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Understanding Regular Grammars and Languages: Theorem and Proof

Learn about regular grammars generating regular languages, such as right-linear and left-linear grammars. Follow the proof showing the interrelation between regular languages and regular grammars. Convert NFAs to regular grammars and explore the generalization of the process.

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Understanding Regular Grammars and Languages: Theorem and Proof

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  1. Regular Grammars GenerateRegular Languages

  2. Theorem • Regular grammars generate exactly the • class of regular languages: • If is a • regular grammar • then is • a regular language If is a regular language then there is a regular grammar with

  3. Proof • First we prove: • If is a • regular grammar • then is • a regular language can be: Right-linear grammar or Left-linear grammar

  4. The case of Right-Linear Grammars • Let be a right-linear grammar • We will show: is regular • The proof: • We will construct an NFA • with

  5. Grammar is right-linear Example:

  6. Construct NFA such that • every state is a variable: special final state

  7. Add edges for each production:

  8. NFA Grammar

  9. In General • A right-linear grammar • has variables: • and productions: or

  10. We construct the NFA such that: • each variable corresponds to a node:

  11. For each production: • we add transitions and intermediate nodes ………

  12. For each production: • we add transitions and intermediate nodes ………

  13. Resulting NFA looks like this:

  14. Now, we need to show:

  15. The Case: • Take We will show: there is a path with label in from state to state …………

  16. Grammar looks like: strings

  17. ……

  18. …… ……

  19. …… …… …… ……

  20. Since: • We have: …… …… …… ……

  21. Since: • We have: • We have: and ……

  22. The Case: Take We will show that in

  23. Since • there is a path ……

  24. Write: • There is a path …… …… …… ……

  25. Since: • This derivation is possible …… …… …… ……

  26. Since: • We have: • We have:

  27. The Case of Left-Linear Grammars • Let be a left-linear grammar • We will show: is regular • The proof: • We will construct a right-linear • grammar with

  28. Since is left-linear grammar the productions look like:

  29. Construct right-linear grammar In : In :

  30. In : In :

  31. It is easy to see that: • Since is right-linear, we have: • is regular language • is regular language (homework) • is regular language

  32. Proof - Part 2 • Now we will prove: If is a regular language then there is a regular grammar with Proof outline: we will take an NFA for and convert it to a regular grammar

  33. Since is regular • There is an NFA such that Example:

  34. Convert to a right-linear grammar

  35. We can generalize this process for • any regular language : • For any regular language • we obtain an right-linear grammar • with

  36. Since is right-linear grammar • is also a regular grammar • with

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