1. Costas Busch - RPI1 Grammars

2. Costas Busch - RPI2 Grammars Grammars express languages Example: the English language

3. Costas Busch - RPI3

4. 4 A derivation of “the dog walks”:

5. Costas Busch - RPI5 A derivation of “a cat runs”:

6. Costas Busch - RPI6 Language of the grammar: L = { “a cat runs”, “a cat walks”, “the cat runs”, “the cat walks”, “a dog runs”, “a dog walks”, “the dog runs”, “the dog walks” }

7. Costas Busch - RPI7 Notation VariableTerminal Production Rules

8. Costas Busch - RPI8 Another Example Grammar: Derivation of sentence :

9. Costas Busch - RPI9 Grammar: Derivation of sentence :

10. Costas Busch - RPI10 Other derivations:

11. Costas Busch - RPI11 Language of the grammar

12. Costas Busch - RPI12 More Notation Grammar Set of variables Set of terminal symbols Start variable Set of Production rules

13. Costas Busch - RPI13 Example Grammar :

14. Costas Busch - RPI14 More Notation Sentential Form: A sentence that contains variables and terminals Example: Sentential Formssentence

15. Costas Busch - RPI15 We write: Instead of:

16. Costas Busch - RPI16 In general we write: If:

17. Costas Busch - RPI17 By default:

18. Costas Busch - RPI18 Example Grammar Derivations

19. Costas Busch - RPI19 Grammar Example Derivations

20. Costas Busch - RPI20 Another Grammar Example Grammar : Derivations:

21. Costas Busch - RPI21 More Derivations

22. Costas Busch - RPI22 Language of a Grammar For a grammar with start variable : String of terminals

23. Costas Busch - RPI23 Example For grammar : Since:

24. Costas Busch - RPI24 A Convenient Notation

25. Costas Busch - RPI25 Linear Grammars

26. Costas Busch - RPI26 Linear Grammars Grammars with at most one variable at the right side of a production Examples:

27. Costas Busch - RPI27 A Non-Linear Grammar Grammar : Number of in string

28. Costas Busch - RPI28 Another Linear Grammar Grammar :

29. Costas Busch - RPI29 Right-Linear Grammars All productions have form: Example: or string of terminals

30. Costas Busch - RPI30 Left-Linear Grammars All productions have form: Example: or string of terminals

31. Costas Busch - RPI31 Regular Grammars

32. Costas Busch - RPI32 Regular Grammars A regular grammar is any right-linear or left-linear grammar Examples:

33. Costas Busch - RPI33 Observation Regular grammars generate regular languages Examples:

34. Costas Busch - RPI34 Regular Grammars Generate Regular Languages

35. Costas Busch - RPI35 Theorem Languages Generated by Regular Grammars Regular Languages

36. Costas Busch - RPI36 Theorem - Part 1 Languages Generated by Regular Grammars Regular Languages Any regular grammar generates a regular language

37. Costas Busch - RPI37 Theorem - Part 2 Languages Generated by Regular Grammars Regular Languages Any regular language is generated by a regular grammar

38. Costas Busch - RPI38 Proof – Part 1 Languages Generated by Regular Grammars Regular Languages The language generated by any regular grammar is regular

39. Costas Busch - RPI39 The case of Right-Linear Grammars Let be a right-linear grammar We will prove: is regular Proof idea: We will construct NFA with

40. Costas Busch - RPI40 Grammar is right-linear Example:

41. Costas Busch - RPI41 Construct NFA such that every state is a grammar variable: special final state

42. Costas Busch - RPI42 Add edges for each production:

43. Costas Busch - RPI43

44. Costas Busch - RPI44

45. Costas Busch - RPI45

46. Costas Busch - RPI46

47. Costas Busch - RPI47

48. Costas Busch - RPI48 Grammar NFA

49. Costas Busch - RPI49 In General A right-linear grammar has variables: and productions: or

50. Costas Busch - RPI50 We construct the NFA such that: each variable corresponds to a node: special final state

51. Costas Busch - RPI51 For each production: we add transitions and intermediate nodes ………

52. Costas Busch - RPI52 For each production: we add transitions and intermediate nodes ………

53. Costas Busch - RPI53 Resulting NFA looks like this: It holds that:

54. Costas Busch - RPI54 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

55. Costas Busch - RPI55 Since is left-linear grammar the productions look like:

56. Costas Busch - RPI56 Construct right-linear grammar Left linear Right linear

57. Costas Busch - RPI57 Construct right-linear grammar Left linear Right linear

58. Costas Busch - RPI58 It is easy to see that: Since is right-linear, we have: Regular Language Regular Language Regular Language

59. Costas Busch - RPI59 Proof - Part 2 Languages Generated by Regular Grammars Regular Languages Any regular language is generated by some regular grammar

60. Costas Busch - RPI60 Proof idea: Let be the NFA with. Construct from a regular grammar such that Any regular language is generated by some regular grammar

61. Costas Busch - RPI61 Since is regular there is an NFA such that Example:

62. Costas Busch - RPI62 Convert to a right-linear grammar

63. Costas Busch - RPI63

64. Costas Busch - RPI64

65. Costas Busch - RPI65

66. Costas Busch - RPI66 In General For any transition: Add production: variableterminalvariable

67. Costas Busch - RPI67 For any final state: Add production:

68. Costas Busch - RPI68 Since is right-linear grammar is also a regular grammar with