact that Post's Correspondence Problem is undecidable to sol… \end{verbatim} ```
874. ph-b22852b6eedb061a74d2 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md ### Plain (markdown context) ions about solvability with regard to context-free languages. We now use ( s 00 , /star s 1 ) to get Theorem 5.10 It is undecidable for arbitrary context-free grammars G 1 and G 2 whether L ( G 1 ) ∩ L ( G 2 ) = ∅ . Proof Let P ⊂ /Sigma1 ∗ × /Sigma1 ∗ be an arbitrary correspondence system with pairs ( u 0 , v 0 ) , ( u 1 , v 1 ) , ( u 2 , v 2 ) , . . . , ( un , v n ). In the following, w -1 will be w with the letters reversed. For example 1101 -1 is 1011. Let G 1 be generated by productions Thus every word in L ( G 1) has the form ui 0… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ions about solvability with regard to context-free languages. We now use ( s 00 , /star s 1 ) to get Theorem 5.10 It is undecidable for arbitrary context-free grammars G 1 and G 2 whether L ( G 1 ) ∩ L ( G 2 ) = ∅ . Proof Let P ⊂ /Sigma1 ∗ × /Sigma1 ∗ be an arbitrary corresponde…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md:offset=37024 \begin{verbatim} ions about solvability with regard to context-free languages. We now use ( s 00 , /star s 1 ) to get Theorem 5.10 It is undecidable for arbitrary context-free grammars G 1 and G 2 whether L ( G 1 ) ∩ L ( G 2 ) = ∅ . Proof Let P ⊂ /Sigma1 ∗ × /Sigma1 ∗ be an arbitrary correspondence system with pairs ( u 0 , v 0 ) , ( u 1 , v 1 ) , ( u 2 , v 2 ) , . . . , ( un , v n ). In the following, w -1 will be w with the letters reversed. For example 1101 -1 is 1011. Let G 1 be generated by productions Thus every word in L ( G 1) has the form ui 0… \end{verbatim} ``` </details>
875. ph-b43a722ef557afe1559c — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md ### Plain (markdown context) gard to context-free languages. We now use ( s 00 , /star s 1 ) to get Theorem 5.10 It is undecidable for arbitrary context-free grammars G 1 and G 2 whether L ( G 1 ) ∩ L ( G 2 ) = ∅ . Proof Let P ⊂ /Sigma1 ∗ × /Sigma1 ∗ be an arbitrary correspondence system with pairs ( u 0 , v 0 ) , ( u 1 , v 1 ) , ( u 2 , v 2 ) , . . . , ( un , v n ). In the following, w -1 will be w with the letters reversed. For example 1101 -1 is 1011. Let G 1 be generated by productions Thus every word in L ( G 1) has the form ui 0 ui 1 ui 2 . . . uim c v -1 im… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{gard to context-free languages. We now use ( s 00 , /star s 1 ) to get Theorem 5.10 It is undecidable for arbitrary context-free grammars G 1 and G 2 whether L ( G 1 ) ∩ L ( G 2 ) = ∅ . Proof Let P ⊂ /Sigma1 ∗ × /Sigma1 ∗ be an arbitrary correspondence system with pairs ( u 0 , …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md:offset=37054 \begin{verbatim} gard to context-free languages. We now use ( s 00 , /star s 1 ) to get Theorem 5.10 It is undecidable for arbitrary context-free grammars G 1 and G 2 whether L ( G 1 ) ∩ L ( G 2 ) = ∅ . Proof Let P ⊂ /Sigma1 ∗ × /Sigma1 ∗ be an arbitrary correspondence system with pairs ( u 0 , v 0 ) , ( u 1 , v 1 ) , ( u 2 , v 2 ) , . . . , ( un , v n ). In the following, w -1 will be w with the letters reversed. For example 1101 -1 is 1011. Let G 1 be generated by productions Thus every word in L ( G 1) has the form ui 0 ui 1 ui 2 . . . uim c v -1 im… \end{verbatim} ``` </details>
876. ph-3844737f5f7e5395d78c — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md ### Plain (markdown context) ly if w = ui 0 ui 1 ui 2 . . . uim = v i 0 v i 1 v i 2 . . . v im which is a solution to the Post's correspondence system. Hence it is undecidable for arbitrary context-free grammars G G whether L ( G ) L ( G ) . 1 and 2 1 ∩ 2 = ∅ /square Definition 5.8 A context-free grammar is ambiguous if there are two leftmost generations of the same word. Example 5.4 Let /Gamma1 = ( N , /Sigma1 , S , P ) be the grammar defined by N = { S , A , B } , /Sigma1 = { a , b } , and P be the set of productions Obviously a n b n can be generated in two different ways. Theorem 5.11 It … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ly if w = ui 0 ui 1 ui 2 . . . uim = v i 0 v i 1 v i 2 . . . v im which is a solution to the Post's correspondence system. Hence it is undecidable for arbitrary context-free grammars G G whether L ( G ) L ( G ) . 1 and 2 1 ∩ 2 = ∅ /square Definition 5.8 A context-free grammar is…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md:offset=37782 \begin{verbatim} ly if w = ui 0 ui 1 ui 2 . . . uim = v i 0 v i 1 v i 2 . . . v im which is a solution to the Post's correspondence system. Hence it is undecidable for arbitrary context-free grammars G G whether L ( G ) L ( G ) . 1 and 2 1 ∩ 2 = ∅ /square Definition 5.8 A context-free grammar is ambiguous if there are two leftmost generations of the same word. Example 5.4 Let /Gamma1 = ( N , /Sigma1 , S , P ) be the grammar defined by N = { S , A , B } , /Sigma1 = { a , b } , and P be the set of productions Obviously a n b n can be generated in two different ways. Theorem 5.11 It … \end{verbatim} ``` </details>
877. ph-8b35bc61886f22eef83e — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md ### Plain (markdown context) /Sigma1 = { a , b } , and P be the set of productions Obviously a n b n can be generated in two different ways. Theorem 5.11 It is undecidable whether an arbitrary context-free grammar is ambiguous. Proof Let P ⊂ /Sigma1 + × /Sigma1 + be an arbitrary correspondence system with pairs ( u 0 , v 0 )( u 1 , v 1 ) , ( u 2 , v 2 ) , . . . , ( un , v n ). Let α 0 , α 1 , α 2 , . . . , α n be symbols not in /Sigma1 ∗ . We construct two grammars G 1 and G 2 as follows: where N 1 = { S 1 } , /Sigma1 a = /Sigma1 ∪ { α 0 , α 1 , α 2 , . . . , α n … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{/Sigma1 = \{ a , b \} , and P be the set of productions Obviously a n b n can be generated in two different ways. Theorem 5.11 It is undecidable whether an arbitrary context-free grammar is ambiguous. Proof Let P ⊂ /Sigma1 + × /Sigma1 + be an arbitrary…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md:offset=38226 \begin{verbatim} /Sigma1 = { a , b } , and P be the set of productions Obviously a n b n can be generated in two different ways. Theorem 5.11 It is undecidable whether an arbitrary context-free grammar is ambiguous. Proof Let P ⊂ /Sigma1 + × /Sigma1 + be an arbitrary correspondence system with pairs ( u 0 , v 0 )( u 1 , v 1 ) , ( u 2 , v 2 ) , . . . , ( un , v n ). Let α 0 , α 1 , α 2 , . . . , α n be symbols not in /Sigma1 ∗ . We construct two grammars G 1 and G 2 as follows: where N 1 = { S 1 } , /Sigma1 a = /Sigma1 ∪ { α 0 , α 1 , α 2 , . . . , α n … \end{verbatim} ``` </details>
878. ph-28d7d2abebdcbc5d1896 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md ### Plain (markdown context) an arbitrary context-free grammar is ambiguous. Proof Let P ⊂ /Sigma1 + × /Sigma1 + be an arbitrary correspondence system with pairs ( u 0 , v 0 )( u 1 , v 1 ) , ( u 2 , v 2 ) , . . . , ( un , v n ). Let α 0 , α 1 , α 2 , . . . , α n be symbols not in /Sigma1 ∗ . We construct two grammars G 1 and G 2 as follows: where N 1 = { S 1 } , /Sigma1 a = /Sigma1 ∪ { α 0 , α 1 , α 2 , . . . , α n } , and P 1 = { S 1 → α i S 1 ui for i = 0 , 1 , . . . , n , and S 1 → λ } . where N 2 = { S 2 } , /Sigma1 a = /Sigma1 ∪ { α 0 , α 1 , α 2 , . . . , α … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{an arbitrary context-free grammar is ambiguous. Proof Let P ⊂ /Sigma1 + × /Sigma1 + be an arbitrary correspondence system with pairs ( u 0 , v 0 )( u 1 , v 1 ) , ( u 2 , v 2 ) , . . . , ( un , v n ). Let α 0 , α 1 , α 2 , . . . , α n be symbols not in /Sigma1 ∗ . We construct tw…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_2.3.md:offset=38410 \begin{verbatim} an arbitrary context-free grammar is ambiguous. Proof Let P ⊂ /Sigma1 + × /Sigma1 + be an arbitrary correspondence system with pairs ( u 0 , v 0 )( u 1 , v 1 ) , ( u 2 , v 2 ) , . . . , ( un , v n ). Let α 0 , α 1 , α 2 , . . . , α n be symbols not in /Sigma1 ∗ . We construct two grammars G 1 and G 2 as follows: where N 1 = { S 1 } , /Sigma1 a = /Sigma1 ∪ { α 0 , α 1 , α 2 , . . . , α n } , and P 1 = { S 1 → α i S 1 ui for i = 0 , 1 , . . . , n , and S 1 → λ } . where N 2 = { S 2 } , /Sigma1 a = /Sigma1 ∪ { α 0 , α 1 , α 2 , . . . , α … \end{verbatim} ``` </details>
879. ph-1699b60134474f9b8f57 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_3.1.md ### Plain (markdown context) λ . Proposition 6.2 For each word u in /Sigma1 + there is a unique pair ( q , n ) , with q in Q and n in N, for which u = q n . Proof For each i with 1 ≤ i ≤ | u | , let u i be the prefix of u of length i . Compute successively the u / ui until a j occurs at which u / u j = ( m , λ ). Such a j will certainly occur since u / u | u | = (1 , λ ). For the pair ( u j , m ) we have u j in Q. Consequently u = u m j has the required form. Suppose now that u = p m = q n , where both p and q are in Q. and Thus pq = qp and by Proposition 6.1, p a… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{λ . Proposition 6.2 For each word u in /Sigma1 + there is a unique pair ( q , n ) , with q in Q and n in N, for which u = q n . Proof For each i with 1 ≤ i ≤ | u | , let u i be the prefix of u of length i . Compute successively the u / ui until a j occurs at which u / u j = ( m …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_3.1.md:offset=3394 \begin{verbatim} λ . Proposition 6.2 For each word u in /Sigma1 + there is a unique pair ( q , n ) , with q in Q and n in N, for which u = q n . Proof For each i with 1 ≤ i ≤ | u | , let u i be the prefix of u of length i . Compute successively the u / ui until a j occurs at which u / u j = ( m , λ ). Such a j will certainly occur since u / u | u | = (1 , λ ). For the pair ( u j , m ) we have u j in Q. Consequently u = u m j has the required form. Suppose now that u = p m = q n , where both p and q are in Q. and Thus pq = qp and by Proposition 6.1, p a… \end{verbatim} ``` </details>
880. ph-4e9ae0426eb655af2cc4 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_3.1.md ### Plain (markdown context) u in /Sigma1 + there is a unique pair ( q , n ) , with q in Q and n in N, for which u = q n . Proof For each i with 1 ≤ i ≤ | u | , let u i be the prefix of u of length i . Compute successively the u / ui until a j occurs at which u / u j = ( m , λ ). Such a j will certainly occur since u / u | u | = (1 , λ ). For the pair ( u j , m ) we have u j in Q. Consequently u = u m j has the required form. Suppose now that u = p m = q n , where both p and q are in Q. and Thus pq = qp and by Proposition 6.1, p and q must be powers of a common wo… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{u in /Sigma1 + there is a unique pair ( q , n ) , with q in Q and n in N, for which u = q n . Proof For each i with 1 ≤ i ≤ | u | , let u i be the prefix of u of length i . Compute successively the u / ui until a j occurs at which u / u j = ( m , λ ). Such a j will certainly occ…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_3.1.md:offset=3429 \begin{verbatim} u in /Sigma1 + there is a unique pair ( q , n ) , with q in Q and n in N, for which u = q n . Proof For each i with 1 ≤ i ≤ | u | , let u i be the prefix of u of length i . Compute successively the u / ui until a j occurs at which u / u j = ( m , λ ). Such a j will certainly occur since u / u | u | = (1 , λ ). For the pair ( u j , m ) we have u j in Q. Consequently u = u m j has the required form. Suppose now that u = p m = q n , where both p and q are in Q. and Thus pq = qp and by Proposition 6.1, p and q must be powers of a common wo… \end{verbatim} ``` </details>
881. ph-28a07591a813294df972 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_3.1.md ### Plain (markdown context) /Sigma1 + is either ∅ or { 1 } . 8. (c) For the language LL , determine the spectra of ab , abab , and ababab . 9. (d) Describe P ( LL ) and Su( LL ). ## 6.5 Visualizing languages In order to spell out the visualization of a language L within Q × N , we begin with the usual x -y plane with each point having associated real number coordinates ( x , y ). We use only the upper half plane , { ( x , y ) : y > 0 } . With each integer i and each positive integer n we associate the unit rectangle In this way the upper half plane is partitioned into nonoverlapping unit … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{/Sigma1 + is either ∅ or \{ 1 \} . 8. (c) For the language LL , determine the spectra of ab , abab , and ababab . 9. (d) Describe P ( LL ) and Su( LL ). \#\# 6.5 Visualizing languages In order to spell out the visualization of a language L within Q × N , we begin with the usual x -y…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_3.1.md:offset=15060 \begin{verbatim} /Sigma1 + is either ∅ or { 1 } . 8. (c) For the language LL , determine the spectra of ab , abab , and ababab . 9. (d) Describe P ( LL ) and Su( LL ). ## 6.5 Visualizing languages In order to spell out the visualization of a language L within Q × N , we begin with the usual x -y plane with each point having associated real number coordinates ( x , y ). We use only the upper half plane , { ( x , y ) : y > 0 } . With each integer i and each positive integer n we associate the unit rectangle In this way the upper half plane is partitioned into nonoverlapping unit … \end{verbatim} ``` </details>
882. ph-2b90b81707bff7176105 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_3.2.md ### Plain (markdown context) f any word is m . The collection of distinct flags { F ( w ) : w ∈ /Sigma1 + } associated with a regular language L is necessarily finite. By a fl ag F of the language L we mean a sequence of states that constitutes the flag, relative to L , of some word in w in /Sigma1 + . With each flag F of L we associate the language I ( F ) = { w ∈ /Sigma1 + : F ( w ) = F } . We call I ( F ) the language of the flag F . For each flag F = { s j : 0 ≤ j ≤ k } , where the s i denote the states in F , we have where each L ( s j , s j + 1) is the language that consists of all word… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{f any word is m . The collection of distinct flags \{ F ( w ) : w ∈ /Sigma1 + \} associated with a regular language L is necessarily finite. By a fl ag F of the language L we mean a sequence of states that constitutes the flag, relative to L , of some word in w in /Sigma1 + . With…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_3.2.md:offset=6848 \begin{verbatim} f any word is m . The collection of distinct flags { F ( w ) : w ∈ /Sigma1 + } associated with a regular language L is necessarily finite. By a fl ag F of the language L we mean a sequence of states that constitutes the flag, relative to L , of some word in w in /Sigma1 + . With each flag F of L we associate the language I ( F ) = { w ∈ /Sigma1 + : F ( w ) = F } . We call I ( F ) the language of the flag F . For each flag F = { s j : 0 ≤ j ≤ k } , where the s i denote the states in F , we have where each L ( s j , s j + 1) is the language that consists of all word… \end{verbatim} ``` </details>
883. ph-7d736290c416c532785f — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_3.2.md ### Plain (markdown context) next proposition it follows that conjugates have the same exponent , which includes the information that the conjugates of primitive words are primitive . This last fact, that conjugates of primitives are primitive, is applied many times in Section 6.9. Proposition 6.3 If u v = p n then v u = q n with q a conjugate of p. Proof Since u v = p n , we may assume that p = u ′′ v ′ where u = p i u ′′ and v = v ′ p j with i and j nonnegative integers for which i + j = n -1. For q = v ′ u ′′ we have Lemma 6.1 Let v be a word for which vv = x v y with x and y nonnull, then… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{next proposition it follows that conjugates have the same exponent , which includes the information that the conjugates of primitive words are primitive . This last fact, that conjugates of primitives are primitive, is applied many times in Section 6.9. Proposition 6.3 If u v = …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_3.2.md:offset=8822 \begin{verbatim} next proposition it follows that conjugates have the same exponent , which includes the information that the conjugates of primitive words are primitive . This last fact, that conjugates of primitives are primitive, is applied many times in Section 6.9. Proposition 6.3 If u v = p n then v u = q n with q a conjugate of p. Proof Since u v = p n , we may assume that p = u ′′ v ′ where u = p i u ′′ and v = v ′ p j with i and j nonnegative integers for which i + j = n -1. For q = v ′ u ′′ we have Lemma 6.1 Let v be a word for which vv = x v y with x and y nonnull, then… \end{verbatim} ``` </details>
884. ph-7b6906526b0ba657683c — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_3.2.md ### Plain (markdown context) e ligase can now paste together the fragments ttttgga ′ and ′ acctttt to yield the dsDNA molecule x ttttggaacctttt. The ligase can also paste together the fragments = ## TTTTGGAACCTTT we know its companion row is ## AAAACCTTGGAAA . Consequently, we need to give only one of the two strands. For efficiency and convenience we will list only one row of each dsDNA molecule. To be certain not to confuse dsDNA and ssDNA, we will use lowercase a, c, g, t to denote the paired deoxyribonucleotides: tttgga ′ and ′ accttt to yield the dsDNA molecule y = tttggaaccttt. The mole… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{e ligase can now paste together the fragments ttttgga ′ and ′ acctttt to yield the dsDNA molecule x ttttggaacctttt. The ligase can also paste together the fragments = \#\# TTTTGGAACCTTT we know its companion row is \#\# AAAACCTTGGAAA . Consequently, we need to give only one of the t…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_3.2.md:offset=34898 \begin{verbatim} e ligase can now paste together the fragments ttttgga ′ and ′ acctttt to yield the dsDNA molecule x ttttggaacctttt. The ligase can also paste together the fragments = ## TTTTGGAACCTTT we know its companion row is ## AAAACCTTGGAAA . Consequently, we need to give only one of the two strands. For efficiency and convenience we will list only one row of each dsDNA molecule. To be certain not to confuse dsDNA and ssDNA, we will use lowercase a, c, g, t to denote the paired deoxyribonucleotides: tttgga ′ and ′ accttt to yield the dsDNA molecule y = tttggaaccttt. The mole… \end{verbatim} ``` </details>
885. ph-98d674eba92d7d555276 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_3.2.md ### Plain (markdown context) nner, possibly allowing ambiguity between molecules and the words used to represent them. The remainder of this chapter deals specifically with words in a free monoid. (However, all results in the chapter have meaningful interpretations for enzymes acting on dsDNA.) Let /Sigma1 be a finite set to be used as an alphabet. Let /Sigma1 ∗ be the set of all strings over /Sigma1 . By a language we mean a subset of /Sigma1 ∗ . A splicing rule is an element r = ( u , u ′ , v ′ , v ) of the product set The action of the rule r on a language L defines the language r ( L ) = … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{nner, possibly allowing ambiguity between molecules and the words used to represent them. The remainder of this chapter deals specifically with words in a free monoid. (However, all results in the chapter have meaningful interpretations for enzymes acting on dsDNA.) Let /Sigma1 …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_3.2.md:offset=38185 \begin{verbatim} nner, possibly allowing ambiguity between molecules and the words used to represent them. The remainder of this chapter deals specifically with words in a free monoid. (However, all results in the chapter have meaningful interpretations for enzymes acting on dsDNA.) Let /Sigma1 be a finite set to be used as an alphabet. Let /Sigma1 ∗ be the set of all strings over /Sigma1 . By a language we mean a subset of /Sigma1 ∗ . A splicing rule is an element r = ( u , u ′ , v ′ , v ) of the product set The action of the rule r on a language L defines the language r ( L ) = … \end{verbatim} ``` </details>
886. ph-d1aa5155163db2c33838 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_3.2.md ### Plain (markdown context) a pair ( σ, I ) , where σ is a splicing scheme and I is a finite initial language contained in /Sigma1 ∗ . The language generated by ( σ, I ) is L ( σ, I ) = σ ∗ ( I ) . A language L is a splicing language if L = L ( σ, I ) for some splicing system ( σ, I ) . Example 7.1 Let /Sigma1 = { a , c , g , t } . Let r = ( u , u ′ , v ′ , v ) where the four words u , u ′ , v ′ , v in /Sigma1 ∗ appearing in the rule r are u = v ′ = gga and u ′ = v = acc . Let R = { r } . This gives the splicing scheme Let Observe that r applied to the ordered pa… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{a pair ( σ, I ) , where σ is a splicing scheme and I is a finite initial language contained in /Sigma1 ∗ . The language generated by ( σ, I ) is L ( σ, I ) = σ ∗ ( I ) . A language L is a splicing language if L = L ( σ, I ) for some splicing system ( σ, I ) . Example 7.1 Let /Si…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_3.2.md:offset=39658 \begin{verbatim} a pair ( σ, I ) , where σ is a splicing scheme and I is a finite initial language contained in /Sigma1 ∗ . The language generated by ( σ, I ) is L ( σ, I ) = σ ∗ ( I ) . A language L is a splicing language if L = L ( σ, I ) for some splicing system ( σ, I ) . Example 7.1 Let /Sigma1 = { a , c , g , t } . Let r = ( u , u ′ , v ′ , v ) where the four words u , u ′ , v ′ , v in /Sigma1 ∗ appearing in the rule r are u = v ′ = gga and u ′ = v = acc . Let R = { r } . This gives the splicing scheme Let Observe that r applied to the ordered pa… \end{verbatim} ``` </details>
887. ph-407c53a659335dc8a6cc — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_3.2.md ### Plain (markdown context) icing scheme and I is a finite initial language contained in /Sigma1 ∗ . The language generated by ( σ, I ) is L ( σ, I ) = σ ∗ ( I ) . A language L is a splicing language if L = L ( σ, I ) for some splicing system ( σ, I ) . Example 7.1 Let /Sigma1 = { a , c , g , t } . Let r = ( u , u ′ , v ′ , v ) where the four words u , u ′ , v ′ , v in /Sigma1 ∗ appearing in the rule r are u = v ′ = gga and u ′ = v = acc . Let R = { r } . This gives the splicing scheme Let Observe that r applied to the ordered pair ( ttttggaaccttt , tttggaacctttt… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{icing scheme and I is a finite initial language contained in /Sigma1 ∗ . The language generated by ( σ, I ) is L ( σ, I ) = σ ∗ ( I ) . A language L is a splicing language if L = L ( σ, I ) for some splicing system ( σ, I ) . Example 7.1 Let /Sigma1 = \{ a , c , g , t \} . Let r =…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_3.2.md:offset=39693 \begin{verbatim} icing scheme and I is a finite initial language contained in /Sigma1 ∗ . The language generated by ( σ, I ) is L ( σ, I ) = σ ∗ ( I ) . A language L is a splicing language if L = L ( σ, I ) for some splicing system ( σ, I ) . Example 7.1 Let /Sigma1 = { a , c , g , t } . Let r = ( u , u ′ , v ′ , v ) where the four words u , u ′ , v ′ , v in /Sigma1 ∗ appearing in the rule r are u = v ′ = gga and u ′ = v = acc . Let R = { r } . This gives the splicing scheme Let Observe that r applied to the ordered pair ( ttttggaaccttt , tttggaacctttt… \end{verbatim} ``` </details>
888. ph-fb28d63473fea8b49396 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_3.2.md ### Plain (markdown context) for some splicing system ( σ, I ) . Example 7.1 Let /Sigma1 = { a , c , g , t } . Let r = ( u , u ′ , v ′ , v ) where the four words u , u ′ , v ′ , v in /Sigma1 ∗ appearing in the rule r are u = v ′ = gga and u ′ = v = acc . Let R = { r } . This gives the splicing scheme Let Observe that r applied to the ordered pair ( ttttggaaccttt , tttggaacctttt ) of words in I gives the word ttttggaacctttt , and r applied to the ordered pair of words in I gives tttggaaccttt . The less interesting actions of r on I must… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{for some splicing system ( σ, I ) . Example 7.1 Let /Sigma1 = \{ a , c , g , t \} . Let r = ( u , u ′ , v ′ , v ) where the four words u , u ′ , v ′ , v in /Sigma1 ∗ appearing in the rule r are u = v ′ = gga and u ′ = v = acc . Let R = \{ r \} . This gives the splicing scheme Let Observe that r applied to the ordered pair ( ttttggaaccttt , tttggaacctttt ) of words in I gives the word ttttggaacctttt , and r applied to the ordered pair of words in I gives tttggaaccttt . The less interesting actions of r on I must… \end{verbatim} ```
889. ph-c82d482ba14f102b9809 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_3.2.md ### Plain (markdown context) ′ = gga and u ′ = v = acc . Let R = { r } . This gives the splicing scheme Let Observe that r applied to the ordered pair ( ttttggaaccttt , tttggaacctttt ) of words in I gives the word ttttggaacctttt , and r applied to the ordered pair of words in I gives tttggaaccttt . The less interesting actions of r on I must be recognized: When r acts on ordered pairs in the 'diagonal' of I × I , for example on the result is merely ttttggaaccttt which appeared as each coordinate of the pair… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{′ = gga and u ′ = v = acc . Let R = \{ r \} . This gives the splicing scheme Let Observe that r applied to the ordered pair ( ttttggaaccttt , tttggaacctttt ) of words in I gives the word ttttggaacctttt , and r applied to th…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_3.2.md:offset=40081 \begin{verbatim} ′ = gga and u ′ = v = acc . Let R = { r } . This gives the splicing scheme Let Observe that r applied to the ordered pair ( ttttggaaccttt , tttggaacctttt ) of words in I gives the word ttttggaacctttt , and r applied to the ordered pair of words in I gives tttggaaccttt . The less interesting actions of r on I must be recognized: When r acts on ordered pairs in the 'diagonal' of I × I , for example on the result is merely ttttggaaccttt which appeared as each coordinate of the pair… \end{verbatim} ``` </details>
890. ph-d47f190d65fdbd820d7f — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_3.2.md ### Plain (markdown context) ot-decoded --> Observe that r applied to the ordered pair ( ttttggaaccttt , tttggaacctttt ) of words in I gives the word ttttggaacctttt , and r applied to the ordered pair of words in I gives tttggaaccttt . The less interesting actions of r on I must be recognized: When r acts on ordered pairs in the 'diagonal' of I × I , for example on the result is merely ttttggaaccttt which appeared as each coordinate of the pair. Here we have - = I ∪ { ttttggaacctttt , tttggaaccttt , ttttggaaccttt , tttggaacctttt } and … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ot-decoded --> Observe that r applied to the ordered pair ( ttttggaaccttt , tttggaacctttt ) of words in I gives the word ttttggaacctttt , and r applied to the ordered pair of words in I gives tttggaaccttt . The less interesting actions of r on I must…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_3.2.md:offset=40207 \begin{verbatim} ot-decoded --> Observe that r applied to the ordered pair ( ttttggaaccttt , tttggaacctttt ) of words in I gives the word ttttggaacctttt , and r applied to the ordered pair of words in I gives tttggaaccttt . The less interesting actions of r on I must be recognized: When r acts on ordered pairs in the 'diagonal' of I × I , for example on the result is merely ttttggaaccttt which appeared as each coordinate of the pair. Here we have - = I ∪ { ttttggaacctttt , tttggaaccttt , ttttggaaccttt , tttggaacctttt } and … \end{verbatim} ``` </details>
891. ph-c0c653b537f07adec6d3 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_3.2.md ### Plain (markdown context) ). Then also σ 3 ( I )) = σ 2 ( I ) = σ 1 ( I ) and in fact σ ∗ ( I ) = σ 1 ( I ). Thus L ( σ, I ) is the finite language ``` σ ∗ ( I ) = { ttttggaacctttt , tttggaaccttt , ttttggaaccttt , tttggaacctttt } . ``` This example connects the formal definitions of splicing systems and languages with the less formal introductory remarks of Sections 7.1 and 7.2. Example 7.2 Let /Sigma1 = { a , c , g , t } . Let r = ( c , cccgg , c , cccgg ), R = { r } , and let I contain only one word of length 30, The rule can be applied to the ordered pair wi… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{). Then also σ 3 ( I )) = σ 2 ( I ) = σ 1 ( I ) and in fact σ ∗ ( I ) = σ 1 ( I ). Thus L ( σ, I ) is the finite language ``` σ ∗ ( I ) = \{ ttttggaacctttt , tttggaaccttt , ttttggaaccttt , tttggaacctttt \} . ``` This example connects the formal definitions of splicing systems and …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_3.2.md:offset=41011 \begin{verbatim} ). Then also σ 3 ( I )) = σ 2 ( I ) = σ 1 ( I ) and in fact σ ∗ ( I ) = σ 1 ( I ). Thus L ( σ, I ) is the finite language ``` σ ∗ ( I ) = { ttttggaacctttt , tttggaaccttt , ttttggaaccttt , tttggaacctttt } . ``` This example connects the formal definitions of splicing systems and languages with the less formal introductory remarks of Sections 7.1 and 7.2. Example 7.2 Let /Sigma1 = { a , c , g , t } . Let r = ( c , cccgg , c , cccgg ), R = { r } , and let I contain only one word of length 30, The rule can be applied to the ordered pair wi… \end{verbatim} ``` </details>
892. ph-60c1699e1a8f876718b9 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_3.2.md ### Plain (markdown context) 1 ( I ). Thus L ( σ, I ) is the finite language ``` σ ∗ ( I ) = { ttttggaacctttt , tttggaaccttt , ttttggaaccttt , tttggaacctttt } . ``` This example connects the formal definitions of splicing systems and languages with the less formal introductory remarks of Sections 7.1 and 7.2. Example 7.2 Let /Sigma1 = { a , c , g , t } . Let r = ( c , cccgg , c , cccgg ), R = { r } , and let I contain only one word of length 30, The rule can be applied to the ordered pair with cuts made using the right occurrence of ccccgg in the first coordinate … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{1 ( I ). Thus L ( σ, I ) is the finite language ``` σ ∗ ( I ) = \{ ttttggaacctttt , tttggaaccttt , ttttggaaccttt , tttggaacctttt \} . ``` This example connects the formal definitions of splicing systems and languages with the less formal introductory remarks of Sections 7.1 and 7.…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_3.2.md:offset=41086 \begin{verbatim} 1 ( I ). Thus L ( σ, I ) is the finite language ``` σ ∗ ( I ) = { ttttggaacctttt , tttggaaccttt , ttttggaaccttt , tttggaacctttt } . ``` This example connects the formal definitions of splicing systems and languages with the less formal introductory remarks of Sections 7.1 and 7.2. Example 7.2 Let /Sigma1 = { a , c , g , t } . Let r = ( c , cccgg , c , cccgg ), R = { r } , and let I contain only one word of length 30, The rule can be applied to the ordered pair with cuts made using the right occurrence of ccccgg in the first coordinate … \end{verbatim} ``` </details>
893. ph-81033b5f332bfc17ec88 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_3.2.md ### Plain (markdown context) and languages with the less formal introductory remarks of Sections 7.1 and 7.2. Example 7.2 Let /Sigma1 = { a , c , g , t } . Let r = ( c , cccgg , c , cccgg ), R = { r } , and let I contain only one word of length 30, The rule can be applied to the ordered pair with cuts made using the right occurrence of ccccgg in the first coordinate and the left occurrence of ccccgg in the second coordinate. This gives the word of length 42: The rule can be also applied to the ordered pair using the left occurrence of … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{and languages with the less formal introductory remarks of Sections 7.1 and 7.2. Example 7.2 Let /Sigma1 = \{ a , c , g , t \} . Let r = ( c , cccgg , c , cccgg ), R = \{ r \} , and let I contain only one word of length 30, The rule can be applied to the…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_3.2.md:offset=41288 \begin{verbatim} and languages with the less formal introductory remarks of Sections 7.1 and 7.2. Example 7.2 Let /Sigma1 = { a , c , g , t } . Let r = ( c , cccgg , c , cccgg ), R = { r } , and let I contain only one word of length 30, The rule can be applied to the ordered pair with cuts made using the right occurrence of ccccgg in the first coordinate and the left occurrence of ccccgg in the second coordinate. This gives the word of length 42: The rule can be also applied to the ordered pair using the left occurrence of … \end{verbatim} ``` </details>
894. ph-c880285955262c177993 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_3.2.md ### Plain (markdown context) rule can be applied to the ordered pair with cuts made using the right occurrence of ccccgg in the first coordinate and the left occurrence of ccccgg in the second coordinate. This gives the word of length 42: The rule can be also applied to the ordered pair using the left occurrence of ccccgg in the first coordinate and the right occurrence of ccccgg in the second coordinate. This gives the word of length 18, a 6 ccccgga 6 . Thus Continuing with similar considerations one finds that L ( σ, I ) = σ ∗ ( I ) … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{rule can be applied to the ordered pair with cuts made using the right occurrence of ccccgg in the first coordinate and the left occurrence of ccccgg in the second coordinate. This gives the word of length 42: The rule ca…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_3.2.md:offset=41544 \begin{verbatim} rule can be applied to the ordered pair with cuts made using the right occurrence of ccccgg in the first coordinate and the left occurrence of ccccgg in the second coordinate. This gives the word of length 42: The rule can be also applied to the ordered pair using the left occurrence of ccccgg in the first coordinate and the right occurrence of ccccgg in the second coordinate. This gives the word of length 18, a 6 ccccgga 6 . Thus Continuing with similar considerations one finds that L ( σ, I ) = σ ∗ ( I ) … \end{verbatim} ``` </details>
895. ph-647a50644a73a32f2f5b — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_3.2.md ### Plain (markdown context) dinate and the left occurrence of ccccgg in the second coordinate. This gives the word of length 42: The rule can be also applied to the ordered pair using the left occurrence of ccccgg in the first coordinate and the right occurrence of ccccgg in the second coordinate. This gives the word of length 18, a 6 ccccgga 6 . Thus Continuing with similar considerations one finds that L ( σ, I ) = σ ∗ ( I ) is the infinite regular language Example 7.3 We may interpret the 30 symbol word given in Example 7.2 as a mo… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{dinate and the left occurrence of ccccgg in the second coordinate. This gives the word of length 42: The rule can be also applied to the ordered pair using the left occurrence of ccccgg in the first coordinate and the right occurrence of ccccgg in th…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_3.2.md:offset=41685 \begin{verbatim} dinate and the left occurrence of ccccgg in the second coordinate. This gives the word of length 42: The rule can be also applied to the ordered pair using the left occurrence of ccccgg in the first coordinate and the right occurrence of ccccgg in the second coordinate. This gives the word of length 18, a 6 ccccgga 6 . Thus Continuing with similar considerations one finds that L ( σ, I ) = σ ∗ ( I ) is the infinite regular language Example 7.3 We may interpret the 30 symbol word given in Example 7.2 as a mo… \end{verbatim} ``` </details>
896. ph-47f0ae91d82710ea66f2 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_3.2.md ### Plain (markdown context) d of length 18, a 6 ccccgga 6 . Thus Continuing with similar considerations one finds that L ( σ, I ) = σ ∗ ( I ) is the infinite regular language Example 7.3 We may interpret the 30 symbol word given in Example 7.2 as a model of a dsDNA molecule as indicated in Section 7.3. The rule r of Example 7.2 represents the cut and paste activity of the restriction enzyme BsaJ I accompanied by a ligase. With these understandings the language obtained in Example 7.2 is a model of the set of all dsDNA molecules (havin… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{d of length 18, a 6 ccccgga 6 . Thus Continuing with similar considerations one finds that L ( σ, I ) = σ ∗ ( I ) is the infinite regular language Example 7.3 We may interpret the 30 symbol word given in Example 7.2 as a …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_3.2.md:offset=42006 \begin{verbatim} d of length 18, a 6 ccccgga 6 . Thus Continuing with similar considerations one finds that L ( σ, I ) = σ ∗ ( I ) is the infinite regular language Example 7.3 We may interpret the 30 symbol word given in Example 7.2 as a model of a dsDNA molecule as indicated in Section 7.3. The rule r of Example 7.2 represents the cut and paste activity of the restriction enzyme BsaJ I accompanied by a ligase. With these understandings the language obtained in Example 7.2 is a model of the set of all dsDNA molecules (havin… \end{verbatim} ``` </details>
897. ph-5c72e14583b26eb60ec1 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_3.2.md ### Plain (markdown context) ( cabaab , cabaaab ), and the analyses caba / ab and cab / aaab , the rule r gives the word cabaaaab . (Note that r provides a form of pumping.) Continuing in this way all words caba n b with n ≥ 3 can be obtained. Since caba n b with 0 ≤ n ≤ 2 were given in I , we have, for R = { r } and σ = ( /Sigma1 , R ), L ( σ, I ) = caba ∗ b as desired. In fact it has been shown [ 12 ] that for any regular language L ′ over any alphabet /Sigma1 , by choosing a symbol not in /Sigma1 , say c , the language is generated by a splicing system that can be specified very much as we… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{( cabaab , cabaaab ), and the analyses caba / ab and cab / aaab , the rule r gives the word cabaaaab . (Note that r provides a form of pumping.) Continuing in this way all words caba n b with n ≥ 3 can be obtained. Since caba n b with 0 ≤ n ≤ 2 were given in I , we have, for R =…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_3.2.md:offset=43534 \begin{verbatim} ( cabaab , cabaaab ), and the analyses caba / ab and cab / aaab , the rule r gives the word cabaaaab . (Note that r provides a form of pumping.) Continuing in this way all words caba n b with n ≥ 3 can be obtained. Since caba n b with 0 ≤ n ≤ 2 were given in I , we have, for R = { r } and σ = ( /Sigma1 , R ), L ( σ, I ) = caba ∗ b as desired. In fact it has been shown [ 12 ] that for any regular language L ′ over any alphabet /Sigma1 , by choosing a symbol not in /Sigma1 , say c , the language is generated by a splicing system that can be specified very much as we… \end{verbatim} ``` </details>
898. ph-2bbb71cb963da87b30ce — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_3.2.md ### Plain (markdown context) ormally speaking, each regular language is almost a splicing language. Example 7.5 Let /Sigma1 = { a , b } . The regular language L = ( aa ) ∗ cannot be generated by a splicing system. As the reader may verify, any finite set of rules that allows every word in L to be generated will also generate strings of odd length as well as the strings of even length. Example 7.6 The regular language L ′ = a ∗ ba ∗ ba ∗ cannot be generated by a finite set of rules either: For any nonnegative integer n , generate a ∗ ba n ba ∗ . Consequently, for a… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ormally speaking, each regular language is almost a splicing language. Example 7.5 Let /Sigma1 = \{ a , b \} . The regular language L = ( aa ) ∗ cannot be generated by a splicing system. As the reader may verify, any finite set of rules that allows every word in L to be generated …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_3.2.md:offset=44197 \begin{verbatim} ormally speaking, each regular language is almost a splicing language. Example 7.5 Let /Sigma1 = { a , b } . The regular language L = ( aa ) ∗ cannot be generated by a splicing system. As the reader may verify, any finite set of rules that allows every word in L to be generated will also generate strings of odd length as well as the strings of even length. Example 7.6 The regular language L ′ = a ∗ ba ∗ ba ∗ cannot be generated by a finite set of rules either: For any nonnegative integer n , generate a ∗ ba n ba ∗ . Consequently, for a… \end{verbatim} ``` </details>
899. ph-87def480c30dea806e25 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_3.2.md ### Plain (markdown context) language is almost a splicing language. Example 7.5 Let /Sigma1 = { a , b } . The regular language L = ( aa ) ∗ cannot be generated by a splicing system. As the reader may verify, any finite set of rules that allows every word in L to be generated will also generate strings of odd length as well as the strings of even length. Example 7.6 The regular language L ′ = a ∗ ba ∗ ba ∗ cannot be generated by a finite set of rules either: For any nonnegative integer n , generate a ∗ ba n ba ∗ . Consequently, for any finite subset F of nonnegati… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{language is almost a splicing language. Example 7.5 Let /Sigma1 = \{ a , b \} . The regular language L = ( aa ) ∗ cannot be generated by a splicing system. As the reader may verify, any finite set of rules that allows every word in L to be generated will also generate strings of o…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_3.2.md:offset=44227 \begin{verbatim} language is almost a splicing language. Example 7.5 Let /Sigma1 = { a , b } . The regular language L = ( aa ) ∗ cannot be generated by a splicing system. As the reader may verify, any finite set of rules that allows every word in L to be generated will also generate strings of odd length as well as the strings of even length. Example 7.6 The regular language L ′ = a ∗ ba ∗ ba ∗ cannot be generated by a finite set of rules either: For any nonnegative integer n , generate a ∗ ba n ba ∗ . Consequently, for any finite subset F of nonnegati… \end{verbatim} ``` </details>
900. ph-d147addbaf2693a37b77 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_3.2.md ### Plain (markdown context) the [ n ( L )] 4 quadruples of syntactic classes determined in /Sigma1 ∗ by L and, from each such quadruple ( W , X , Y , Z ), we choose one word from each class to obtain one rule ( w, x , y , z ) and then decide whether it respects L . If it does then every rule in W × X × Y × Z respects L . If it does not respect L then no rule in W × X × Y × Z respects L . This discussion has justified the following: Proposition 7.1 Let L be a regular language. The set of rules that respect L has the form where m is a nonnegative integer and each of the sets where m is a nonnegative integer and each of the sets where m is a nonnegative integer and each of the sets is an element of the syntactic monoid of L. Since each syntactic class of … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{om each such quadruple ( W , X , Y , Z ), we choose one word from each class to obtain one rule ( w, x , y , z ) and then decide whether it respects L . If it does then every rule in W × X × Y × Z respects L . If it does not respect L then no rule in W × X × Y × Z respects L . T…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_3.2.md:offset=51546 \begin{verbatim} om each such quadruple ( W , X , Y , Z ), we choose one word from each class to obtain one rule ( w, x , y , z ) and then decide whether it respects L . If it does then every rule in W × X × Y × Z respects L . If it does not respect L then no rule in W × X × Y × Z respects L . This discussion has justified the following: Proposition 7.1 Let L be a regular language. The set of rules that respect L has the form where m is a nonnegative integer and each of the sets is an element of the syntactic monoid of L. Since each syntactic class of … \end{verbatim} ``` </details>
902. ph-d66fc13a79b02a1a7410 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_3.2.md ### Plain (markdown context) - formula-not-decoded --> where m is a nonnegative integer and each of the sets is an element of the syntactic monoid of L. Since each syntactic class of a regular language L is itself a regular language, one can list all the strings of length at most k in the class. Consequently when the representation in the proposition has been constructed, the set of all rules of radius at most k that preserve L can be listed with no additional testing: For each of the sets in the representation, list all of the rules ( w, x , y , z ) in where m is a nonnegative integer and each of the sets is an element of the syntactic monoid of L. Since each syntactic class of a regular language L is itself a regular language, one can list all the strings of length at mos…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_3.2.md:offset=51964 \begin{verbatim} - formula-not-decoded --> where m is a nonnegative integer and each of the sets is an element of the syntactic monoid of L. Since each syntactic class of a regular language L is itself a regular language, one can list all the strings of length at most k in the class. Consequently when the representation in the proposition has been constructed, the set of all rules of radius at most k that preserve L can be listed with no additional testing: For each of the sets in the representation, list all of the rules ( w, x , y , z ) in is an element of the syntactic monoid of L. Since each syntactic class of a regular language L is itself a regular language, one can list all the strings of length at most k in the class. Consequently when the representation in the proposition has been constructed, the set of all rules of radius at most k that preserve L can be listed with no additional testing: For each of the sets in the representation, list all of the rules ( w, x , y , z ) in of radius at most k . In order to create such a list without using the synt… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ot-decoded --> is an element of the syntactic monoid of L. Since each syntactic class of a regular language L is itself a regular language, one can list all the strings of length at most k in the class. Consequently when the representation in the proposition has been constructed…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_3.2.md:offset=52060 \begin{verbatim} ot-decoded --> is an element of the syntactic monoid of L. Since each syntactic class of a regular language L is itself a regular language, one can list all the strings of length at most k in the class. Consequently when the representation in the proposition has been constructed, the set of all rules of radius at most k that preserve L can be listed with no additional testing: For each of the sets in the representation, list all of the rules ( w, x , y , z ) in of radius at most k . In order to create such a list without using the synt… \end{verbatim} ``` </details>
904. ph-e1bcc2c9325085359d4a — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_3.2.md ### Plain (markdown context) ) are also in R . When R is reflexive we say the same of any scheme or system having R as its rule set. In fact, splicing systems that model the cut and paste action of restriction enzymes and a ligase are necessarily reflexive. Consequently, from a modeling perspective, it is the reflexive splicing systems that are of prime interest. Section 7.6 provides the tools to construct, for each regular language L and each positive integer k , the following finite reflexive set Tk of splicing rules: Recall that Tk ( L ) = ∪{ r ( L ) : r ∈ Tk } , which is regular since Tk … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{) are also in R . When R is reflexive we say the same of any scheme or system having R as its rule set. In fact, splicing systems that model the cut and paste action of restriction enzymes and a ligase are necessarily reflexive. Consequently, from a modeling perspective, it is t…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_3.2.md:offset=53339 \begin{verbatim} ) are also in R . When R is reflexive we say the same of any scheme or system having R as its rule set. In fact, splicing systems that model the cut and paste action of restriction enzymes and a ligase are necessarily reflexive. Consequently, from a modeling perspective, it is the reflexive splicing systems that are of prime interest. Section 7.6 provides the tools to construct, for each regular language L and each positive integer k , the following finite reflexive set Tk of splicing rules: Recall that Tk ( L ) = ∪{ r ( L ) : r ∈ Tk } , which is regular since Tk … \end{verbatim} ``` </details>
905. ph-1e90dbd3b066f0fcfa26 — automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_3.2.md ### Plain (markdown context) ss, there is no f -1 ( t i ). Let S 1 be the elements of S for which the first result occurs. Let S 2 be the elements of S for which the second result occurs. Let S 3 be the elements of S for which the third result occurs. Obviously these sets are disjoint. Similarly form T 1 , T 2, and T 3 as subsets of T . f is a one-to-one correspondence from S 1 to T 1 . f is also a one-to-one correspondence from S 2 to T 2 . g -1 is a one-to-one correspondence from S 3 to T 3 . Let θ : S → T be defined by θ is a one to one correspondence from S to T . Theorem A.2 For any set … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ss, there is no f -1 ( t i ). Let S 1 be the elements of S for which the first result occurs. Let S 2 be the elements of S for which the second result occurs. Let S 3 be the elements of S for which the third result occurs. Obviously these sets are disjoint. Similarly form T 1 , …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/AutomataTheory_chapters/AutomataTheory_chapter_3.2.md:offset=57812 \begin{verbatim} ss, there is no f -1 ( t i ). Let S 1 be the elements of S for which the first result occurs. Let S 2 be the elements of S for which the second result occurs. Let S 3 be the elements of S for which the third result occurs. Obviously these sets are disjoint. Similarly form T 1 , T 2, and T 3 as subsets of T . f is a one-to-one correspondence from S 1 to T 1 . f is also a one-to-one correspondence from S 2 to T 2 . g -1 is a one-to-one correspondence from S 3 to T 3 . Let θ : S → T be defined by θ is a one to one correspondence from S to T . Theorem A.2 For any set … \end{verbatim} ``` </details>
906. ph-bfb4066574dc6b9e3c9a — automata/docling_md/Computational Inteligence for Modelling Control.md ### Plain (markdown context) w an edge on the game board to be selected and used to perform an action in the game. Where actions refer to a long chain symbol l n , an additional property defines the position along the chain to be filled, starting from the end described by the partner node in the action clause. A textual representation of the graph rule can be produced by listing each node in the graph and each of the partner nodes that it is connected to, and a pair of nodes to describe the edge to be filled. For example describes a rule that will match a collection of nodes, or board squares… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{w an edge on the game board to be selected and used to perform an action in the game. Where actions refer to a long chain symbol l n , an additional property defines the position along the chain to be filled, starting from the end described by the partner node in the action clau…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/Computational Inteligence for Modelling Control.md:offset=7571 \begin{verbatim} w an edge on the game board to be selected and used to perform an action in the game. Where actions refer to a long chain symbol l n , an additional property defines the position along the chain to be filled, starting from the end described by the partner node in the action clause. A textual representation of the graph rule can be produced by listing each node in the graph and each of the partner nodes that it is connected to, and a pair of nodes to describe the edge to be filled. For example describes a rule that will match a collection of nodes, or board squares… \end{verbatim} ``` </details>
907. ph-8c260682bfcb175f65ae — automata/docling_md/Computational Inteligence for Modelling Control.md ### Plain (markdown context) t has been used, or the expected utility from using the agent. When an agent is created this is given an initial constant value, Q initial . To determine which agent rule should act at a given time, each agent is examined to see if its rule matches the given game state, and the winning agent is selected probabilistically from the set of matching agents. The probability of each is determined from the Q value using a Boltzmann distribution according to the following formula, as described in [7]. where x i is the probability agent i is chosen, β is a constant control… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{t has been used, or the expected utility from using the agent. When an agent is created this is given an initial constant value, Q initial . To determine which agent rule should act at a given time, each agent is examined to see if its rule matches the given game state, and the …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/Computational Inteligence for Modelling Control.md:offset=13494 \begin{verbatim} t has been used, or the expected utility from using the agent. When an agent is created this is given an initial constant value, Q initial . To determine which agent rule should act at a given time, each agent is examined to see if its rule matches the given game state, and the winning agent is selected probabilistically from the set of matching agents. The probability of each is determined from the Q value using a Boltzmann distribution according to the following formula, as described in [7]. where x i is the probability agent i is chosen, β is a constant control… \end{verbatim} ``` </details>
908. ph-4e2b8cea2c9bc296a8dc — automata/docling_md/Computational Inteligence for Modelling Control.md ### Plain (markdown context) st action, N is size of the set of matching agents, and Q n is the utility value of agent n , where agent n is a member of the matching agent set. When a game is completed, each agent that performed an action in the game is given a reward. Each is rewarded ![Image](./Computational Inteligence for Modelling Control_artifacts/image_000002_d7351d8cc9ac5bb7ac2316251345f9fc253ffb66c7fbe459991f4d321b60e481.png) equally and the Q value of each is updated using the following formula, also from [7]: where r is the reward given and α and T are constants. Agents which did no… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{st action, N is size of the set of matching agents, and Q n is the utility value of agent n , where agent n is a member of the matching agent set. When a game is completed, each agent that performed an action in the game is given a reward. Each is rewarded ![Image](./Computation…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/Computational Inteligence for Modelling Control.md:offset=14157 \begin{verbatim} st action, N is size of the set of matching agents, and Q n is the utility value of agent n , where agent n is a member of the matching agent set. When a game is completed, each agent that performed an action in the game is given a reward. Each is rewarded ![Image](./Computational Inteligence for Modelling Control_artifacts/image_000002_d7351d8cc9ac5bb7ac2316251345f9fc253ffb66c7fbe459991f4d321b60e481.png) equally and the Q value of each is updated using the following formula, also from [7]: where r is the reward given and α and T are constants. Agents which did no… \end{verbatim} ``` </details>
909. ph-f529850eb6ff3ee20bfb — automata/docling_md/Computational Inteligence for Modelling Control.md ### Plain (markdown context) eted, each agent that performed an action in the game is given a reward. Each is rewarded ![Image](./Computational Inteligence for Modelling Control_artifacts/image_000002_d7351d8cc9ac5bb7ac2316251345f9fc253ffb66c7fbe459991f4d321b60e481.png) equally and the Q value of each is updated using the following formula, also from [7]: where r is the reward given and α and T are constants. Agents which did not act during the game are updated using the following formula: which is identical to equation 2 without the reward. A number of reward met… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{eted, each agent that performed an action in the game is given a reward. Each is rewarded ![Image](./Computational Inteligence for Modelling Control\_artifacts/image\_000002\_d7351d8cc9ac5bb7ac2316251345f9fc253ffb66c7fbe459991f4d321b60e481.png) equally and the Q value of each is up…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/Computational Inteligence for Modelling Control.md:offset=14325 \begin{verbatim} eted, each agent that performed an action in the game is given a reward. Each is rewarded ![Image](./Computational Inteligence for Modelling Control_artifacts/image_000002_d7351d8cc9ac5bb7ac2316251345f9fc253ffb66c7fbe459991f4d321b60e481.png) equally and the Q value of each is updated using the following formula, also from [7]: where r is the reward given and α and T are constants. Agents which did not act during the game are updated using the following formula: which is identical to equation 2 without the reward. A number of reward met… \end{verbatim} ``` </details>
910. ph-18369a0700f106880c1e — automata/docling_md/indexed grammars and logic programs.md ### Plain (markdown context) for indexed languages based on the constructions presented before. Possibilities and limitations of the approach are discussed. ## 2. DEFINITION OF INDEXED GRAMMARS An indexedgrammar G is a 5tuple G = (N, T, F, R, S) where N, T, F are alphabets whose elements are called nonterminals, terminals, and indices (or flags), respectively. A finite set Rf of index rules A + w (A E N, w E (T U N)* ) is associated to each index SE F. R is a finite set of rewriting rules whereAEN,x,EN~T,z~EF*,andz,=Eifx~ET(i<m).SENiscalledthe initial nonterminal. Let M = NF* and K = (A4 U… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{for indexed languages based on the constructions presented before. Possibilities and limitations of the approach are discussed. \#\# 2. DEFINITION OF INDEXED GRAMMARS An indexedgrammar G is a 5tuple G = (N, T, F, R, S) where N, T, F are alphabets whose elements are called nontermi…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/indexed grammars and logic programs.md:offset=5310 \begin{verbatim} for indexed languages based on the constructions presented before. Possibilities and limitations of the approach are discussed. ## 2. DEFINITION OF INDEXED GRAMMARS An indexedgrammar G is a 5tuple G = (N, T, F, R, S) where N, T, F are alphabets whose elements are called nonterminals, terminals, and indices (or flags), respectively. A finite set Rf of index rules A + w (A E N, w E (T U N)* ) is associated to each index SE F. R is a finite set of rewriting rules whereAEN,x,EN~T,z~EF*,andz,=Eifx~ET(i<m).SENiscalledthe initial nonterminal. Let M = NF* and K = (A4 U… \end{verbatim} ``` </details>
911. ph-02eae59e424fdd88214a — automata/docling_md/indexed grammars and logic programs.md ### Plain (markdown context) T, F are alphabets whose elements are called nonterminals, terminals, and indices (or flags), respectively. A finite set Rf of index rules A + w (A E N, w E (T U N)* ) is associated to each index SE F. R is a finite set of rewriting rules whereAEN,x,EN~T,z~EF*,andz,=Eifx~ET(i<m).SENiscalledthe initial nonterminal. Let M = NF* and K = (A4 U T>* as standard notations throughout this paper. We write x + ,y for x, y E K if either The reflexive, transitive closure *o of jG combined with the notion of termi… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{T, F are alphabets whose elements are called nonterminals, terminals, and indices (or flags), respectively. A finite set Rf of index rules A + w (A E N, w E (T U N)* ) is associated to each index SE F. R is a finite set of rewriting rules whereAEN,x,…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/indexed grammars and logic programs.md:offset=5551 \begin{verbatim} T, F are alphabets whose elements are called nonterminals, terminals, and indices (or flags), respectively. A finite set Rf of index rules A + w (A E N, w E (T U N)* ) is associated to each index SE F. R is a finite set of rewriting rules whereAEN,x,EN~T,z~EF*,andz,=Eifx~ET(i<m).SENiscalledthe initial nonterminal. Let M = NF* and K = (A4 U T>* as standard notations throughout this paper. We write x + ,y for x, y E K if either The reflexive, transitive closure *o of jG combined with the notion of termi… \end{verbatim} ``` </details>
912. ph-04757e552dd4807999c2 — automata/docling_md/indexed grammars and logic programs.md ### Plain (markdown context) lements are called nonterminals, terminals, and indices (or flags), respectively. A finite set Rf of index rules A + w (A E N, w E (T U N)* ) is associated to each index SE F. R is a finite set of rewriting rules whereAEN,x,EN~T,z~EF*,andz,=Eifx~ET(i<m).SENiscalledthe initial nonterminal. Let M = NF* and K = (A4 U T>* as standard notations throughout this paper. We write x + ,y for x, y E K if either The reflexive, transitive closure *o of jG combined with the notion of terminal symbol gives us the la… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{lements are called nonterminals, terminals, and indices (or flags), respectively. A finite set Rf of index rules A + w (A E N, w E (T U N)* ) is associated to each index SE F. R is a finite set of rewriting rules whereAEN,x,EN~T,z~EF*,andz,=Eifx~ET(i…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/indexed grammars and logic programs.md:offset=5581 \begin{verbatim} lements are called nonterminals, terminals, and indices (or flags), respectively. A finite set Rf of index rules A + w (A E N, w E (T U N)* ) is associated to each index SE F. R is a finite set of rewriting rules whereAEN,x,EN~T,z~EF*,andz,=Eifx~ET(i<m).SENiscalledthe initial nonterminal. Let M = NF* and K = (A4 U T>* as standard notations throughout this paper. We write x + ,y for x, y E K if either The reflexive, transitive closure *o of jG combined with the notion of terminal symbol gives us the la… \end{verbatim} ``` </details>
913. ph-67422a21750431dec94e — automata/docling_md/indexed grammars and logic programs.md ### Plain (markdown context) ed to each index SE F. R is a finite set of rewriting rules whereAEN,x,EN~T,z~EF*,andz,=Eifx~ET(i<m).SENiscalledthe initial nonterminal. Let M = NF* and K = (A4 U T>* as standard notations throughout this paper. We write x + ,y for x, y E K if either The reflexive, transitive closure *o of jG combined with the notion of terminal symbol gives us the language of a grammar To discuss intermediate stages of indexed derivations, we shall make use of the set of sentential forms … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ed to each index SE F. R is a finite set of rewriting rules whereAEN,x,EN~T,z~EF*,andz,=Eifx~ET(i\<m).SENiscalledthe initial nonterminal. Let M = NF* and K = (A4 U T\>* as standard notations throughout this paper. We write x + ,y for x, y E K if …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/indexed grammars and logic programs.md:offset=5743 \begin{verbatim} ed to each index SE F. R is a finite set of rewriting rules whereAEN,x,EN~T,z~EF*,andz,=Eifx~ET(i<m).SENiscalledthe initial nonterminal. Let M = NF* and K = (A4 U T>* as standard notations throughout this paper. We write x + ,y for x, y E K if either The reflexive, transitive closure *o of jG combined with the notion of terminal symbol gives us the language of a grammar To discuss intermediate stages of indexed derivations, we shall make use of the set of sentential forms … \end{verbatim} ``` </details>
914. ph-29d0023f00a9e50875ef — automata/docling_md/indexed grammars and logic programs.md ### Plain (markdown context) Niscalledthe initial nonterminal. Let M = NF* and K = (A4 U T>* as standard notations throughout this paper. We write x + ,y for x, y E K if either The reflexive, transitive closure *o of jG combined with the notion of terminal symbol gives us the language of a grammar To discuss intermediate stages of indexed derivations, we shall make use of the set of sentential forms and the set of terminal-free sentential forms Furthermore, for subsets N' of … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{Niscalledthe initial nonterminal. Let M = NF* and K = (A4 U T\>* as standard notations throughout this paper. We write x + ,y for x, y E K if either The reflexive, transitive closure *o of jG combined with the notion of…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/indexed grammars and logic programs.md:offset=5886 \begin{verbatim} Niscalledthe initial nonterminal. Let M = NF* and K = (A4 U T>* as standard notations throughout this paper. We write x + ,y for x, y E K if either The reflexive, transitive closure *o of jG combined with the notion of terminal symbol gives us the language of a grammar To discuss intermediate stages of indexed derivations, we shall make use of the set of sentential forms and the set of terminal-free sentential forms Furthermore, for subsets N' of … \end{verbatim} ``` </details>
915. ph-7be57ea3c99c35768e4a — automata/docling_md/indexed grammars and logic programs.md ### Plain (markdown context) ard notations throughout this paper. We write x + ,y for x, y E K if either The reflexive, transitive closure *o of jG combined with the notion of terminal symbol gives us the language of a grammar To discuss intermediate stages of indexed derivations, we shall make use of the set of sentential forms and the set of terminal-free sentential forms Furthermore, for subsets N' of N, it is useful to have the following subsets of Sent(G). The reflexive, transitive closure *o of jG combined with the notion of terminal symbol gives us the language of a grammar The reflexive, transitive closure *o of jG combined with the notion of terminal symbol gives us the language of a grammar To discuss intermediate stages of indexed derivations, we shall make use of the set of sentential forms and the set of terminal-free sentential forms Furthermore, for subsets N' of N, it is useful to have the following subsets of Sent(G). The reflexive, transitive closure *o of jG combined with the notion of terminal symbol gives us the language of a grammar To discuss intermediate stages of indexed derivations, we shall make use of the set of sentential forms and the set of terminal-free sentential forms Furthermore, for subsets N' of N, it is useful to have the following subsets of Sent(G). As an informal explanation of the derivation process in indexed grammars, please note that an index rule … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{-not-decoded --> The reflexive, transitive closure *o of jG combined with the notion of terminal symbol gives us the language of a grammar To discuss intermediate stages of indexed derivations, we shall make use of the set of sentential forms The reflexive, transitive closure *o of jG combined with the notion of terminal symbol gives us the language of a grammar To discuss intermediate stages of indexed derivations, we shall make use of the set of sentential forms and the set of terminal-free sentential forms Furthermore, for subsets N' of N, it is useful to have the following subsets of Sent(G). As an informal explanation of the derivation process in indexed grammars, please note that an index rule … \end{verbatim} ```
917. ph-4c02b17fef6fd394e426 — automata/docling_md/indexed grammars and logic programs.md ### Plain (markdown context) of stacking control information for later derivation steps. In [2], various interesting properties of indexed languages are demonstrated. [3] gives an automata-theoretic characterization. An efficient parsing procedure for a subclass characterized in terms of unambiguity conditions is presented in [41. A standard example of a noncontext-free language will serve to illustrate the core idea: The language L = {a'b'a'ln r 1) is generated by the indexed grammar with index rule sets The flags f serve as a counter of unindexed derivation step… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{of stacking control information for later derivation steps. In [2], various interesting properties of indexed languages are demonstrated. [3] gives an automata-theoretic characterization. An efficient parsing procedure for a subclass characterized in terms of unambiguity conditi…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/indexed grammars and logic programs.md:offset=6982 \begin{verbatim} of stacking control information for later derivation steps. In [2], various interesting properties of indexed languages are demonstrated. [3] gives an automata-theoretic characterization. An efficient parsing procedure for a subclass characterized in terms of unambiguity conditions is presented in [41. A standard example of a noncontext-free language will serve to illustrate the core idea: The language L = {a'b'a'ln r 1) is generated by the indexed grammar with index rule sets The flags f serve as a counter of unindexed derivation step… \end{verbatim} ``` </details>
918. ph-02184b4bdd1787ad805e — automata/docling_md/indexed grammars and logic programs.md ### Plain (markdown context) n steps. In [2], various interesting properties of indexed languages are demonstrated. [3] gives an automata-theoretic characterization. An efficient parsing procedure for a subclass characterized in terms of unambiguity conditions is presented in [41. A standard example of a noncontext-free language will serve to illustrate the core idea: The language L = {a'b'a'ln r 1) is generated by the indexed grammar with index rule sets The flags f serve as a counter of unindexed derivation steps. When T has been rewritten as U, fs can be taken … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{n steps. In [2], various interesting properties of indexed languages are demonstrated. [3] gives an automata-theoretic characterization. An efficient parsing procedure for a subclass characterized in terms of unambiguity conditions is presented in [41. A standard example of a no…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/indexed grammars and logic programs.md:offset=7037 \begin{verbatim} n steps. In [2], various interesting properties of indexed languages are demonstrated. [3] gives an automata-theoretic characterization. An efficient parsing procedure for a subclass characterized in terms of unambiguity conditions is presented in [41. A standard example of a noncontext-free language will serve to illustrate the core idea: The language L = {a'b'a'ln r 1) is generated by the indexed grammar with index rule sets The flags f serve as a counter of unindexed derivation steps. When T has been rewritten as U, fs can be taken … \end{verbatim} ``` </details>
919. ph-6e06ebe1e8364d801597 — automata/docling_md/indexed grammars and logic programs.md ### Plain (markdown context) defined in the following straightforward way. A variable X is a unay term. If t is a unary term and f is a unary function symbol, then f(t) is a unay term. We also say that f, t, and the terms and function symbols occurring in t occur in f(t). If t is a unary term and p is a unary predicate symbol, then p(t) is a unay atom. We also say that p, t, and the terms and function symbols occurring in t OCCUY in p(t). For unary atoms A, B,, . . . , B,, is a unay clause if the same variable, say X, occurs in all atoms. (The body B ,, . . . , B, may be empty.> A unay pro… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{defined in the following straightforward way. A variable X is a unay term. If t is a unary term and f is a unary function symbol, then f(t) is a unay term. We also say that f, t, and the terms and function symbols occurring in t occur in f(t). If t is a unary term and p is a una…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/indexed grammars and logic programs.md:offset=7948 \begin{verbatim} defined in the following straightforward way. A variable X is a unay term. If t is a unary term and f is a unary function symbol, then f(t) is a unay term. We also say that f, t, and the terms and function symbols occurring in t occur in f(t). If t is a unary term and p is a unary predicate symbol, then p(t) is a unay atom. We also say that p, t, and the terms and function symbols occurring in t OCCUY in p(t). For unary atoms A, B,, . . . , B,, is a unay clause if the same variable, say X, occurs in all atoms. (The body B ,, . . . , B, may be empty.> A unay pro… \end{verbatim} ``` </details>
920. ph-807b82a62f1a302b934c — automata/docling_md/indexed grammars and logic programs.md ### Plain (markdown context) and Pred(P) denote the set of function symbols and predicate symbols, respectively, which occur in the atoms of its clauses. To discuss goals, subgoals, and SLD-derivations, we shall use a unique constant term c. Please note that c is not a component of unary programs. It will occur only in goals which are processed by applying unary clauses. Given sets Func and Pred containing unary function and predicate symbols, respectively, every ground atom has the form for f,,..., f, E Func, a E Pred (including a(c) as a special case), and every ground goal has the form <!-… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{and Pred(P) denote the set of function symbols and predicate symbols, respectively, which occur in the atoms of its clauses. To discuss goals, subgoals, and SLD-derivations, we shall use a unique constant term c. Please note that c is not a component of unary programs. It will o…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/indexed grammars and logic programs.md:offset=8687 \begin{verbatim} and Pred(P) denote the set of function symbols and predicate symbols, respectively, which occur in the atoms of its clauses. To discuss goals, subgoals, and SLD-derivations, we shall use a unique constant term c. Please note that c is not a component of unary programs. It will occur only in goals which are processed by applying unary clauses. Given sets Func and Pred containing unary function and predicate symbols, respectively, every ground atom has the form for f,,..., f, E Func, a E Pred (including a(c) as a special case), and every ground goal has the form <!-… \end{verbatim} ``` </details>
921. ph-d8e9009a377972229235 — automata/docling_md/indexed grammars and logic programs.md ### Plain (markdown context) iscuss goals, subgoals, and SLD-derivations, we shall use a unique constant term c. Please note that c is not a component of unary programs. It will occur only in goals which are processed by applying unary clauses. Given sets Func and Pred containing unary function and predicate symbols, respectively, every ground atom has the form for f,,..., f, E Func, a E Pred (including a(c) as a special case), and every ground goal has the form where P,, . \_ . , P,,, are ground atoms (including the empty goal as a special case). Let us call the … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{iscuss goals, subgoals, and SLD-derivations, we shall use a unique constant term c. Please note that c is not a component of unary programs. It will occur only in goals which are processed by applying unary clauses. Given sets Func and Pred containing unary function and predicat…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/indexed grammars and logic programs.md:offset=8827 \begin{verbatim} iscuss goals, subgoals, and SLD-derivations, we shall use a unique constant term c. Please note that c is not a component of unary programs. It will occur only in goals which are processed by applying unary clauses. Given sets Func and Pred containing unary function and predicate symbols, respectively, every ground atom has the form for f,,..., f, E Func, a E Pred (including a(c) as a special case), and every ground goal has the form where P,, . \_ . , P,,, are ground atoms (including the empty goal as a special case). Let us call the … \end{verbatim} ``` </details>
922. ph-80e05551cd944a3e3b0d — automata/docling_md/indexed grammars and logic programs.md ### Plain (markdown context) in indexed grammars. Partial solutions to this problem are offered in a later section. ## 5. SIMULATING INDEXED GRAMMARS BY LOGIC PROGRAMS In this section, we describe a simple construction which yields a unique definite logic program P(G) for a given indexed grammar G. We claim that P(G) simulates the generative capability of G. Proposition 1. Let G = (N, T, F, R, S) be an indexed grammar. Then there exists a unary program P(G) and a bijective mapping such that if and only if PROOF. To avoid notational ove… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{in indexed grammars. Partial solutions to this problem are offered in a later section. \#\# 5. SIMULATING INDEXED GRAMMARS BY LOGIC PROGRAMS In this section, we describe a simple construction which yields a unique definite logic program P(G) for a given indexed grammar G. We claim…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/indexed grammars and logic programs.md:offset=13462 \begin{verbatim} in indexed grammars. Partial solutions to this problem are offered in a later section. ## 5. SIMULATING INDEXED GRAMMARS BY LOGIC PROGRAMS In this section, we describe a simple construction which yields a unique definite logic program P(G) for a given indexed grammar G. We claim that P(G) simulates the generative capability of G. Proposition 1. Let G = (N, T, F, R, S) be an indexed grammar. Then there exists a unary program P(G) and a bijective mapping such that if and only if PROOF. To avoid notational ove… \end{verbatim} ``` </details>
923. ph-090c82fc12fbcc30d6a9 — automata/docling_md/indexed grammars and logic programs.md ### Plain (markdown context) to this problem are offered in a later section. ## 5. SIMULATING INDEXED GRAMMARS BY LOGIC PROGRAMS In this section, we describe a simple construction which yields a unique definite logic program P(G) for a given indexed grammar G. We claim that P(G) simulates the generative capability of G. Proposition 1. Let G = (N, T, F, R, S) be an indexed grammar. Then there exists a unary program P(G) and a bijective mapping such that if and only if PROOF. To avoid notational overloading, we distinguish between a bar-… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{to this problem are offered in a later section. \#\# 5. SIMULATING INDEXED GRAMMARS BY LOGIC PROGRAMS In this section, we describe a simple construction which yields a unique definite logic program P(G) for a given indexed grammar G. We claim that P(G) simulates the generative cap…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/indexed grammars and logic programs.md:offset=13504 \begin{verbatim} to this problem are offered in a later section. ## 5. SIMULATING INDEXED GRAMMARS BY LOGIC PROGRAMS In this section, we describe a simple construction which yields a unique definite logic program P(G) for a given indexed grammar G. We claim that P(G) simulates the generative capability of G. Proposition 1. Let G = (N, T, F, R, S) be an indexed grammar. Then there exists a unary program P(G) and a bijective mapping such that if and only if PROOF. To avoid notational overloading, we distinguish between a bar-… \end{verbatim} ``` </details>
924. ph-8fc8e67c5d9960d52581 — automata/docling_md/indexed grammars and logic programs.md ### Plain (markdown context) . ## 5. SIMULATING INDEXED GRAMMARS BY LOGIC PROGRAMS In this section, we describe a simple construction which yields a unique definite logic program P(G) for a given indexed grammar G. We claim that P(G) simulates the generative capability of G. Proposition 1. Let G = (N, T, F, R, S) be an indexed grammar. Then there exists a unary program P(G) and a bijective mapping such that if and only if PROOF. To avoid notational overloading, we distinguish between a bar-mapping p defined on an alphabet of symbols an… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{. \#\# 5. SIMULATING INDEXED GRAMMARS BY LOGIC PROGRAMS In this section, we describe a simple construction which yields a unique definite logic program P(G) for a given indexed grammar G. We claim that P(G) simulates the generative capability of G. Proposition 1. Let G = (N, T, F,…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/indexed grammars and logic programs.md:offset=13552 \begin{verbatim} . ## 5. SIMULATING INDEXED GRAMMARS BY LOGIC PROGRAMS In this section, we describe a simple construction which yields a unique definite logic program P(G) for a given indexed grammar G. We claim that P(G) simulates the generative capability of G. Proposition 1. Let G = (N, T, F, R, S) be an indexed grammar. Then there exists a unary program P(G) and a bijective mapping such that if and only if PROOF. To avoid notational overloading, we distinguish between a bar-mapping p defined on an alphabet of symbols an… \end{verbatim} ``` </details>
925. ph-a815d9a03ab50cad7035 — automata/docling_md/indexed grammars and logic programs.md ### Plain (markdown context) every flag k E F. We call the set of all such predicate symbols Pred, and the set of all such function symbols Func. Please note that we are leaving terminal symbols out of consideration. The results about smooth grammars in the special section on terminal symbols indicate the importance of this restriction. A brief discussion of the treatment of terminals in a parsing framework is given in the final part of the paper. We construct a bijective mapping by first letting for si = ak, . . . k, E M and fixed constant c and extending c to p … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{every flag k E F. We call the set of all such predicate symbols Pred, and the set of all such function symbols Func. Please note that we are leaving terminal symbols out of consideration. The results about smooth grammars in the special section on terminal symbols indicate the i…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/indexed grammars and logic programs.md:offset=14500 \begin{verbatim} every flag k E F. We call the set of all such predicate symbols Pred, and the set of all such function symbols Func. Please note that we are leaving terminal symbols out of consideration. The results about smooth grammars in the special section on terminal symbols indicate the importance of this restriction. A brief discussion of the treatment of terminals in a parsing framework is given in the final part of the paper. We construct a bijective mapping by first letting for si = ak, . . . k, E M and fixed constant c and extending c to p … \end{verbatim} ``` </details>
926. ph-3e08c4c39c1db6d05cda — automata/docling_md/indexed grammars and logic programs.md ### Plain (markdown context) l such predicate symbols Pred, and the set of all such function symbols Func. Please note that we are leaving terminal symbols out of consideration. The results about smooth grammars in the special section on terminal symbols indicate the importance of this restriction. A brief discussion of the treatment of terminals in a parsing framework is given in the final part of the paper. We construct a bijective mapping by first letting for si = ak, . . . k, E M and fixed constant c and extending c to p by p(sl.. . s,) = :- j.xs,>, . . .) … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{l such predicate symbols Pred, and the set of all such function symbols Func. Please note that we are leaving terminal symbols out of consideration. The results about smooth grammars in the special section on terminal symbols indicate the importance of this restriction. A brief …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/indexed grammars and logic programs.md:offset=14548 \begin{verbatim} l such predicate symbols Pred, and the set of all such function symbols Func. Please note that we are leaving terminal symbols out of consideration. The results about smooth grammars in the special section on terminal symbols indicate the importance of this restriction. A brief discussion of the treatment of terminals in a parsing framework is given in the final part of the paper. We construct a bijective mapping by first letting for si = ak, . . . k, E M and fixed constant c and extending c to p by p(sl.. . s,) = :- j.xs,>, . . .) … \end{verbatim} ``` </details>
927. ph-72e9b7e773b103884ca4 — automata/docling_md/indexed grammars and logic programs.md ### Plain (markdown context) by first letting for si = ak, . . . k, E M and fixed constant c and extending c to p by p(sl.. . s,) = :- j.xs,>, . . .) jXs,> where si EM (1 I i < n). As a special case, we let P.(E) be the empty goal (:- ). The decomposition of words in M* is obviously well defined because each S, starts with a single nonterminal. We need to consider two kinds of rules. Let us first turn to rewriting rules (Rules producing terminal symbols do not need to be considered here because they lead outside of M*.) Each z… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ by first letting for si = ak, . . . k, E M and fixed constant c and extending c to p by p(sl.. . s,) = :- j.xs,\>, . . .) jXs,\> where si EM (1 I i \< n). As a special case, we let P.(E) be the empty goal (:- ). The…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/indexed grammars and logic programs.md:offset=15000 \begin{verbatim} by first letting for si = ak, . . . k, E M and fixed constant c and extending c to p by p(sl.. . s,) = :- j.xs,>, . . .) jXs,> where si EM (1 I i < n). As a special case, we let P.(E) be the empty goal (:- ). The decomposition of words in M* is obviously well defined because each S, starts with a single nonterminal. We need to consider two kinds of rules. Let us first turn to rewriting rules (Rules producing terminal symbols do not need to be considered here because they lead outside of M*.) Each z… \end{verbatim} ``` </details>
928. ph-a2ea5cd1a73889e4b981 — automata/docling_md/indexed grammars and logic programs.md ### Plain (markdown context) the empty goal (:- ). The decomposition of words in M* is obviously well defined because each S, starts with a single nonterminal. We need to consider two kinds of rules. Let us first turn to rewriting rules (Rules producing terminal symbols do not need to be considered here because they lead outside of M*.) Each zi is composed of individual flag symbols zr,,, . . ., zi, n, or is the empty string. Then we construct the clause where p:cZi (1 I i 2 m) stands for or else Empty righ… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{the empty goal (:- ). The decomposition of words in M* is obviously well defined because each S, starts with a single nonterminal. We need to consider two kinds of rules. Let us first turn to rewriting rules (Rules producing terminal symbols do not n…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/indexed grammars and logic programs.md:offset=15270 \begin{verbatim} the empty goal (:- ). The decomposition of words in M* is obviously well defined because each S, starts with a single nonterminal. We need to consider two kinds of rules. Let us first turn to rewriting rules (Rules producing terminal symbols do not need to be considered here because they lead outside of M*.) Each zi is composed of individual flag symbols zr,,, . . ., zi, n, or is the empty string. Then we construct the clause where p:cZi (1 I i 2 m) stands for or else Empty righ… \end{verbatim} ``` </details>
929. ph-be2817c4787a37d8f44f — automata/docling_md/indexed grammars and logic programs.md ### Plain (markdown context) usly well defined because each S, starts with a single nonterminal. We need to consider two kinds of rules. Let us first turn to rewriting rules (Rules producing terminal symbols do not need to be considered here because they lead outside of M*.) Each zi is composed of individual flag symbols zr,,, . . ., zi, n, or is the empty string. Then we construct the clause where p:cZi (1 I i 2 m) stands for or else Empty right sides of rules r do not require special treatment. Thus, each… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{usly well defined because each S, starts with a single nonterminal. We need to consider two kinds of rules. Let us first turn to rewriting rules (Rules producing terminal symbols do not need to be considered here because they lead outside of M*.) Eac…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/indexed grammars and logic programs.md:offset=15340 \begin{verbatim} usly well defined because each S, starts with a single nonterminal. We need to consider two kinds of rules. Let us first turn to rewriting rules (Rules producing terminal symbols do not need to be considered here because they lead outside of M*.) Each zi is composed of individual flag symbols zr,,, . . ., zi, n, or is the empty string. Then we construct the clause where p:cZi (1 I i 2 m) stands for or else Empty right sides of rules r do not require special treatment. Thus, each… \end{verbatim} ``` </details>
930. ph-c68430f8f409c1cf8f5d — automata/docling_md/indexed grammars and logic programs.md ### Plain (markdown context) arts with a single nonterminal. We need to consider two kinds of rules. Let us first turn to rewriting rules (Rules producing terminal symbols do not need to be considered here because they lead outside of M*.) Each zi is composed of individual flag symbols zr,,, . . ., zi, n, or is the empty string. Then we construct the clause where p:cZi (1 I i 2 m) stands for or else Empty right sides of rules r do not require special treatment. Thus, each nonempty flag sequence is coded by … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{arts with a single nonterminal. We need to consider two kinds of rules. Let us first turn to rewriting rules (Rules producing terminal symbols do not need to be considered here because they lead outside of M*.) Each zi is composed of individual flag …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/indexed grammars and logic programs.md:offset=15379 \begin{verbatim} arts with a single nonterminal. We need to consider two kinds of rules. Let us first turn to rewriting rules (Rules producing terminal symbols do not need to be considered here because they lead outside of M*.) Each zi is composed of individual flag symbols zr,,, . . ., zi, n, or is the empty string. Then we construct the clause where p:cZi (1 I i 2 m) stands for or else Empty right sides of rules r do not require special treatment. Thus, each nonempty flag sequence is coded by … \end{verbatim} ``` </details>
931. ph-a237df50c8f3be5ad856 — automata/docling_md/indexed grammars and logic programs.md ### Plain (markdown context) ,,, . . ., zi, n, or is the empty string. Then we construct the clause where p:cZi (1 I i 2 m) stands for or else Empty right sides of rules r do not require special treatment. Thus, each nonempty flag sequence is coded by a term containing unary function symbols. The variable X occurs in each such term at the innermost level of nesting. On the other hand, index rules lead to the construction of clauses p(r,) of the form Here, the flag symbol which i… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{,,, . . ., zi, n, or is the empty string. Then we construct the clause where p:cZi (1 I i 2 m) stands for or else Empty right sides of rules r do not require special treatment. Thus, each none…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/indexed grammars and logic programs.md:offset=15694 \begin{verbatim} ,,, . . ., zi, n, or is the empty string. Then we construct the clause where p:cZi (1 I i 2 m) stands for or else Empty right sides of rules r do not require special treatment. Thus, each nonempty flag sequence is coded by a term containing unary function symbols. The variable X occurs in each such term at the innermost level of nesting. On the other hand, index rules lead to the construction of clauses p(r,) of the form Here, the flag symbol which i… \end{verbatim} ``` </details>
932. ph-7b8922799cb4ad5d946f — automata/docling_md/indexed grammars and logic programs.md ### Plain (markdown context) decoded --> where p:cZi (1 I i 2 m) stands for or else Empty right sides of rules r do not require special treatment. Thus, each nonempty flag sequence is coded by a term containing unary function symbols. The variable X occurs in each such term at the innermost level of nesting. On the other hand, index rules lead to the construction of clauses p(r,) of the form Here, the flag symbol which is used up in applications of the rule rf appears as a single function symbol in the term… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{decoded --> where p:cZi (1 I i 2 m) stands for or else Empty right sides of rules r do not require special treatment. Thus, each nonempty flag sequence is coded by a term containing unary function symbols. The variable X …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/indexed grammars and logic programs.md:offset=15787 \begin{verbatim} decoded --> where p:cZi (1 I i 2 m) stands for or else Empty right sides of rules r do not require special treatment. Thus, each nonempty flag sequence is coded by a term containing unary function symbols. The variable X occurs in each such term at the innermost level of nesting. On the other hand, index rules lead to the construction of clauses p(r,) of the form Here, the flag symbol which is used up in applications of the rule rf appears as a single function symbol in the term… \end{verbatim} ``` </details>
933. ph-0eb54acb4cc2ab5ccc6f — automata/docling_md/indexed grammars and logic programs.md ### Plain (markdown context) lead to a goal which is the image of a string derived by one application of some rule of the indexed grammar from the previous string. The induction on the number of derivation steps contains a case distinction depending on whether flags are added or taken off. Suppose a string x = wlwu (u, v EM*, A EN, w E F') has been derived in no more than k steps and is rewritten by a rule r = (A +x,zl.. . x,z,). The result is (by definition of the rewriting process) where zj=ziw (1 li<m). In p.(x), we consider the subgoal fi(Aw). From the induction hypothesis, we get that… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{lead to a goal which is the image of a string derived by one application of some rule of the indexed grammar from the previous string. The induction on the number of derivation steps contains a case distinction depending on whether flags are added or taken off. Suppose a string …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/indexed grammars and logic programs.md:offset=17496 \begin{verbatim} lead to a goal which is the image of a string derived by one application of some rule of the indexed grammar from the previous string. The induction on the number of derivation steps contains a case distinction depending on whether flags are added or taken off. Suppose a string x = wlwu (u, v EM*, A EN, w E F') has been derived in no more than k steps and is rewritten by a rule r = (A +x,zl.. . x,z,). The result is (by definition of the rewriting process) where zj=ziw (1 li<m). In p.(x), we consider the subgoal fi(Aw). From the induction hypothesis, we get that… \end{verbatim} ``` </details>
934. ph-5739b55ea97975de5165 — automata/docling_md/indexed grammars and logic programs.md ### Plain (markdown context) ation steps contains a case distinction depending on whether flags are added or taken off. Suppose a string x = wlwu (u, v EM*, A EN, w E F') has been derived in no more than k steps and is rewritten by a rule r = (A +x,zl.. . x,z,). The result is (by definition of the rewriting process) where zj=ziw (1 li<m). In p.(x), we consider the subgoal fi(Aw). From the induction hypothesis, we get that for w=w,... w,, (w~EF, 1 <isn) is a subgoal of p(x). Performing a resolution step with the clause p(r), the most general unifier of where zj=ziw (1 li<m). In p.(x), we consider the subgoal fi(Aw). From the induction hypothesis, we get that for w=w,... w,, (w~EF, 1 <isn) is a subgoal of p(x). Performing a resolution step with the clause p(r), the most general unifier of where zj=ziw (1 li<m). In p.(x), we consider the subgoal fi(Aw). From the induction hypothesis, we get that for w=w,... w,, (w~EF, 1 <isn) is a subgoal of p(x). Performing a resolution step with the clause p(r), the most general unifier of is given by substituting the term jXwi)(. . . in.. .> occurring in the latter atom for X. It follows that th… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{w E F') has been derived in no more than k steps and is rewritten by a rule r = (A +x,zl.. . x,z,). The result is (by definition of the rewriting process) where zj=ziw (1 li\<m). In p.(x), we consider the subgoal fi(Aw). From the induction hypothes…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/indexed grammars and logic programs.md:offset=17827 \begin{verbatim} w E F') has been derived in no more than k steps and is rewritten by a rule r = (A +x,zl.. . x,z,). The result is (by definition of the rewriting process) where zj=ziw (1 li<m). In p.(x), we consider the subgoal fi(Aw). From the induction hypothesis, we get that for w=w,... w,, (w~EF, 1 <isn) is a subgoal of p(x). Performing a resolution step with the clause p(r), the most general unifier of is given by substituting the term jXwi)(. . . in.. .> occurring in the latter atom for X. It follows that th… \end{verbatim} ``` </details>
936. ph-4cba47ac8910b0decfef — automata/docling_md/indexed grammars and logic programs.md ### Plain (markdown context) hypothesis, we get that for w=w,... w,, (w~EF, 1 <isn) is a subgoal of p(x). Performing a resolution step with the clause p(r), the most general unifier of is given by substituting the term jXwi)(. . . in.. .> occurring in the latter atom for X. It follows that the atoms in the body of p(r) become new subgoals after substituting the same term for their respective occurrences of X. This gives us subgoals for llirm. Applying our mapping p, the sequence of subgoals j.Xx,zj> (i = 1,. . . , ml results f… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{hypothesis, we get that for w=w,... w,, (w~EF, 1 \<isn) is a subgoal of p(x). Performing a resolution step with the clause p(r), the most general unifier of is given by substituting the term jXwi)(. . . in.. .\> occur…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/indexed grammars and logic programs.md:offset=18124 \begin{verbatim} hypothesis, we get that for w=w,... w,, (w~EF, 1 <isn) is a subgoal of p(x). Performing a resolution step with the clause p(r), the most general unifier of is given by substituting the term jXwi)(. . . in.. .> occurring in the latter atom for X. It follows that the atoms in the body of p(r) become new subgoals after substituting the same term for their respective occurrences of X. This gives us subgoals for llirm. Applying our mapping p, the sequence of subgoals j.Xx,zj> (i = 1,. . . , ml results f… \end{verbatim} ``` </details>
937. ph-30e10df006d30f5a80e4 — automata/docling_md/indexed grammars and logic programs.md ### Plain (markdown context) (N'F* )* such that gEDeriu(P,:n(c)) ESent(G,N'). ``` ifandonlyif /3(g) ``` PROOF. Similarly to the procedure used in showing Proposition 1, we introduce mappings on an alphabet, on the set of atomic subgoals, and on the set of goals, built successively on top of each other. We define p to map Bed(P) bijectively into a set ,N' of symbols and to map Func(P) bijectively into a set F of flags. Furthermore, p mapping atoms on strings in N'F' is constructed by letting for atoms of the form p, = s(t,(t,( . . . (t,(c)). . . >>I such that f,, t,, . . . , t, E FLUX, a… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{(N'F* )* such that gEDeriu(P,:n(c)) ESent(G,N'). ``` ifandonlyif /3(g) ``` PROOF. Similarly to the procedure used in showing Proposition 1, we introduce mappings on an alphabet, on the set of atomic subgoals, and on the set of goals, built successively on top of each other. We d…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/indexed grammars and logic programs.md:offset=22070 \begin{verbatim} (N'F* )* such that gEDeriu(P,:n(c)) ESent(G,N'). ``` ifandonlyif /3(g) ``` PROOF. Similarly to the procedure used in showing Proposition 1, we introduce mappings on an alphabet, on the set of atomic subgoals, and on the set of goals, built successively on top of each other. We define p to map Bed(P) bijectively into a set ,N' of symbols and to map Func(P) bijectively into a set F of flags. Furthermore, p mapping atoms on strings in N'F' is constructed by letting for atoms of the form p, = s(t,(t,( . . . (t,(c)). . . >>I such that f,, t,, . . . , t, E FLUX, a… \end{verbatim} ``` </details>
938. ph-691ae75ef7162254ca27 — automata/docling_md/indexed grammars and logic programs.md ### Plain (markdown context) ppings on an alphabet, on the set of atomic subgoals, and on the set of goals, built successively on top of each other. We define p to map Bed(P) bijectively into a set ,N' of symbols and to map Func(P) bijectively into a set F of flags. Furthermore, p mapping atoms on strings in N'F' is constructed by letting for atoms of the form p, = s(t,(t,( . . . (t,(c)). . . >>I such that f,, t,, . . . , t, E FLUX, and finally, we let where pi are atoms (1 -< i in). As a special case, the empty goal maps on the empty string. This gives u… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ppings on an alphabet, on the set of atomic subgoals, and on the set of goals, built successively on top of each other. We define p to map Bed(P) bijectively into a set ,N' of symbols and to map Func(P) bijectively into a set F of flags. Furthermore, p mapping atoms on strings i…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/indexed grammars and logic programs.md:offset=22233 \begin{verbatim} ppings on an alphabet, on the set of atomic subgoals, and on the set of goals, built successively on top of each other. We define p to map Bed(P) bijectively into a set ,N' of symbols and to map Func(P) bijectively into a set F of flags. Furthermore, p mapping atoms on strings in N'F' is constructed by letting for atoms of the form p, = s(t,(t,( . . . (t,(c)). . . >>I such that f,, t,, . . . , t, E FLUX, and finally, we let where pi are atoms (1 -< i in). As a special case, the empty goal maps on the empty string. This gives u… \end{verbatim} ``` </details>
939. ph-8bb3f00e6e07f6a0b5ef — automata/docling_md/indexed grammars and logic programs.md ### Plain (markdown context) (1 -< i in). As a special case, the empty goal maps on the empty string. This gives us a mapping: /3: Ground + (N'F' >*. Each string in (N'F*>* is uniquely decomposable into individual symbols and flags. For-each such item, there is a unique predicate or function symbol mapped on it by p. We construct the sets of rules and flag rules of an indexed grammar G(P) as follows: Let a unary program P be given. Suppose clause cl in P has the following form: Define in the set of flag rules I?P(~,,,~) a new rule in the sets of flag rule… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{(1 -\< i in). As a special case, the empty goal maps on the empty string. This gives us a mapping: /3: Ground + (N'F' \>*. Each string in (N'F*\>* is uniquely decomposable into individual symbols and flags. For-each such item, there is a unique predicate or function symbol…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/indexed grammars and logic programs.md:offset=22784 \begin{verbatim} (1 -< i in). As a special case, the empty goal maps on the empty string. This gives us a mapping: /3: Ground + (N'F' >*. Each string in (N'F*>* is uniquely decomposable into individual symbols and flags. For-each such item, there is a unique predicate or function symbol mapped on it by p. We construct the sets of rules and flag rules of an indexed grammar G(P) as follows: Let a unary program P be given. Suppose clause cl in P has the following form: Define in the set of flag rules I?P(~,,,~) a new rule in the sets of flag rule… \end{verbatim} ``` </details>
940. ph-3a0f10929cc186583323 — automata/docling_md/indexed grammars and logic programs.md ### Plain (markdown context) is gives us a mapping: /3: Ground + (N'F' >*. Each string in (N'F*>* is uniquely decomposable into individual symbols and flags. For-each such item, there is a unique predicate or function symbol mapped on it by p. We construct the sets of rules and flag rules of an indexed grammar G(P) as follows: Let a unary program P be given. Suppose clause cl in P has the following form: Define in the set of flag rules I?P(~,,,~) a new rule in the sets of flag rules RP(~,~,,) (2 I i I 2) new rules and finally, in… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{is gives us a mapping: /3: Ground + (N'F' \>*. Each string in (N'F*\>* is uniquely decomposable into individual symbols and flags. For-each such item, there is a unique predicate or function symbol mapped on it by p. We construct the sets of rules and flag rules of an indexe…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/indexed grammars and logic programs.md:offset=22873 \begin{verbatim} is gives us a mapping: /3: Ground + (N'F' >*. Each string in (N'F*>* is uniquely decomposable into individual symbols and flags. For-each such item, there is a unique predicate or function symbol mapped on it by p. We construct the sets of rules and flag rules of an indexed grammar G(P) as follows: Let a unary program P be given. Suppose clause cl in P has the following form: Define in the set of flag rules I?P(~,,,~) a new rule in the sets of flag rules RP(~,~,,) (2 I i I 2) new rules and finally, in… \end{verbatim} ``` </details>
941. ph-83e1a0527fc75a2f8dcf — automata/docling_md/indexed grammars and logic programs.md ### Plain (markdown context) ecomposable into individual symbols and flags. For-each such item, there is a unique predicate or function symbol mapped on it by p. We construct the sets of rules and flag rules of an indexed grammar G(P) as follows: Let a unary program P be given. Suppose clause cl in P has the following form: Define in the set of flag rules I?P(~,,,~) a new rule in the sets of flag rules RP(~,~,,) (2 I i I 2) new rules and finally, in the set of rules R a new rule The special case that the le… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ecomposable into individual symbols and flags. For-each such item, there is a unique predicate or function symbol mapped on it by p. We construct the sets of rules and flag rules of an indexed grammar G(P) as follows: Let a unary program P be given. Suppose clause cl in P has th…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/indexed grammars and logic programs.md:offset=22967 \begin{verbatim} ecomposable into individual symbols and flags. For-each such item, there is a unique predicate or function symbol mapped on it by p. We construct the sets of rules and flag rules of an indexed grammar G(P) as follows: Let a unary program P be given. Suppose clause cl in P has the following form: Define in the set of flag rules I?P(~,,,~) a new rule in the sets of flag rules RP(~,~,,) (2 I i I 2) new rules and finally, in the set of rules R a new rule The special case that the le… \end{verbatim} ``` </details>
942. ph-0bb67e3fbb4e4e9ffb7c — automata/docling_md/indexed grammars and logic programs.md ### Plain (markdown context) unique predicate or function symbol mapped on it by p. We construct the sets of rules and flag rules of an indexed grammar G(P) as follows: Let a unary program P be given. Suppose clause cl in P has the following form: Define in the set of flag rules I?P(~,,,~) a new rule in the sets of flag rules RP(~,~,,) (2 I i I 2) new rules and finally, in the set of rules R a new rule The special case that the left-hand side of cl contains no functionsymbol (p,(X)) is conveniently dealt wi… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{unique predicate or function symbol mapped on it by p. We construct the sets of rules and flag rules of an indexed grammar G(P) as follows: Let a unary program P be given. Suppose clause cl in P has the following form: Define in the set of flag rules…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/indexed grammars and logic programs.md:offset=23048 \begin{verbatim} unique predicate or function symbol mapped on it by p. We construct the sets of rules and flag rules of an indexed grammar G(P) as follows: Let a unary program P be given. Suppose clause cl in P has the following form: Define in the set of flag rules I?P(~,,,~) a new rule in the sets of flag rules RP(~,~,,) (2 I i I 2) new rules and finally, in the set of rules R a new rule The special case that the left-hand side of cl contains no functionsymbol (p,(X)) is conveniently dealt wi… \end{verbatim} ``` </details>
943. ph-28eebe1873280f7fede1 — automata/docling_md/indexed grammars and logic programs.md ### Plain (markdown context) defined to be the set of all sentential forms WE (N'F' >* that can be reached from S by S =x0 +o x1 . . . -jG x, = W Cm 2 1) such that no more than k 5 m elements of the sequence x1 . . . x, are contained in (N'F*)*. In this way, intermediate stages containing nonterminals not in N' are skipped in the step count. As an induction hypothesis, we now assume that the proposition holds for Deriu,(P :- V(C)) and Sent,(G(P), N'). Then for a given goal reached from the initial goal after k or fewer resolution steps, the sentential form P(G1) is indeed derivable from S … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{defined to be the set of all sentential forms WE (N'F' \>* that can be reached from S by S =x0 +o x1 . . . -jG x, = W Cm 2 1) such that no more than k 5 m elements of the sequence x1 . . . x, are contained in (N'F*)*. In this way, intermediate stages containing nonterminals no…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/indexed grammars and logic programs.md:offset=24507 \begin{verbatim} defined to be the set of all sentential forms WE (N'F' >* that can be reached from S by S =x0 +o x1 . . . -jG x, = W Cm 2 1) such that no more than k 5 m elements of the sequence x1 . . . x, are contained in (N'F*)*. In this way, intermediate stages containing nonterminals not in N' are skipped in the step count. As an induction hypothesis, we now assume that the proposition holds for Deriu,(P :- V(C)) and Sent,(G(P), N'). Then for a given goal reached from the initial goal after k or fewer resolution steps, the sentential form P(G1) is indeed derivable from S … \end{verbatim} ``` </details>
944. ph-7f3959e2793901beeed1 — automata/docling_md/indexed grammars and logic programs.md ### Plain (markdown context) the proposition holds for Deriu,(P :- V(C)) and Sent,(G(P), N'). Then for a given goal reached from the initial goal after k or fewer resolution steps, the sentential form P(G1) is indeed derivable from S in the indexed grammar G(P) and is an element of Sent,(GW, N'). Suppose that the next resolution step uses Pl(fi,,(. . . f,, n!~>. . .>> and that, furthermore, P,(f,,,(. . . f,, .!c>. . .I) unifies with the head of a rule For this to be possible, Pi(fi 1(. . . f, .(c). . .I) must have the sequence of function symbols h,,, … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{the proposition holds for Deriu,(P :- V(C)) and Sent,(G(P), N'). Then for a given goal reached from the initial goal after k or fewer resolution steps, the sentential form P(G1) is indeed derivable from S in the indexed grammar G(P) and is an element…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/indexed grammars and logic programs.md:offset=24910 \begin{verbatim} the proposition holds for Deriu,(P :- V(C)) and Sent,(G(P), N'). Then for a given goal reached from the initial goal after k or fewer resolution steps, the sentential form P(G1) is indeed derivable from S in the indexed grammar G(P) and is an element of Sent,(GW, N'). Suppose that the next resolution step uses Pl(fi,,(. . . f,, n!~>. . .>> and that, furthermore, P,(f,,,(. . . f,, .!c>. . .I) unifies with the head of a rule For this to be possible, Pi(fi 1(. . . f, .(c). . .I) must have the sequence of function symbols h,,, … \end{verbatim} ``` </details>
945. ph-4ae72cf1e0837e0d7504 — automata/docling_md/indexed grammars and logic programs.md ### Plain (markdown context) e that the next resolution step uses Pl(fi,,(. . . f,, n!~>. . .>> and that, furthermore, P,(f,,,(. . . f,, .!c>. . .I) unifies with the head of a rule For this to be possible, Pi(fi 1(. . . f, .(c). . .I) must have the sequence of function symbols h,,, I,. . . , ho, 4,, as an initial subsequence of length q. of its own sequence of function symbols fi, 1,. . . , fi, ',. Then we know that the resulting goal will have the form NOW, by construction, the flag rules in G(P) have been defined in such a way that for .z = qo, the s… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{e that the next resolution step uses Pl(fi,,(. . . f,, n!~\>. . .\>\> and that, furthermore, P,(f,,,(. . . f,, .!c\>. . .I) unifies with the head of a rule For this to be possible, Pi(fi 1(. . . f, .(c). . .I) must have the sequence of functi…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/indexed grammars and logic programs.md:offset=25236 \begin{verbatim} e that the next resolution step uses Pl(fi,,(. . . f,, n!~>. . .>> and that, furthermore, P,(f,,,(. . . f,, .!c>. . .I) unifies with the head of a rule For this to be possible, Pi(fi 1(. . . f, .(c). . .I) must have the sequence of function symbols h,,, I,. . . , ho, 4,, as an initial subsequence of length q. of its own sequence of function symbols fi, 1,. . . , fi, ',. Then we know that the resulting goal will have the form NOW, by construction, the flag rules in G(P) have been defined in such a way that for .z = qo, the s… \end{verbatim} ``` </details>
946. ph-1a868c277b93ea9d24b1 — automata/docling_md/indexed grammars and logic programs.md ### Plain (markdown context) as an initial subsequence of length q. of its own sequence of function symbols fi, 1,. . . , fi, ',. Then we know that the resulting goal will have the form NOW, by construction, the flag rules in G(P) have been defined in such a way that for .z = qo, the sets &, ,), . . . , Rp(f,, ,) contain rules taking off the indices h &,,'', h,, z and leading to'an auxiliary nonterminal p,'' specific to that rule. Furthermore, the rule adds index symbols at its jth nonterminal in agreement with the individual functioi symbols h, 1, . . . ,… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{as an initial subsequence of length q. of its own sequence of function symbols fi, 1,. . . , fi, ',. Then we know that the resulting goal will have the form NOW, by construction, the flag rules in G(P) have been defined in such a way that for .z = qo…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/indexed grammars and logic programs.md:offset=25565 \begin{verbatim} as an initial subsequence of length q. of its own sequence of function symbols fi, 1,. . . , fi, ',. Then we know that the resulting goal will have the form NOW, by construction, the flag rules in G(P) have been defined in such a way that for .z = qo, the sets &, ,), . . . , Rp(f,, ,) contain rules taking off the indices h &,,'', h,, z and leading to'an auxiliary nonterminal p,'' specific to that rule. Furthermore, the rule adds index symbols at its jth nonterminal in agreement with the individual functioi symbols h, 1, . . . ,… \end{verbatim} ``` </details>
947. ph-be142a3d13c10af8f169 — automata/docling_md/indexed grammars and logic programs.md ### Plain (markdown context) the formulation of Proposition 2 plays only the minor role of determining a start symbol in the corresponding grammar. If we are willing to deal with strings generated from other symbols, the following version of the result may be preferred. Corollary. Let P be a unaq logic program. Then there is an indexed grammar G = (N, T, F, R, S>, and an inject&e mapping 5: Ground + M* such that for any predicate symbol rr, any constant c, and any goal g REMARK. The only difference in the construction is that instead of choosing an arbitrary predicate symbol which is t… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{the formulation of Proposition 2 plays only the minor role of determining a start symbol in the corresponding grammar. If we are willing to deal with strings generated from other symbols, the following version of the result may be preferred. Corollary. Let P be a unaq logic prog…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/indexed grammars and logic programs.md:offset=28530 \begin{verbatim} the formulation of Proposition 2 plays only the minor role of determining a start symbol in the corresponding grammar. If we are willing to deal with strings generated from other symbols, the following version of the result may be preferred. Corollary. Let P be a unaq logic program. Then there is an indexed grammar G = (N, T, F, R, S>, and an inject&e mapping 5: Ground + M* such that for any predicate symbol rr, any constant c, and any goal g REMARK. The only difference in the construction is that instead of choosing an arbitrary predicate symbol which is t… \end{verbatim} ``` </details>
948. ph-3b6ec0f82e91ab459446 — automata/docling_md/indexed grammars and logic programs.md ### Plain (markdown context) rn the only successfully derived final product. To introduce a distinction similar to the one between terminals and nonterminals would require an artificial nd otherwise unmotivated separation between two kinds of atoms of a logic program. Apart from this aspect, a less fundamental but challenging problem arises from the different ways in which flag symbols are consumed in indexed grammar derivations. To see this, consider the rule set consisting of two rewrite rules where S may be assumed to be the start symbol, f the only available flag, and a the only terminal … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{rn the only successfully derived final product. To introduce a distinction similar to the one between terminals and nonterminals would require an artificial nd otherwise unmotivated separation between two kinds of atoms of a logic program. Apart from this aspect, a less fundamen…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/indexed grammars and logic programs.md:offset=30447 \begin{verbatim} rn the only successfully derived final product. To introduce a distinction similar to the one between terminals and nonterminals would require an artificial nd otherwise unmotivated separation between two kinds of atoms of a logic program. Apart from this aspect, a less fundamental but challenging problem arises from the different ways in which flag symbols are consumed in indexed grammar derivations. To see this, consider the rule set consisting of two rewrite rules where S may be assumed to be the start symbol, f the only available flag, and a the only terminal … \end{verbatim} ``` </details>
949. ph-2efbd197c0b58a6bcd39 — automata/docling_md/indexed grammars and logic programs.md ### Plain (markdown context) en two kinds of atoms of a logic program. Apart from this aspect, a less fundamental but challenging problem arises from the different ways in which flag symbols are consumed in indexed grammar derivations. To see this, consider the rule set consisting of two rewrite rules where S may be assumed to be the start symbol, f the only available flag, and a the only terminal symbol. By repeated application of S + Sf, we get sentential forms for arbitrarily long sequences of flags. Using the construction presented in the previous section, the… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{en two kinds of atoms of a logic program. Apart from this aspect, a less fundamental but challenging problem arises from the different ways in which flag symbols are consumed in indexed grammar derivations. To see this, consider the rule set consisting of two rewrite rules where S may be assumed to be the start symbol, f the only available flag, and a the only terminal symbol. By repeated application of S + Sf, we get sentential forms for arbitrarily long sequences of flags. Using the construction presented in the previous section, the… \end{verbatim} ```
950. ph-77dcf45303cd4b512519 — automata/docling_md/indexed grammars and logic programs.md ### Plain (markdown context) ammar derivations. To see this, consider the rule set consisting of two rewrite rules where S may be assumed to be the start symbol, f the only available flag, and a the only terminal symbol. By repeated application of S + Sf, we get sentential forms for arbitrarily long sequences of flags. Using the construction presented in the previous section, the goal corresponding to this by one-to-one mapping has the form Now, the decisive point is that such sentential forms can be replaced by the terminal symbol a i… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ammar derivations. To see this, consider the rule set consisting of two rewrite rules where S may be assumed to be the start symbol, f the only available flag, and a the only terminal symbol. By repeated application of S + Sf, we get sentential forms…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/indexed grammars and logic programs.md:offset=30852 \begin{verbatim} ammar derivations. To see this, consider the rule set consisting of two rewrite rules where S may be assumed to be the start symbol, f the only available flag, and a the only terminal symbol. By repeated application of S + Sf, we get sentential forms for arbitrarily long sequences of flags. Using the construction presented in the previous section, the goal corresponding to this by one-to-one mapping has the form Now, the decisive point is that such sentential forms can be replaced by the terminal symbol a i… \end{verbatim} ``` </details>
951. ph-af586fa6fcc3d35b3c30 — automata/docling_md/indexed grammars and logic programs.md ### Plain (markdown context) g having these intermediate stages of resolution as images. The purpose of this section is to show that by using a certain normalized version of indexed grammar, bijective mappings between goals and sentential forms are again possible. We introduce the notion of 'smooth' indexed grammars. Call an indexed grammar smooth if the start symbol S never occurs on the right side of any rule or flag rule and occurs on the left side of only one rule which has the form where T is a nonterminal and I is a flag not generated by any other rule, and if the set of flag rules (ind… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{g having these intermediate stages of resolution as images. The purpose of this section is to show that by using a certain normalized version of indexed grammar, bijective mappings between goals and sentential forms are again possible. We introduce the notion of 'smooth' indexed…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/indexed grammars and logic programs.md:offset=32358 \begin{verbatim} g having these intermediate stages of resolution as images. The purpose of this section is to show that by using a certain normalized version of indexed grammar, bijective mappings between goals and sentential forms are again possible. We introduce the notion of 'smooth' indexed grammars. Call an indexed grammar smooth if the start symbol S never occurs on the right side of any rule or flag rule and occurs on the left side of only one rule which has the form where T is a nonterminal and I is a flag not generated by any other rule, and if the set of flag rules (ind… \end{verbatim} ``` </details>
952. ph-769f874c7b353140b5b2 — automata/docling_md/indexed grammars and logic programs.md ### Plain (markdown context) tial forms are again possible. We introduce the notion of 'smooth' indexed grammars. Call an indexed grammar smooth if the start symbol S never occurs on the right side of any rule or flag rule and occurs on the left side of only one rule which has the form where T is a nonterminal and I is a flag not generated by any other rule, and if the set of flag rules (index rules) associated to the flag I consists of rules of the form where A is a nonterminal, a is a terminal, and there are no other rules or flag rules with terminal symbols on … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{tial forms are again possible. We introduce the notion of 'smooth' indexed grammars. Call an indexed grammar smooth if the start symbol S never occurs on the right side of any rule or flag rule and occurs on the left side of only one rule which has the form where T is a nonterminal and I is a flag not generated by any other rule, and if the set of flag rules (index rules) associated to the flag I consists of rules of the form where A is a nonterminal, a is a terminal, and there are no other rules or flag rules with terminal symbols on … \end{verbatim} ```
953. ph-e3aba7cfc03f8ca8046a — automata/docling_md/indexed grammars and logic programs.md ### Plain (markdown context) terminal symbols on their right-hand sides. We can now claim the following. Proposition 3. Every indexed language is generated by a smooth indexed grammar. Furthermore, for smooth indexed grammars, an analog of Proposition 1 holds even if sentential forms containing terminals are included in the domain of the mapping. Proposition 4. Let G = (N, T, F, R, S> be a smooth indexed grammar. Then there exists a wary program P(G) and a bijectiue mapping The motivation of the smoothne… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{terminal symbols on their right-hand sides. We can now claim the following. Proposition 3. Every indexed language is generated by a smooth indexed grammar. Furthermore, for smooth indexed grammars, an analog of Proposition 1 holds even if sentential forms containing terminals ar…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/indexed grammars and logic programs.md:offset=33204 \begin{verbatim} terminal symbols on their right-hand sides. We can now claim the following. Proposition 3. Every indexed language is generated by a smooth indexed grammar. Furthermore, for smooth indexed grammars, an analog of Proposition 1 holds even if sentential forms containing terminals are included in the domain of the mapping. Proposition 4. Let G = (N, T, F, R, S> be a smooth indexed grammar. Then there exists a wary program P(G) and a bijectiue mapping The motivation of the smoothne… \end{verbatim} ``` </details>
954. ph-4e42bef0cfe5c3ac5cfd — automata/docling_md/indexed grammars and logic programs.md ### Plain (markdown context) ght-hand sides. We can now claim the following. Proposition 3. Every indexed language is generated by a smooth indexed grammar. Furthermore, for smooth indexed grammars, an analog of Proposition 1 holds even if sentential forms containing terminals are included in the domain of the mapping. Proposition 4. Let G = (N, T, F, R, S> be a smooth indexed grammar. Then there exists a wary program P(G) and a bijectiue mapping The motivation of the smoothness construction is that rewr… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ght-hand sides. We can now claim the following. Proposition 3. Every indexed language is generated by a smooth indexed grammar. Furthermore, for smooth indexed grammars, an analog of Proposition 1 holds even if sentential forms containing terminals are included in the domain of …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/indexed grammars and logic programs.md:offset=33234 \begin{verbatim} ght-hand sides. We can now claim the following. Proposition 3. Every indexed language is generated by a smooth indexed grammar. Furthermore, for smooth indexed grammars, an analog of Proposition 1 holds even if sentential forms containing terminals are included in the domain of the mapping. Proposition 4. Let G = (N, T, F, R, S> be a smooth indexed grammar. Then there exists a wary program P(G) and a bijectiue mapping The motivation of the smoothness construction is that rewr… \end{verbatim} ``` </details>
955. ph-5c6d3822e318bf0afbd4 — automata/docling_md/indexed grammars and logic programs.md ### Plain (markdown context) claim the following. Proposition 3. Every indexed language is generated by a smooth indexed grammar. Furthermore, for smooth indexed grammars, an analog of Proposition 1 holds even if sentential forms containing terminals are included in the domain of the mapping. Proposition 4. Let G = (N, T, F, R, S> be a smooth indexed grammar. Then there exists a wary program P(G) and a bijectiue mapping The motivation of the smoothness construction is that rewriting rules with terminal s… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{claim the following. Proposition 3. Every indexed language is generated by a smooth indexed grammar. Furthermore, for smooth indexed grammars, an analog of Proposition 1 holds even if sentential forms containing terminals are included in the domain of the mapping. Proposition 4.…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/indexed grammars and logic programs.md:offset=33264 \begin{verbatim} claim the following. Proposition 3. Every indexed language is generated by a smooth indexed grammar. Furthermore, for smooth indexed grammars, an analog of Proposition 1 holds even if sentential forms containing terminals are included in the domain of the mapping. Proposition 4. Let G = (N, T, F, R, S> be a smooth indexed grammar. Then there exists a wary program P(G) and a bijectiue mapping The motivation of the smoothness construction is that rewriting rules with terminal s… \end{verbatim} ``` </details>
