1. Costas Busch - RPI1 Single Final State for NFAs

2. Costas Busch - RPI2 Any NFA can be converted to an equivalent NFA with a single final state

3. Costas Busch - RPI3 NFA Equivalent NFA Example

4. Costas Busch - RPI4 NFA In General Equivalent NFA Single final state

5. Costas Busch - RPI5 Extreme Case NFA without final state Add a final state Without transitions

6. Costas Busch - RPI6 Properties of Regular Languages

7. Costas Busch - RPI7 Concatenation: Star: Union: Are regular Languages For regular languages and we will prove that: Complement: Intersection: Reversal:

8. Costas Busch - RPI8 We say: Regular languages are closed under Concatenation: Star: Union: Complement: Intersection: Reversal:

9. Costas Busch - RPI9 Regular language Single final state NFA Single final state Regular language NFA

10. Costas Busch - RPI10 Example

11. Costas Busch - RPI11 Union NFA for

12. Costas Busch - RPI12 Example NFA for

13. Costas Busch - RPI13 Concatenation NFA for

14. Costas Busch - RPI14 Example NFA for

15. Costas Busch - RPI15 Star Operation NFA for

16. Costas Busch - RPI16 Example NFA for

17. Costas Busch - RPI17 Reverse NFA for 1. Reverse all transitions 2. Make initial state final state and vice versa

18. Costas Busch - RPI18 Example

19. Costas Busch - RPI19 Complement 1. Take the DFA that accepts 2. Make final states non-final, and vice-versa

20. Costas Busch - RPI20 Example

21. Costas Busch - RPI21 Intersection DeMorgan’s Law: regular

22. Costas Busch - RPI22 Example regular

23. Costas Busch - RPI23 Regular Expressions

24. Costas Busch - RPI24 Regular Expressions Regular expressions describe regular languages Example: describes the language

25. Costas Busch - RPI25 Recursive Definition Are regular expressions Primitive regular expressions: Given regular expressions and

26. Costas Busch - RPI26 Examples A regular expression: Not a regular expression:

27. Costas Busch - RPI27 Languages of Regular Expressions : language of regular expression Example

28. Costas Busch - RPI28 Definition For primitive regular expressions:

29. Costas Busch - RPI29 Definition (continued) For regular expressions and

30. Costas Busch - RPI30 Example Regular expression:

31. Costas Busch - RPI31 Example Regular expression

32. Costas Busch - RPI32 Example Regular expression

33. Costas Busch - RPI33 Example Regular expression = { all strings with at least two consecutive 0 }

34. Costas Busch - RPI34 Example Regular expression = { all strings without two consecutive 0 }

35. Costas Busch - RPI35 Equivalent Regular Expressions Definition: Regular expressions and are equivalent if

36. Costas Busch - RPI36 Example = { all strings without two consecutive 0 } and are equivalent regular expr.

37. Costas Busch - RPI37 Regular Expressions and Regular Languages

38. Costas Busch - RPI38 Theorem Languages Generated by Regular Expressions Regular Languages

39. Costas Busch - RPI39 Theorem - Part 1 1. For any regular expression the language is regular Languages Generated by Regular Expressions Regular Languages

40. Costas Busch - RPI40 Theorem - Part 2 Languages Generated by Regular Expressions Regular Languages 2. For any regular language there is a regular expression with

41. Costas Busch - RPI41 Proof - Part 1 1. For any regular expression the language is regular Proof by induction on the size of

42. Costas Busch - RPI42 Induction Basis Primitive Regular Expressions: NFAs regular languages

43. Costas Busch - RPI43 Inductive Hypothesis Assume for regular expressions and that and are regular languages

44. Costas Busch - RPI44 Inductive Step We will prove: Are regular Languages

45. Costas Busch - RPI45 By definition of regular expressions:

46. Costas Busch - RPI46 By inductive hypothesis we know: and are regular languages Regular languages are closed under: Union Concatenation Star We also know:

47. Costas Busch - RPI47 Therefore: Are regular languages

48. Costas Busch - RPI48 And trivially: is a regular language

49. Costas Busch - RPI49 Proof – Part 2 2. For any regular language there is a regular expression with Proof by construction of regular expression

50. Costas Busch - RPI50 Since is regular take the NFA that accepts it Single final state

51. Costas Busch - RPI51 From construct the equivalent Generalized Transition Graph in which transition labels are regular expressions Example:

52. Costas Busch - RPI52 Another Example:

53. Costas Busch - RPI53 Reducing the states:

54. Costas Busch - RPI54 Resulting Regular Expression:

55. Costas Busch - RPI55 In General Removing states:

56. Costas Busch - RPI56 The final transition graph: The resulting regular expression: