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Reverse of a Regular Language

Reverse of a Regular Language. Proof idea:. Construct NFA that accepts :. invert the transitions of the NFA that accepts. Theorem:. The reverse of a regular language is a regular language. Proof. Since is regular, there is NFA that accepts. Example:.

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Reverse of a Regular Language

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  1. Reverse of a Regular Language

  2. Proof idea: Construct NFA that accepts : invert the transitions of the NFA that accepts Theorem: The reverse of a regular language is a regular language

  3. Proof Since is regular, there is NFA that accepts Example:

  4. Invert Transitions

  5. Make old initial state a final state

  6. Add a new initial state

  7. Resulting machine accepts is regular

  8. Grammars

  9. Grammars • Grammars express languages • Example: the English language

  10. A derivation of “the boy walks”:

  11. A derivation of “a dog runs”:

  12. Language of the grammar: L = { “a boy runs”, “a boy walks”, “the boy runs”, “the boy walks”, “a dog runs”, “a dog walks”, “the dog runs”, “the dog walks” }

  13. Notation Variable or Non-terminal Terminal Production rule

  14. Another Example • Grammar: • Derivation of sentence :

  15. Grammar: • Derivation of sentence :

  16. Other derivations:

  17. Language of the grammar

  18. More Notation • Grammar Set of variables Set of terminal symbols Start variable Set of Production rules

  19. Example • Grammar :

  20. More Notation • Sentential Form: • A sentence that contains • variables and terminals • Example: Sentential Forms sentence

  21. We write: • Instead of:

  22. In general we write: • If:

  23. By default:

  24. Example Grammar Derivations

  25. Example Grammar Derivations

  26. Another Grammar Example • Grammar : Derivations:

  27. More Derivations

  28. Language of a Grammar • For a grammar • with start variable : String of terminals

  29. Example • For grammar : Since:

  30. A Convenient Notation

  31. Linear Grammars

  32. Linear Grammars • Grammars with • at most one variable at the right side • of a production • Examples:

  33. A Non-Linear Grammar Grammar :

  34. Another Linear Grammar • Grammar :

  35. Right-Linear Grammars • All productions have form: • Example: or

  36. Left-Linear Grammars • All productions have form: • Example: or

  37. Regular Grammars

  38. Regular Grammars • A regular grammaris any • right-linear or left-linear grammar • Examples:

  39. Observation • Regular grammars generate regular languages • Examples:

  40. Regular Grammars GenerateRegular Languages

  41. Theorem Languages Generated by Regular Grammars Regular Languages

  42. Theorem - Part 1 Languages Generated by Regular Grammars Regular Languages Any regular grammar generates a regular language

  43. Theorem - Part 2 Languages Generated by Regular Grammars Regular Languages Any regular language is generated by a regular grammar

  44. Proof – Part 1 Languages Generated by Regular Grammars Regular Languages The language generated by any regular grammar is regular

  45. The case of Right-Linear Grammars • Let be a right-linear grammar • We will prove: is regular • Proof idea: We will construct NFA • with

  46. Grammar is right-linear Example:

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

  48. Add edges for each production:

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