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Single Final State for NFAs

Single Final State for NFAs. Any NFA can be converted to an equivalent NFA with a single final state. Example. NFA. Equivalent NFA. In General. NFA. Equivalent NFA. Single final state. Add a final state Without transitions. Extreme Case. NFA without final state.

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Single Final State for NFAs

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  1. Single Final State for NFAs Costas Busch - RPI

  2. Any NFA can be converted • to an equivalent NFA • with a single final state Costas Busch - RPI

  3. Example NFA Equivalent NFA Costas Busch - RPI

  4. In General NFA Equivalent NFA Single final state Costas Busch - RPI

  5. Add a final state Without transitions Extreme Case NFA without final state Costas Busch - RPI

  6. Properties of Regular Languages Costas Busch - RPI

  7. Union: Concatenation: Are regular Languages For regular languages and we will prove that: Star: Reversal: Complement: Intersection: Costas Busch - RPI

  8. Union: Concatenation: We say: Regular languages are closed under Star: Reversal: Complement: Intersection: Costas Busch - RPI

  9. Regular language Regular language NFA NFA Single final state Single final state Costas Busch - RPI

  10. Example Costas Busch - RPI

  11. Union • NFA for Costas Busch - RPI

  12. Example NFA for Costas Busch - RPI

  13. Concatenation • NFA for Costas Busch - RPI

  14. Example • NFA for Costas Busch - RPI

  15. Star Operation • NFA for Costas Busch - RPI

  16. Example • NFA for Costas Busch - RPI

  17. Reverse NFA for 1. Reverse all transitions 2. Make initial state final state and vice versa Costas Busch - RPI

  18. Example Costas Busch - RPI

  19. Complement 1. Take the DFA that accepts 2. Make final states non-final, and vice-versa Costas Busch - RPI

  20. Example Costas Busch - RPI

  21. regular regular regular regular regular Intersection DeMorgan’s Law: Costas Busch - RPI

  22. Example regular regular regular Costas Busch - RPI

  23. Regular Expressions Costas Busch - RPI

  24. Regular Expressions • Regular expressions • describe regular languages • Example: • describes the language Costas Busch - RPI

  25. Given regular expressions and Are regular expressions Recursive Definition Primitive regular expressions: Costas Busch - RPI

  26. Not a regular expression: Examples A regular expression: Costas Busch - RPI

  27. Languages of Regular Expressions • : language of regular expression • Example Costas Busch - RPI

  28. Definition • For primitive regular expressions: Costas Busch - RPI

  29. Definition (continued) • For regular expressions and Costas Busch - RPI

  30. Example • Regular expression: Costas Busch - RPI

  31. Example • Regular expression Costas Busch - RPI

  32. Example • Regular expression Costas Busch - RPI

  33. = { all strings with at least two consecutive 0 } Example • Regular expression Costas Busch - RPI

  34. = { all strings without two consecutive 0 } Example • Regular expression Costas Busch - RPI

  35. Equivalent Regular Expressions • Definition: • Regular expressions and • are equivalent if Costas Busch - RPI

  36. and are equivalent regular expr. Example = { all strings without two consecutive 0 } Costas Busch - RPI

  37. Regular ExpressionsandRegular Languages Costas Busch - RPI

  38. Theorem Languages Generated by Regular Expressions Regular Languages Costas Busch - RPI

  39. 1. For any regular expression the language is regular Theorem - Part 1 Languages Generated by Regular Expressions Regular Languages Costas Busch - RPI

  40. 2. For any regular language there is a regular expression with Theorem - Part 2 Languages Generated by Regular Expressions Regular Languages Costas Busch - RPI

  41. 1. For any regular expression the language is regular Proof by induction on the size of Proof - Part 1 Costas Busch - RPI

  42. NFAs regular languages Induction Basis • Primitive Regular Expressions: Costas Busch - RPI

  43. Inductive Hypothesis • Assume • for regular expressions and • that • and are regular languages Costas Busch - RPI

  44. Inductive Step • We will prove: Are regular Languages Costas Busch - RPI

  45. By definition of regular expressions: Costas Busch - RPI

  46. We also know: Regular languages are closed under: Union Concatenation Star By inductive hypothesis we know: and are regular languages Costas Busch - RPI

  47. Therefore: Are regular languages Costas Busch - RPI

  48. And trivially: is a regular language Costas Busch - RPI

  49. Proof – Part 2 2. For any regular language there is a regular expression with Proof by construction of regular expression Costas Busch - RPI

  50. Since is regular take the • NFA that accepts it Single final state Costas Busch - RPI

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