1. 1 Reverse of a Regular Language

2. 2 Theorem: The reverse of a regular language is a regular language Proof idea: Construct NFA that accepts : invert the transitions of the NFA that accepts

3. 3 Proof Since is regular, there is NFA that accepts Example:

4. 4 Invert Transitions

5. 5 Make old initial state a final state

6. 6 Add a new initial state

7. 7 Resulting machine accepts is regular

8. 8 Grammars

9. 9 Grammars express languages Example: the English language

10. 10

11. 11 A derivation of “the boy walks”:

12. 12 A derivation of “a dog runs”:

13. 13 Language of the grammar: L = { “a boy runs”, “a boy walks”, “the boy runs”, “the boy walks”, “a dog runs”, “a dog walks”, “the dog runs”, “the dog walks” }

14. 14 Notation Variable or Non-terminal Terminal Production rule

15. 15 Another Example Grammar: Derivation of sentence :

16. 16 Grammar: Derivation of sentence :

17. 17 Other derivations:

18. 18 Language of the grammar

19. 19 More Notation Grammar Set of variables Set of terminal symbols Start variable Set of Production rules

20. 20 Example Grammar :

21. 21 More Notation Sentential Form: A sentence that contains variables and terminals Example: Sentential Formssentence

22. 22 We write: Instead of:

23. 23 In general we write: If:

24. 24 By default:

25. 25 Example Grammar Derivations

26. 26 Grammar Example Derivations

27. 27 Another Grammar Example Grammar : Derivations:

28. 28 More Derivations

29. 29 Language of a Grammar For a grammar with start variable : String of terminals

30. 30 Example For grammar : Since:

31. 31 A Convenient Notation

32. 32 Linear Grammars

33. 33 Linear Grammars Grammars with at most one variable at the right side of a production Examples:

34. 34 A Non-Linear Grammar Grammar :

35. 35 Another Linear Grammar Grammar :

36. 36 Right-Linear Grammars All productions have form: Example: or

37. 37 Left-Linear Grammars All productions have form: Example: or

38. 38 Regular Grammars

39. 39 Regular Grammars A regular grammar is any right-linear or left-linear grammar Examples:

40. 40 Observation Regular grammars generate regular languages Examples:

41. 41 Regular Grammars Generate Regular Languages

42. 42 Theorem Languages Generated by Regular Grammars Regular Languages

43. 43 Theorem - Part 1 Languages Generated by Regular Grammars Regular Languages Any regular grammar generates a regular language

44. 44 Theorem - Part 2 Languages Generated by Regular Grammars Regular Languages Any regular language is generated by a regular grammar

45. 45 Proof – Part 1 Languages Generated by Regular Grammars Regular Languages The language generated by any regular grammar is regular

46. 46 The case of Right-Linear Grammars Let be a right-linear grammar We will prove: is regular Proof idea: We will construct NFA with

47. 47 Grammar is right-linear Example:

48. 48 Construct NFA such that every state is a grammar variable: special final state

49. 49 Add edges for each production:

50. 50

51. 51

52. 52

53. 53

54. 54

55. 55 Grammar NFA

56. 56 In General A right-linear grammar has variables: and productions: or

57. 57 We construct the NFA such that: each variable corresponds to a node: special final state

58. 58 For each production: we add transitions and intermediate nodes ………

59. 59 For each production: we add transitions and intermediate nodes ………

60. 60 Resulting NFA looks like this: It holds that:

61. 61 The case of Left-Linear Grammars Let be a left-linear grammar We will prove: is regular Proof idea: We will construct a right-linear grammar with

62. 62 Since is left-linear grammar the productions look like:

63. 63 Construct right-linear grammar In :

64. 64 Construct right-linear grammar In :

65. 65 It is easy to see that: Since is right-linear, we have: Regular Language Regular Language Regular Language

66. 66 Proof - Part 2 Languages Generated by Regular Grammars Regular Languages Any regular language is generated by some regular grammar

67. 67 Proof idea: Let be the NFA with. Construct from a regular grammar such that Any regular language is generated by some regular grammar

68. 68 Since is regular there is an NFA such that Example:

69. 69 Convert to a right-linear grammar

70. 70

71. 71

72. 72

73. 73 In General For any transition: Add production: variableterminalvariable

74. 74 For any final state: Add production:

75. 75 Since is right-linear grammar is also a regular grammar with