Single Final State for NFAs and DFAs

1. Single Final State for NFAs and DFAs

2. Observation • Any Finite Automaton (NFA or DFA) • can be converted to an equivalent NFA • with a single final state

3. Equivalent NFA Example NFA

4. Equivalent NFA Single final state In General NFA

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

6. Some Properties of Regular Languages

7. Union: Concatenation: Are regular Languages Star: Properties For regular languages and we will prove that:

8. Union: Concatenation: Star: We Say: Regular languages are closed under

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

10. Example

11. Union • NFA for

12. Example NFA for

13. Concatenation • NFA for

14. Example • NFA for

15. Star Operation • NFA for

16. Example • NFA for

17. Regular Expressions

18. Regular Expressions • Regular expressions • describe regular languages • Example: • describes the language

19. Given regular expressions and Are regular expressions Recursive Definition Primitive regular expressions:

20. Not a regular expression: Examples A regular expression:

21. Languages of Regular Expressions • : language of regular expression • Example

22. Definition • For primitive regular expressions:

23. Definition (continued) • For regular expressions and

24. Example • Regular expression:

25. Example • Regular expression

26. Example • Regular expression

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

28. = { all strings without two consecutive 0 } Example • Regular expression

29. Equivalent Regular Expressions • Definition: • Regular expressions and • are equivalent if

30. and are equivalent regular expr. Example = { all strings with at least two consecutive 0 }

31. Regular ExpressionsandRegular Languages

32. Theorem Languages Generated by Regular Expressions Regular Languages

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

34. 2. For any regular language there is a regular expression with Theorem - Part 2 Languages Generated by Regular Expressions Regular Languages

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

36. NFAs regular languages Induction Basis • Primitive Regular Expressions:

37. Inductive Hypothesis • Assume • for regular expressions and • that • and are regular languages

38. Inductive Step • We will prove: Are regular Languages

39. By definition of regular expressions:

40. We also know: Regular languages are closed under union concatenation star By inductive hypothesis we know: and are regular languages

41. Therefore: Are regular languages

42. And trivially: is a regular language

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

44. Since is regular take the • NFA that accepts it Single final state

45. From construct the equivalent • Generalized Transition Graph • transition labels • are regular expressions Example:

46. Another Example:

47. Reducing the states:

48. Resulting Regular Expression:

49. In General • Removing states:

50. Obtaining the final regular expression: