1. 1 A Single Final State for Finite Accepters

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

3. 3 Example NFA Equivalent NFA

4. 4 In General NFA Equivalent NFA Single final state

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

6. 6 Properties of Regular Languages

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

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

9. 9 For regular languages and take NFAs and with Single final state

10. 10 Example

11. 11 Union NFA for

12. 12 Example NFA for

13. 13 Concatenation NFA for

14. 14 Example NFA for

15. 15 Star Operation NFA for

16. 16 Example NFA for

17. 17 Complement For the complement of regular language : Take the DFA that accepts Construct such that: –Each final state of is nonfinal in nonfinal final We have:

18. 18 Example

19. 19 Intersection For regular languages and : regular

20. 20 Example Regular languages: The languageis regular

21. 21 Regular Expressions

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

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

24. 24 Examples A regular expression Not a regular expression

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

26. 26 Definition For primitive regular expressions:

27. 27 Definition (continued) For regular expressions and

28. 28 Example Regular expression:

29. 29 Example Regular expression

30. 30 Example Regular expression

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

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

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

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

35. 35 Regular Expressions and Regular Languages

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

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

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

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

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

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

42. 42 By definition of regular expressions:

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

44. 44 Therefore: Are regular languages

45. 45 And trivially: is a regular language

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

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

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

49. 49 Another Example:

50. 50 Reducing the states:

51. 51 Resulting Regular Expression:

52. 52 In General Removing states:

53. 53 Obtaining the final regular expression: