A Single Final State for Finite Accepters

1. A Single Final Statefor Finite Accepters

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

3. Example NFA Equivalent NFA

4. In General NFA Equivalent NFA Single final state

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

6. Properties of Regular Languages

7. Properties • Take any regular languages and • We will prove: Union: Concatenation: Are regular Languages Star: Complement: Intersection:

8. We Say • Regular Languages are closed: • Under union: • Under concatenation: • Under the star operation: • Under complement: • Under intersection:

9. For regular languages and • take NFAs and with 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. Complement • For the complement of regular language : • Take the DFA that accepts • Construct such that: • Each final state of is nonfinal in • nonfinalfinal • We have:

18. Example

19. Intersection • For regular languages and : regular regular regular regular regular

20. Example • Regular languages: The language is regular

21. Regular Expressions

22. Regular Expressions • Regular expressions • are another way of expressing • regular languages • Example: • Stands for the language

23. Recursive Definition • Regular Expressions: • Primitive regular expressions: • Given regular expressions and Are regular expressions

24. Examples • A regular expression • Not a regular expression

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

26. Definition • For primitive regular expressions:

27. Definition (continued) • For regular expressions and

28. Example • Regular expression:

29. Example • Regular expression

30. Example • Regular expression

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

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

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

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

35. Regular ExpressionsandRegular Languages

36. Theorem • The class of languages • described by Regular expressions • is identical • to the Regular languages

37. In Other Words • For any regular expression • the language is regular • For any regular language there is • a regular expression with

38. Proof • First we prove: • For any regular expression • the language is regular

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

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

41. Inductive Step • We will prove that: Are regular Languages

42. By definition of regular expressions:

43. By inductive hypothesis • and are regular languages • We know: Regular languages are closed under union concatenation star operation

44. Therefore: Are regular languages

45. And trivially: is a regular language

46. Proof - Second Part • Now we want to prove: • For any regular language there is • a regular expression with

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

48. From Construct the equivalent • Generalized Transition Graph • labels of transitions • are regular expressions Example:

49. Another Example:

50. Reducing the states: