Single Final State for NFA

1. Single Final State for NFA COMP 335

2. Any NFA can be converted • to an equivalent NFA • with a single final state COMP 335

3. Example NFA Equivalent NFA COMP 335

4. In General NFA Equivalent NFA Single final state COMP 335

5. Add a final state Without transitions Extreme Case NFA without final state COMP 335

6. Properties of Regular Languages COMP 335

7. Union: Concatenation: Are regular Languages For regular languages and we will prove that: Star: Reversal: Complement: Intersection: COMP 335

8. Union: Concatenation: We say: Regular languages are closed under Star: Reversal: Complement: Intersection: COMP 335

9. Regular language Regular language NFA NFA Single final state Single final state COMP 335

10. Example COMP 335

11. Union • NFA for COMP 335

12. Example NFA for COMP 335

13. Concatenation • NFA for COMP 335

14. Example • NFA for COMP 335

15. Star Operation • NFA for COMP 335

16. Example • NFA for COMP 335

17. Reverse NFA for 1. Reverse all transitions 2. Make initial state final state and vice versa COMP 335

18. Example COMP 335

19. Complement 1. Take the DFA that accepts 2. Make final states non-final, and vice-versa COMP 335

20. Example COMP 335

21. regular regular regular regular regular Intersection DeMorgan’s Law: COMP 335

22. Example regular regular regular COMP 335

23. Regular Expressions COMP 335

24. Regular Expressions • Regular expressions • describe regular languages • Example: • describes the language COMP 335

25. Given regular expressions and Are regular expressions Recursive Definition Primitive regular expressions: COMP 335

26. Not a regular expression: Examples A regular expression: COMP 335

27. Languages of Regular Expressions • : language of regular expression • Example: COMP 335

28. Definition • For primitive regular expressions : COMP 335

29. Definition (continued) • For regular expressions and COMP 335

30. Example • Regular expression: COMP 335

31. Example • Regular expression COMP 335

32. Example • Regular expression COMP 335

33. = {all strings with at least two consecutive 0} Example • Regular expression COMP 335

34. = { all strings without two consecutive 0 } Example • Regular expression COMP 335

35. Equivalent Regular Expressions • Definition: • Regular expressions and • are equivalent if COMP 335

36. and are equivalent Reg. expressions Example = { all strings without two consecutive 0 } COMP 335

37. Regular ExpressionsandRegular Languages COMP 335

38. Theorem Languages Generated by Regular Expressions Regular Languages COMP 335

39. 1. For any regular expression the language is regular Theorem - Part 1 Languages Generated by Regular Expressions Regular Languages COMP 335

40. 2. For any regular language , there is a regular expression with Theorem - Part 2 Languages Generated by Regular Expressions Regular Languages COMP 335

41. 1. For any regular expression the language is regular Proof by induction on the size of Proof - Part 1 COMP 335

42. NFAs regular languages Induction Basis • Primitive Regular Expressions: COMP 335

43. Inductive Hypothesis • Assume for regular expressions and • that and are regular languages COMP 335

44. Inductive Step • We will prove: are regular Languages. COMP 335

45. By definition of regular expressions: COMP 335

46. We also know: Regular languages are closed under: Union Concatenation Star By inductive hypothesis we know: and are regular languages COMP 335

47. Therefore: Are regular languages COMP 335

48. And trivially: is a regular language COMP 335

49. Proof – Part 2 2. For any regular language there is a regular expression with Proof by construction of regular expression COMP 335

50. Since is regular, take an • NFA that accepts it Single final state COMP 335