956. ph-12ffdc06346dae65547f — automata/docling_md/indexed grammars and logic programs.md ### Plain (markdown context) sition 3. Every indexed language is generated by a smooth indexed grammar. Furthermore, for smooth indexed grammars, an analog of Proposition 1 holds even if sentential forms containing terminals are included in the domain of the mapping. Proposition 4. Let G = (N, T, F, R, S> be a smooth indexed grammar. Then there exists a wary program P(G) and a bijectiue mapping The motivation of the smoothness construction is that rewriting rules with terminal symbols must somehow be for… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{sition 3. Every indexed language is generated by a smooth indexed grammar. Furthermore, for smooth indexed grammars, an analog of Proposition 1 holds even if sentential forms containing terminals are included in the domain of the mapping. Proposition 4. Let G = (N, T, F, R, S\&gt…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/indexed grammars and logic programs.md:offset=33294 \begin{verbatim} sition 3. Every indexed language is generated by a smooth indexed grammar. Furthermore, for smooth indexed grammars, an analog of Proposition 1 holds even if sentential forms containing terminals are included in the domain of the mapping. Proposition 4. Let G = (N, T, F, R, S> be a smooth indexed grammar. Then there exists a wary program P(G) and a bijectiue mapping The motivation of the smoothness construction is that rewriting rules with terminal symbols must somehow be for… \end{verbatim} ``` </details>
957. ph-76dfcd53d82b2aaa335e — automata/docling_md/indexed grammars and logic programs.md ### Plain (markdown context) able only after the flag sequences which they might otherwise delete in a single step have been cancelled flag by flag. This is achieved by placing a unique bottom symbol at the start of every flag sequence. Terminal replacements are restricted to the context of a reemergent bottom flag. To get to that point, nonterminal place-holders for the terminals must be capable of cancelling all other flags in individual auxiliary derivation steps. To show Proposition 3, let be an indexed grammar. Choose a new start symbol S' and construct a rule be an indexed grammar. Choose a new start symbol S' and construct a rule be an indexed grammar. Choose a new start symbol S' and construct a rule where I is a new flag symbol. For every terminal t of G, provide a new unique nonterminal N,. Repl… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{elled flag by flag. This is achieved by placing a unique bottom symbol at the start of every flag sequence. Terminal replacements are restricted to the context of a reemergent bottom flag. To get to that point, nonterminal place-holders for the terminals must be capable of cance…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/indexed grammars and logic programs.md:offset=34066 \begin{verbatim} elled flag by flag. This is achieved by placing a unique bottom symbol at the start of every flag sequence. Terminal replacements are restricted to the context of a reemergent bottom flag. To get to that point, nonterminal place-holders for the terminals must be capable of cancelling all other flags in individual auxiliary derivation steps. To show Proposition 3, let be an indexed grammar. Choose a new start symbol S' and construct a rule where I is a new flag symbol. For every terminal t of G, provide a new unique nonterminal N,. Repl… \end{verbatim} ``` </details>
959. ph-1d7a352e57e9ec08c6fe — automata/docling_md/indexed grammars and logic programs.md ### Plain (markdown context) new start symbol S' and construct a rule where I is a new flag symbol. For every terminal t of G, provide a new unique nonterminal N,. Replace every occurrence of t in some rule or index rule by N,. Additionally, let N, + N, be an element of the index rule set Rf for all flags f # J\_ and all new symbols N,, and let N, + t be an element of the index rule set R i for all new symbols N,. As a consequence, for every derivation step where (Y, p E K, N a nonterminal, 4 E F*, the modified grammar permits a derivation step where I is a new flag symbol. For every terminal t of G, provide a new unique nonterminal N,. Replace every occurrence of t in some rule or index rule by N,. Additionally, let N, + N, be an element of the inde…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/indexed grammars and logic programs.md:offset=34520 \begin{verbatim} new start symbol S' and construct a rule where I is a new flag symbol. For every terminal t of G, provide a new unique nonterminal N,. Replace every occurrence of t in some rule or index rule by N,. Additionally, let N, + N, be an element of the index rule set Rf for all flags f # J\_ and all new symbols N,, and let N, + t be an element of the index rule set R i for all new symbols N,. As a consequence, for every derivation step where (Y, p E K, N a nonterminal, 4 E F*, the modified grammar permits a derivation step where (Y, p E K, N a nonterminal, 4 E F*, the modified grammar permits a derivation step It should be noted that the final flag in $ is always I by construction of the starting rule. The … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{y terminal t of G, provide a new unique nonterminal N,. Replace every occurrence of t in some rule or index rule by N,. Additionally, let N, + N, be an element of the index rule set Rf for all flags f \# J\textbackslash \_ and all new symbols N,, and let N, + t be an element of the index rule s…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/indexed grammars and logic programs.md:offset=34640 \begin{verbatim} y terminal t of G, provide a new unique nonterminal N,. Replace every occurrence of t in some rule or index rule by N,. Additionally, let N, + N, be an element of the index rule set Rf for all flags f # J\_ and all new symbols N,, and let N, + t be an element of the index rule set R i for all new symbols N,. As a consequence, for every derivation step where (Y, p E K, N a nonterminal, 4 E F*, the modified grammar permits a derivation step It should be noted that the final flag in $ is always I by construction of the starting rule. The … \end{verbatim} ``` </details>
961. ph-0e7533e2b7162f7d0c40 — automata/docling_md/indexed grammars and logic programs.md ### Plain (markdown context) he index rule set Rf for all flags f # J\_ and all new symbols N,, and let N, + t be an element of the index rule set R i for all new symbols N,. As a consequence, for every derivation step where (Y, p E K, N a nonterminal, 4 E F*, the modified grammar permits a derivation step It should be noted that the final flag in $ is always I by construction of the starting rule. The N, -+ N, rules permit a sequence As R, contains N, -+ t, c+ I P * at/3 holds. Alternative ways of deriving strings containing terminals… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{he index rule set Rf for all flags f \# J\textbackslash \_ and all new symbols N,, and let N, + t be an element of the index rule set R i for all new symbols N,. As a consequence, for every derivation step where (Y, p E K, N a nonterminal, 4 E F*, the modified gramm…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/indexed grammars and logic programs.md:offset=34814 \begin{verbatim} he index rule set Rf for all flags f # J\_ and all new symbols N,, and let N, + t be an element of the index rule set R i for all new symbols N,. As a consequence, for every derivation step where (Y, p E K, N a nonterminal, 4 E F*, the modified grammar permits a derivation step It should be noted that the final flag in $ is always I by construction of the starting rule. The N, -+ N, rules permit a sequence As R, contains N, -+ t, c+ I P * at/3 holds. Alternative ways of deriving strings containing terminals… \end{verbatim} ``` </details>
962. ph-b7b962d2aab48e668c57 — automata/docling_md/indexed grammars and logic programs.md ### Plain (markdown context) > As R, contains N, -+ t, c+ I P * at/3 holds. Alternative ways of deriving strings containing terminals have been excluded by the construction. On the other hand, every terminal word of the language is generated in this way. To complete our discussion, we sketch the way in which a one-to-one mapping between derivation steps and resolution steps is obtained. This gives us Proposition 4. As stated, derivation steps producing terminal symbols have the general form Clearly, we can have clauses in logic programs of the type Considering a s… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{> As R, contains N, -+ t, c+ I P * at/3 holds. Alternative ways of deriving strings containing terminals have been excluded by the construction. On the other hand, every terminal word of the language is generated in this way. To complete our discussion, we sketch the way in whic…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/indexed grammars and logic programs.md:offset=35341 \begin{verbatim} > As R, contains N, -+ t, c+ I P * at/3 holds. Alternative ways of deriving strings containing terminals have been excluded by the construction. On the other hand, every terminal word of the language is generated in this way. To complete our discussion, we sketch the way in which a one-to-one mapping between derivation steps and resolution steps is obtained. This gives us Proposition 4. As stated, derivation steps producing terminal symbols have the general form Clearly, we can have clauses in logic programs of the type Considering a s… \end{verbatim} ``` </details>
963. ph-8ec080bcee8f23a0033f — automata/docling_md/indexed grammars and logic programs.md ### Plain (markdown context) ining terminals have been excluded by the construction. On the other hand, every terminal word of the language is generated in this way. To complete our discussion, we sketch the way in which a one-to-one mapping between derivation steps and resolution steps is obtained. This gives us Proposition 4. As stated, derivation steps producing terminal symbols have the general form Clearly, we can have clauses in logic programs of the type Considering a smooth indexed grammar constructed as shown, we establish a bijection which associates, fo… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ining terminals have been excluded by the construction. On the other hand, every terminal word of the language is generated in this way. To complete our discussion, we sketch the way in which a one-to-one mapping between derivation steps and resolution steps is obtained. This gi…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/indexed grammars and logic programs.md:offset=35435 \begin{verbatim} ining terminals have been excluded by the construction. On the other hand, every terminal word of the language is generated in this way. To complete our discussion, we sketch the way in which a one-to-one mapping between derivation steps and resolution steps is obtained. This gives us Proposition 4. As stated, derivation steps producing terminal symbols have the general form Clearly, we can have clauses in logic programs of the type Considering a smooth indexed grammar constructed as shown, we establish a bijection which associates, fo… \end{verbatim} ``` </details>
964. ph-bd976d776648bb19dd46 — automata/docling_md/indexed grammars and logic programs.md ### Plain (markdown context) t in all rules by uniquely associated new nonterminals N(t). Let us call the resulting grammar G'. Then the construction yielding a unary program P(G') is used. Quite obviously, the predicate symbols ji(N(t)) do not occur in the head of any clause. Without wanting to get into technical details at this point, the subsequent development essentially requires two additional arguments for each predicate, giving us ternary (3-ary) instead of unary predicate symbols. Generally, is transformed into In the case of empty bodies, L is taken inste… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{t in all rules by uniquely associated new nonterminals N(t). Let us call the resulting grammar G'. Then the construction yielding a unary program P(G') is used. Quite obviously, the predicate symbols ji(N(t)) do not occur in the head of any clause. Without wanting to get into te…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/indexed grammars and logic programs.md:offset=44373 \begin{verbatim} t in all rules by uniquely associated new nonterminals N(t). Let us call the resulting grammar G'. Then the construction yielding a unary program P(G') is used. Quite obviously, the predicate symbols ji(N(t)) do not occur in the head of any clause. Without wanting to get into technical details at this point, the subsequent development essentially requires two additional arguments for each predicate, giving us ternary (3-ary) instead of unary predicate symbols. Generally, is transformed into In the case of empty bodies, L is taken inste… \end{verbatim} ``` </details>
965. ph-c344b2ea9aee79468976 — automata/docling_md/indexed grammars and logic programs.md ### Plain (markdown context) minals N(t). Let us call the resulting grammar G'. Then the construction yielding a unary program P(G') is used. Quite obviously, the predicate symbols ji(N(t)) do not occur in the head of any clause. Without wanting to get into technical details at this point, the subsequent development essentially requires two additional arguments for each predicate, giving us ternary (3-ary) instead of unary predicate symbols. Generally, is transformed into In the case of empty bodies, L is taken instead of L,,. This provides for the case of e-produ… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{minals N(t). Let us call the resulting grammar G'. Then the construction yielding a unary program P(G') is used. Quite obviously, the predicate symbols ji(N(t)) do not occur in the head of any clause. Without wanting to get into technical details at this point, the subsequent de…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/indexed grammars and logic programs.md:offset=44425 \begin{verbatim} minals N(t). Let us call the resulting grammar G'. Then the construction yielding a unary program P(G') is used. Quite obviously, the predicate symbols ji(N(t)) do not occur in the head of any clause. Without wanting to get into technical details at this point, the subsequent development essentially requires two additional arguments for each predicate, giving us ternary (3-ary) instead of unary predicate symbols. Generally, is transformed into In the case of empty bodies, L is taken instead of L,,. This provides for the case of e-produ… \end{verbatim} ``` </details>
966. ph-05669f5cdbd5abc74515 — automata/docling_md/indexed grammars and logic programs.md ### Plain (markdown context) lause. Without wanting to get into technical details at this point, the subsequent development essentially requires two additional arguments for each predicate, giving us ternary (3-ary) instead of unary predicate symbols. Generally, is transformed into In the case of empty bodies, L is taken instead of L,,. This provides for the case of e-productions. Furthermore, for all terminals t, we need a clause of the form where goals check( L,, t, L,) are defined to succeed if the first argument is a list, the seco… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{lause. Without wanting to get into technical details at this point, the subsequent development essentially requires two additional arguments for each predicate, giving us ternary (3-ary) instead of unary predicate symbols. Generally, is transformed i…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/indexed grammars and logic programs.md:offset=44627 \begin{verbatim} lause. Without wanting to get into technical details at this point, the subsequent development essentially requires two additional arguments for each predicate, giving us ternary (3-ary) instead of unary predicate symbols. Generally, is transformed into In the case of empty bodies, L is taken instead of L,,. This provides for the case of e-productions. Furthermore, for all terminals t, we need a clause of the form where goals check( L,, t, L,) are defined to succeed if the first argument is a list, the seco… \end{verbatim} ``` </details>
967. ph-045a7739fbc60a9843a0 — automata/docling_md/indexed grammars and logic programs.md ### Plain (markdown context) in PROLOG systems. Instead of :- y(S)(c) as the initial goal, we now have ``` :- p(S) (c, TerminalString, [ 1) ``` provided that TerrninalStting is instantiated to a list of terminals. To demonstrate this procedure, we take the example grammar shown before. The terminal symbols a and b are replaced by nonterminals A and B in all rules and flag rules. With function symbols f and g and predicate symbols S, T, U, A, B, the set of clauses P(G') may be written as The above augmentation yields S(X, L,,, L):-A(X, L,,, L,), T@(X), L,, L), and … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{in PROLOG systems. Instead of :- y(S)(c) as the initial goal, we now have ``` :- p(S) (c, TerminalString, [ 1) ``` provided that TerrninalStting is instantiated to a list of terminals. To demonstrate this procedure, we take the example grammar shown before. The terminal symbols …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/indexed grammars and logic programs.md:offset=45386 \begin{verbatim} in PROLOG systems. Instead of :- y(S)(c) as the initial goal, we now have ``` :- p(S) (c, TerminalString, [ 1) ``` provided that TerrninalStting is instantiated to a list of terminals. To demonstrate this procedure, we take the example grammar shown before. The terminal symbols a and b are replaced by nonterminals A and B in all rules and flag rules. With function symbols f and g and predicate symbols S, T, U, A, B, the set of clauses P(G') may be written as The above augmentation yields S(X, L,,, L):-A(X, L,,, L,), T@(X), L,, L), and … \end{verbatim} ``` </details>
968. ph-d56efc3eb0c1ff3209c0 — automata/docling_md/indexed grammars and logic programs.md ### Plain (markdown context) :- y(S)(c) as the initial goal, we now have ``` :- p(S) (c, TerminalString, [ 1) ``` provided that TerrninalStting is instantiated to a list of terminals. To demonstrate this procedure, we take the example grammar shown before. The terminal symbols a and b are replaced by nonterminals A and B in all rules and flag rules. With function symbols f and g and predicate symbols S, T, U, A, B, the set of clauses P(G') may be written as The above augmentation yields S(X, L,,, L):-A(X, L,,, L,), T@(X), L,, L), and likewise for the other clauses… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{:- y(S)(c) as the initial goal, we now have ``` :- p(S) (c, TerminalString, [ 1) ``` provided that TerrninalStting is instantiated to a list of terminals. To demonstrate this procedure, we take the example grammar shown before. The terminal symbols a and b are replaced by nonter…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/indexed grammars and logic programs.md:offset=45416 \begin{verbatim} :- y(S)(c) as the initial goal, we now have ``` :- p(S) (c, TerminalString, [ 1) ``` provided that TerrninalStting is instantiated to a list of terminals. To demonstrate this procedure, we take the example grammar shown before. The terminal symbols a and b are replaced by nonterminals A and B in all rules and flag rules. With function symbols f and g and predicate symbols S, T, U, A, B, the set of clauses P(G') may be written as The above augmentation yields S(X, L,,, L):-A(X, L,,, L,), T@(X), L,, L), and likewise for the other clauses… \end{verbatim} ``` </details>
969. ph-a3111ecd664fb74f1a70 — automata/docling_md/the Aho-Corasick machine.md ### Plain (markdown context) s K\_SET= {"he", "she", "his", "hers", "we"}. Here, ~{'h', 's', 'w'} denotes all input symbols other than 'h', 's' and 'w'. For example, the transition labeled 'h' from state 0 to state 1 in Fig. l(a) indicates that g(O,'h') = 1. The absence of the arc indicates fail. The AC machine has the -~{~w} (a) The goto function. ![Image](./the Aho-Corasick machine_artifacts/image_000002_656f1c8a3c06899591bcd0416f40a27c4bbe5ae7bcb0ada4e8253603d8c1ee62.png) (c) The output function. ## (b) The failure function. Fig. 1. The AC machine for K\_SET. property that g(0,'a') ~ fail … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{s K\textbackslash \_SET= \{"he", "she", "his", "hers", "we"\}. Here, ~\{'h', 's', 'w'\} denotes all input symbols other than 'h', 's' and 'w'. For example, the transition labeled 'h' from state 0 to state 1 in Fig. l(a) indicates that g(O,'h') = 1. The absence of the arc indicates fail. The AC mac…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/the Aho-Corasick machine.md:offset=5036 \begin{verbatim} s K\_SET= {"he", "she", "his", "hers", "we"}. Here, ~{'h', 's', 'w'} denotes all input symbols other than 'h', 's' and 'w'. For example, the transition labeled 'h' from state 0 to state 1 in Fig. l(a) indicates that g(O,'h') = 1. The absence of the arc indicates fail. The AC machine has the -~{~w} (a) The goto function. ![Image](./the Aho-Corasick machine_artifacts/image_000002_656f1c8a3c06899591bcd0416f40a27c4bbe5ae7bcb0ada4e8253603d8c1ee62.png) (c) The output function. ## (b) The failure function. Fig. 1. The AC machine for K\_SET. property that g(0,'a') ~ fail … \end{verbatim} ``` </details>
970. ph-8fdd1050fa829ce56015 — automata/docling_md/the Aho-Corasick machine.md ### Plain (markdown context) The failure function. Fig. 1. The AC machine for K\_SET. property that g(0,'a') ~ fail for all input symbols a. The behavior of the AC machine is summarized below. ## Algorithm 1. The AC machine [Inputl: A target text TX---clc2...cn where each ci, for 1 ~ i<<.n, is an input symbol and an AC machine with goto function g, failure function f, and output function output. [Outputl: Locations at which the keywords occur in TX. Step 1.1: {Initialization} Step 1.2: {State transitions} if g(s, ci) ~ fail then goto Step 1.3; s ~ f(s); goto Step 1.2; Step 1.3: {Output … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{The failure function. Fig. 1. The AC machine for K\textbackslash \_SET. property that g(0,'a') ~ fail for all input symbols a. The behavior of the AC machine is summarized below. \#\# Algorithm 1. The AC machine [Inputl: A target text TX---clc2...cn where each ci, for 1 ~ i\<\<.n, is an inpu…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/the Aho-Corasick machine.md:offset=5573 \begin{verbatim} The failure function. Fig. 1. The AC machine for K\_SET. property that g(0,'a') ~ fail for all input symbols a. The behavior of the AC machine is summarized below. ## Algorithm 1. The AC machine [Inputl: A target text TX---clc2...cn where each ci, for 1 ~ i<<.n, is an input symbol and an AC machine with goto function g, failure function f, and output function output. [Outputl: Locations at which the keywords occur in TX. Step 1.1: {Initialization} Step 1.2: {State transitions} if g(s, ci) ~ fail then goto Step 1.3; s ~ f(s); goto Step 1.2; Step 1.3: {Output … \end{verbatim} ``` </details>
971. ph-2fc27a299fc12e96390e — automata/docling_md/the Aho-Corasick machine.md ### Plain (markdown context) here each ci, for 1 ~ i<<.n, is an input symbol and an AC machine with goto function g, failure function f, and output function output. [Outputl: Locations at which the keywords occur in TX. Step 1.1: {Initialization} Step 1.2: {State transitions} if g(s, ci) ~ fail then goto Step 1.3; s ~ f(s); goto Step 1.2; Step 1.3: {Output operation} s ~ g(s, ci); if output(s) ~ ck then print i, output(s); Step 1.4: {Operation control} Consider the behavior of the AC machine that uses the functions in Fig. 1 to process the text string "usher… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{here each ci, for 1 ~ i\<\<.n, is an input symbol and an AC machine with goto function g, failure function f, and output function output. [Outputl: Locations at which the keywords occur in TX. Step 1.1: \{Initialization\} Step 1.2: \{State transitio…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/the Aho-Corasick machine.md:offset=5828 \begin{verbatim} here each ci, for 1 ~ i<<.n, is an input symbol and an AC machine with goto function g, failure function f, and output function output. [Outputl: Locations at which the keywords occur in TX. Step 1.1: {Initialization} Step 1.2: {State transitions} if g(s, ci) ~ fail then goto Step 1.3; s ~ f(s); goto Step 1.2; Step 1.3: {Output operation} s ~ g(s, ci); if output(s) ~ ck then print i, output(s); Step 1.4: {Operation control} Consider the behavior of the AC machine that uses the functions in Fig. 1 to process the text string "usher… \end{verbatim} ``` </details>
972. ph-d05ad5a110bb23bf65b0 — automata/docling_md/the Aho-Corasick machine.md ### Plain (markdown context) ## 3. Rules for the multi-attribute information Let A TTR be the attribute name and let VALUE be the attribute value. Let R be a finite set of pairs (A TTR, VALUE ), then we shall call R a rule structure. For example, the following attributes are considered: STR (string, that is, word spelling); CAT (category, or, a part of speech); SEM (semantic information such as concepts and categories). For example, the rule structure R of "student" is defined as follows: If Rule is the matching rule consisting of a sequence of rule structures, Rule is defined as follows: <!-… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{\#\# 3. Rules for the multi-attribute information Let A TTR be the attribute name and let VALUE be the attribute value. Let R be a finite set of pairs (A TTR, VALUE ), then we shall call R a rule structure. For example, the following attributes are considered: STR (string, that is…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/the Aho-Corasick machine.md:offset=7452 \begin{verbatim} ## 3. Rules for the multi-attribute information Let A TTR be the attribute name and let VALUE be the attribute value. Let R be a finite set of pairs (A TTR, VALUE ), then we shall call R a rule structure. For example, the following attributes are considered: STR (string, that is, word spelling); CAT (category, or, a part of speech); SEM (semantic information such as concepts and categories). For example, the rule structure R of "student" is defined as follows: If Rule is the matching rule consisting of a sequence of rule structures, Rule is defined as follows: <!-… \end{verbatim} ``` </details>
973. ph-fa9068f4a837ca081e7b — automata/docling_md/the Aho-Corasick machine.md ### Plain (markdown context) finite set of pairs (A TTR, VALUE ), then we shall call R a rule structure. For example, the following attributes are considered: STR (string, that is, word spelling); CAT (category, or, a part of speech); SEM (semantic information such as concepts and categories). For example, the rule structure R of "student" is defined as follows: If Rule is the matching rule consisting of a sequence of rule structures, Rule is defined as follows: Let R\_SET be a set of Rule. Consider the following R\_SET. For example, R… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{finite set of pairs (A TTR, VALUE ), then we shall call R a rule structure. For example, the following attributes are considered: STR (string, that is, word spelling); CAT (category, or, a part of speech); SEM (semantic information such as concepts and categories). For example, …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/the Aho-Corasick machine.md:offset=7591 \begin{verbatim} finite set of pairs (A TTR, VALUE ), then we shall call R a rule structure. For example, the following attributes are considered: STR (string, that is, word spelling); CAT (category, or, a part of speech); SEM (semantic information such as concepts and categories). For example, the rule structure R of "student" is defined as follows: If Rule is the matching rule consisting of a sequence of rule structures, Rule is defined as follows: Let R\_SET be a set of Rule. Consider the following R\_SET. For example, R… \end{verbatim} ``` </details>
974. ph-bd8a0a8d1f580a5b40aa — automata/docling_md/the Aho-Corasick machine.md ### Plain (markdown context) , the following attributes are considered: STR (string, that is, word spelling); CAT (category, or, a part of speech); SEM (semantic information such as concepts and categories). For example, the rule structure R of "student" is defined as follows: If Rule is the matching rule consisting of a sequence of rule structures, Rule is defined as follows: Let R\_SET be a set of Rule. Consider the following R\_SET. For example, Rulel can detect the input "wife of the professor", "picture of a dog", or "arrival of t… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{, the following attributes are considered: STR (string, that is, word spelling); CAT (category, or, a part of speech); SEM (semantic information such as concepts and categories). For example, the rule structure R of "student" is defined as follows: I…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/the Aho-Corasick machine.md:offset=7685 \begin{verbatim} , the following attributes are considered: STR (string, that is, word spelling); CAT (category, or, a part of speech); SEM (semantic information such as concepts and categories). For example, the rule structure R of "student" is defined as follows: If Rule is the matching rule consisting of a sequence of rule structures, Rule is defined as follows: Let R\_SET be a set of Rule. Consider the following R\_SET. For example, Rulel can detect the input "wife of the professor", "picture of a dog", or "arrival of t… \end{verbatim} ``` </details>
975. ph-ef19373776d1f440e4c8 — automata/docling_md/the Aho-Corasick machine.md ### Plain (markdown context) ain state to nextstate using the relationship, we define a function CHECK(state, N) as follows: ## [Function CHECK(state, N)I For each transition ge(state, R)--next\_state in the goto graph, if some transition labeled R which satisfies the inclusion relationship (N \_~ R) exists, then it returns all next\_state, otherwise it returns fail. (Function END) Suppose that ge(Sl, R1) ----$2 and ge(Sl, R2)--s3 are defined in the goto graph and R1 and R2 are defined as follows: Consider the behavior of the function CHECK to process the following N1 and N2. N1 = {(STR, "stu… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ain state to nextstate using the relationship, we define a function CHECK(state, N) as follows: \#\# [Function CHECK(state, N)I For each transition ge(state, R)--next\textbackslash \_state in the goto graph, if some transition labeled R which satisfies the inclusion relationship (N \textbackslash \_~ R) exists…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/the Aho-Corasick machine.md:offset=10719 \begin{verbatim} ain state to nextstate using the relationship, we define a function CHECK(state, N) as follows: ## [Function CHECK(state, N)I For each transition ge(state, R)--next\_state in the goto graph, if some transition labeled R which satisfies the inclusion relationship (N \_~ R) exists, then it returns all next\_state, otherwise it returns fail. (Function END) Suppose that ge(Sl, R1) ----$2 and ge(Sl, R2)--s3 are defined in the goto graph and R1 and R2 are defined as follows: Consider the behavior of the function CHECK to process the following N1 and N2. N1 = {(STR, "stu… \end{verbatim} ``` </details>
976. ph-70720a6c62a47af35564 — automata/docling_md/the Aho-Corasick machine.md ### Plain (markdown context) ate in the goto graph, if some transition labeled R which satisfies the inclusion relationship (N \_~ R) exists, then it returns all next\_state, otherwise it returns fail. (Function END) Suppose that ge(Sl, R1) ----$2 and ge(Sl, R2)--s3 are defined in the goto graph and R1 and R2 are defined as follows: Consider the behavior of the function CHECK to process the following N1 and N2. N1 = {(STR, "student"), (CAT, Noun), (SEM, Human)}, ``` (CAT,Adjective)}. ``` Then, CHECK(sl, Nl) returns {s2, s3} and CHECK(sl, N2) returns the message fa… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ate in the goto graph, if some transition labeled R which satisfies the inclusion relationship (N \textbackslash \_~ R) exists, then it returns all next\textbackslash \_state, otherwise it returns fail. (Function END) Suppose that ge(Sl, R1) ----$2 and ge(Sl, R2)--s3 are defined in the goto graph and R1 and …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/the Aho-Corasick machine.md:offset=10893 \begin{verbatim} ate in the goto graph, if some transition labeled R which satisfies the inclusion relationship (N \_~ R) exists, then it returns all next\_state, otherwise it returns fail. (Function END) Suppose that ge(Sl, R1) ----$2 and ge(Sl, R2)--s3 are defined in the goto graph and R1 and R2 are defined as follows: Consider the behavior of the function CHECK to process the following N1 and N2. N1 = {(STR, "student"), (CAT, Noun), (SEM, Human)}, ``` (CAT,Adjective)}. ``` Then, CHECK(sl, Nl) returns {s2, s3} and CHECK(sl, N2) returns the message fa… \end{verbatim} ``` </details>
977. ph-7156df07f644cff53d4d — automata/docling_md/the Aho-Corasick machine.md ### Plain (markdown context) ded by the inclusion relationship (N 2 R). Therefore, it has the possibility that some R such that N includes R exist among those transitions. In order to solve this problem, Algorithm 2 stores all states returned by the function CHECK in a first-in-first-out lis denoted by the variable queuel, and the MAPM machine continues to process for each state in queuel. Fig. 2 shows the functions used by the MAPM machine for a R\_SET= {Rulel, Rule2, Rule3, Rule4, Rules}. ``` Rule1 = R1R2R3R3, Rule2 = RIR2R4, Rule3 = R3R5RIR3, Rule4 = R3RsR3, Rule5 = R3RsRIR6 R1 : {(CAT,Art… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{ded by the inclusion relationship (N 2 R). Therefore, it has the possibility that some R such that N includes R exist among those transitions. In order to solve this problem, Algorithm 2 stores all states returned by the function CHECK in a first-in-first-out lis denoted by the …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/the Aho-Corasick machine.md:offset=12793 \begin{verbatim} ded by the inclusion relationship (N 2 R). Therefore, it has the possibility that some R such that N includes R exist among those transitions. In order to solve this problem, Algorithm 2 stores all states returned by the function CHECK in a first-in-first-out lis denoted by the variable queuel, and the MAPM machine continues to process for each state in queuel. Fig. 2 shows the functions used by the MAPM machine for a R\_SET= {Rulel, Rule2, Rule3, Rule4, Rules}. ``` Rule1 = R1R2R3R3, Rule2 = RIR2R4, Rule3 = R3R5RIR3, Rule4 = R3RsR3, Rule5 = R3RsRIR6 R1 : {(CAT,Art… \end{verbatim} ``` </details>
978. ph-1b20cbc408790fe4e8d0 — automata/docling_md/the Aho-Corasick machine.md ### Plain (markdown context) hows the functions used by the MAPM machine for a R\_SET= {Rulel, Rule2, Rule3, Rule4, Rules}. ``` Rule1 = R1R2R3R3, Rule2 = RIR2R4, Rule3 = R3R5RIR3, Rule4 = R3RsR3, Rule5 = R3RsRIR6 R1 : {(CAT,Article)}, R2 = {(CAT, Adjective)}, R4 : {(CAT, Noun), (SEM, Human)}, R5 = { (STR, "of"), (CA T, Preposition) }, ``` ![Image](./the Aho-Corasick machine_artifacts/image_000003_a2e12129bdcd876a9dc8b1bdb090637d2d5c6a80478c0c0ef2ffb63b7771aa29.png) ## (b) The failure function. | s | outputs(s) | |-----|----------------… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{hows the functions used by the MAPM machine for a R\textbackslash \_SET= \{Rulel, Rule2, Rule3, Rule4, Rules\}. ``` Rule1 = R1R2R3R3, Rule2 = RIR2R4, Rule3 = R3R5RIR3, Rule4 = R3RsR3, Rule5 = R3RsRIR6 R1 : \{(CAT,Article)\}, R2 = \{(CAT, Adjective)\}, R4 : \{(CAT, Noun), …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/the Aho-Corasick machine.md:offset=13187 \begin{verbatim} hows the functions used by the MAPM machine for a R\_SET= {Rulel, Rule2, Rule3, Rule4, Rules}. ``` Rule1 = R1R2R3R3, Rule2 = RIR2R4, Rule3 = R3R5RIR3, Rule4 = R3RsR3, Rule5 = R3RsRIR6 R1 : {(CAT,Article)}, R2 = {(CAT, Adjective)}, R4 : {(CAT, Noun), (SEM, Human)}, R5 = { (STR, "of"), (CA T, Preposition) }, ``` ![Image](./the Aho-Corasick machine_artifacts/image_000003_a2e12129bdcd876a9dc8b1bdb090637d2d5c6a80478c0c0ef2ffb63b7771aa29.png) ## (b) The failure function. | s | outputs(s) | |-----|----------------… \end{verbatim} ``` </details>
979. ph-dd57b9c397f6e57ac0e9 — automata/docling_md/the Aho-Corasick machine.md ### Plain (markdown context) the MAPM machine for a R\_SET= {Rulel, Rule2, Rule3, Rule4, Rules}. ``` Rule1 = R1R2R3R3, Rule2 = RIR2R4, Rule3 = R3R5RIR3, Rule4 = R3RsR3, Rule5 = R3RsRIR6 R1 : {(CAT,Article)}, R2 = {(CAT, Adjective)}, R4 : {(CAT, Noun), (SEM, Human)}, R5 = { (STR, "of"), (CA T, Preposition) }, ``` ![Image](./the Aho-Corasick machine_artifacts/image_000003_a2e12129bdcd876a9dc8b1bdb090637d2d5c6a80478c0c0ef2ffb63b7771aa29.png) ## (b) The failure function. | s | outputs(s) | |-----|------------------| | 4 | {Rj, R2, R3, R3} … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{the MAPM machine for a R\textbackslash \_SET= \{Rulel, Rule2, Rule3, Rule4, Rules\}. ``` Rule1 = R1R2R3R3, Rule2 = RIR2R4, Rule3 = R3R5RIR3, Rule4 = R3RsR3, Rule5 = R3RsRIR6 R1 : \{(CAT,Article)\}, R2 = \{(CAT, Adjective)\}, R4 : \{(CAT, Noun), (SEM, Human)\}, R5 = \{ (STR,…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/the Aho-Corasick machine.md:offset=13217 \begin{verbatim} the MAPM machine for a R\_SET= {Rulel, Rule2, Rule3, Rule4, Rules}. ``` Rule1 = R1R2R3R3, Rule2 = RIR2R4, Rule3 = R3R5RIR3, Rule4 = R3RsR3, Rule5 = R3RsRIR6 R1 : {(CAT,Article)}, R2 = {(CAT, Adjective)}, R4 : {(CAT, Noun), (SEM, Human)}, R5 = { (STR, "of"), (CA T, Preposition) }, ``` ![Image](./the Aho-Corasick machine_artifacts/image_000003_a2e12129bdcd876a9dc8b1bdb090637d2d5c6a80478c0c0ef2ffb63b7771aa29.png) ## (b) The failure function. | s | outputs(s) | |-----|------------------| | 4 | {Rj, R2, R3, R3} … \end{verbatim} ``` </details>
980. ph-4b61442b4192d84956c6 — automata/docling_md/the Aho-Corasick machine.md ### Plain (markdown context) position) }, ``` ![Image](./the Aho-Corasick machine_artifacts/image_000003_a2e12129bdcd876a9dc8b1bdb090637d2d5c6a80478c0c0ef2ffb63b7771aa29.png) ## (b) The failure function. | s | outputs(s) | |-----|------------------| | 4 | {Rj, R2, R3, R3} | | 5 | {Rj, R2, R4} | | 9 | {Rs, Rs, R~, R3} | | 10 | {R3,Rs,Rs} | | I1 | {R3, Rs, R~, R6} | ## (e) The output function. Fig. 2. The MAPM machine for R\_SET. Consider the behavior of the MAPM machine that uses the functions in Fig. 2 to process the sequence of the in… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{position) \}, ``` ![Image](./the Aho-Corasick machine\_artifacts/image\_000003\_a2e12129bdcd876a9dc8b1bdb090637d2d5c6a80478c0c0ef2ffb63b7771aa29.png) \#\# (b) The failure function. | s | outputs(s) | |-----|------------------| …}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/the Aho-Corasick machine.md:offset=13539 \begin{verbatim} position) }, ``` ![Image](./the Aho-Corasick machine_artifacts/image_000003_a2e12129bdcd876a9dc8b1bdb090637d2d5c6a80478c0c0ef2ffb63b7771aa29.png) ## (b) The failure function. | s | outputs(s) | |-----|------------------| | 4 | {Rj, R2, R3, R3} | | 5 | {Rj, R2, R4} | | 9 | {Rs, Rs, R~, R3} | | 10 | {R3,Rs,Rs} | | I1 | {R3, Rs, R~, R6} | ## (e) The output function. Fig. 2. The MAPM machine for R\_SET. Consider the behavior of the MAPM machine that uses the functions in Fig. 2 to process the sequence of the in… \end{verbatim} ``` </details>
981. ph-9c4c636e88ce54183309 — automata/docling_md/the Aho-Corasick machine.md ### Plain (markdown context) la-not-decoded --> ## (b) The failure function. | s | outputs(s) | |-----|------------------| | 4 | {Rj, R2, R3, R3} | | 5 | {Rj, R2, R4} | | 9 | {Rs, Rs, R~, R3} | | 10 | {R3,Rs,Rs} | | I1 | {R3, Rs, R~, R6} | ## (e) The output function. Fig. 2. The MAPM machine for R\_SET. Consider the behavior of the MAPM machine that uses the functions in Fig. 2 to process the sequence of the input structures c~ = N1NzN3N4Ns. N2 : { (STR, "of"), N4 : { (STR, "friendly"), (CA T, N5 : { (STR, … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{la-not-decoded --> \#\# (b) The failure function. | s | outputs(s) | |-----|------------------| | 4 | \{Rj, R2, R3, R3\} | | 5 | \{Rj, R2, R4\} | | 9 | \{Rs, Rs, R~, R3\} | | 10 | \{R3,Rs,Rs\} | | I1 | \{R3, Rs, R~, R6\} | \#\# (e) The output function. Fig. 2. The MAPM machine for R\textbackslash \_SET. <!-…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/the Aho-Corasick machine.md:offset=13727 \begin{verbatim} la-not-decoded --> ## (b) The failure function. | s | outputs(s) | |-----|------------------| | 4 | {Rj, R2, R3, R3} | | 5 | {Rj, R2, R4} | | 9 | {Rs, Rs, R~, R3} | | 10 | {R3,Rs,Rs} | | I1 | {R3, Rs, R~, R6} | ## (e) The output function. Fig. 2. The MAPM machine for R\_SET. Consider the behavior of the MAPM machine that uses the functions in Fig. 2 to process the sequence of the input structures c~ = N1NzN3N4Ns. N2 : { (STR, "of"), N4 : { (STR, "friendly"), (CA T, N5 : { (STR, … \end{verbatim} ``` </details>
982. ph-f2266369f620e3bb2436 — automata/docling_md/the Aho-Corasick machine.md ### Plain (markdown context) | outputs(s) | |-----|------------------| | 4 | {Rj, R2, R3, R3} | | 5 | {Rj, R2, R4} | | 9 | {Rs, Rs, R~, R3} | | 10 | {R3,Rs,Rs} | | I1 | {R3, Rs, R~, R6} | ## (e) The output function. Fig. 2. The MAPM machine for R\_SET. Consider the behavior of the MAPM machine that uses the functions in Fig. 2 to process the sequence of the input structures c~ = N1NzN3N4Ns. N2 : { (STR, "of"), N4 : { (STR, "friendly"), (CA T, N5 : { (STR, "professor"), (CA T, Noun), (SEM, Human)}. Since N1 … ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{| outputs(s) | |-----|------------------| | 4 | \{Rj, R2, R3, R3\} | | 5 | \{Rj, R2, R4\} | | 9 | \{Rs, Rs, R~, R3\} | | 10 | \{R3,Rs,Rs\} | | I1 | \{R3, Rs, R~, R6\} | \#\# (e) The output function. Fig. 2. The MAPM machine for R\textbackslash \_SET. Consider the behavior of t…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/the Aho-Corasick machine.md:offset=13781 \begin{verbatim} | outputs(s) | |-----|------------------| | 4 | {Rj, R2, R3, R3} | | 5 | {Rj, R2, R4} | | 9 | {Rs, Rs, R~, R3} | | 10 | {R3,Rs,Rs} | | I1 | {R3, Rs, R~, R6} | ## (e) The output function. Fig. 2. The MAPM machine for R\_SET. Consider the behavior of the MAPM machine that uses the functions in Fig. 2 to process the sequence of the input structures c~ = N1NzN3N4Ns. N2 : { (STR, "of"), N4 : { (STR, "friendly"), (CA T, N5 : { (STR, "professor"), (CA T, Noun), (SEM, Human)}. Since N1 … \end{verbatim} ``` </details>
983. ph-42042b20c31847ab78e6 — automata/docling_md/the Aho-Corasick machine.md ### Plain (markdown context) R3, R3} | | 5 | {Rj, R2, R4} | | 9 | {Rs, Rs, R~, R3} | | 10 | {R3,Rs,Rs} | | I1 | {R3, Rs, R~, R6} | ## (e) The output function. Fig. 2. The MAPM machine for R\_SET. Consider the behavior of the MAPM machine that uses the functions in Fig. 2 to process the sequence of the input structures c~ = N1NzN3N4Ns. N2 : { (STR, "of"), N4 : { (STR, "friendly"), (CA T, N5 : { (STR, "professor"), (CA T, Noun), (SEM, Human)}. Since N1 includes R3 and N2 includes Rs, CHECK(0, N1)=6 and CHECK(… ### Rendered (KaTeX / MathJax-style $ / $$) $$\text{R3, R3\} | | 5 | \{Rj, R2, R4\} | | 9 | \{Rs, Rs, R~, R3\} | | 10 | \{R3,Rs,Rs\} | | I1 | \{R3, Rs, R~, R6\} | \#\# (e) The output function. Fig. 2. The MAPM machine for R\textbackslash \_SET. Consider the behavior of the MAPM machine that uses the functions in Fig. 2 to proc…}$$ ### Raw LaTeX (paste into a .tex document) ```latex % Placeholder: formula region was not decoded in the original Docling run. % Re-run: docling --to md --enrich-formula % context @ automata/docling_md/the Aho-Corasick machine.md:offset=13849 \begin{verbatim} R3, R3} | | 5 | {Rj, R2, R4} | | 9 | {Rs, Rs, R~, R3} | | 10 | {R3,Rs,Rs} | | I1 | {R3, Rs, R~, R6} | ## (e) The output function. Fig. 2. The MAPM machine for R\_SET. Consider the behavior of the MAPM machine that uses the functions in Fig. 2 to process the sequence of the input structures c~ = N1NzN3N4Ns. N2 : { (STR, "of"), N4 : { (STR, "friendly"), (CA T, N5 : { (STR, "professor"), (CA T, Noun), (SEM, Human)}. Since N1 includes R3 and N2 includes Rs, CHECK(0, N1)=6 and CHECK(… \end{verbatim} ``` </details>