1. 1 Applications of Regular Closure

2. 2 The intersection of a context-free language and a regular language is a context-free language context free regular context-free Regular Closure

3. 3 An Application of Regular Closure Prove that: is context-free

4. 4 We know: is context-free

5. 5 is regular We also know:

6. 6 regularcontext-free is context-free context-free (regular closure)

7. 7 Another Application of Regular Closure Prove that: is not context-free

8. 8 context-freeregularcontext-free If is context-free Then Impossible!!! Therefore, is not context free (regular closure)

9. 9 Decidable Properties of Context-Free Languages

10. 10 Membership Question: for context-free grammar find if string Membership Algorithms: Parsers Exhaustive search parser CYK parsing algorithm

11. 11 Empty Language Question: for context-free grammar find if Algorithm: 1.Remove useless variables 2.Check if start variable is useless

12. 12 Infinite Language Question: for context-free grammar find if is infinite Algorithm: 1. Remove useless variables 2. Remove unit and productions 3. Create dependency graph for variables 4. If there is a loop in the dependency graph then the language is infinite

13. 13 Example: Dependency graph Infinite language

14. 14

15. 15 The Pumping Lemma for Context-Free Languages

16. 16 Take an infinite context-free language Example: Generates an infinite number of different strings

17. 17 A derivation:

18. 18 Derivation treestring

19. 19 repeated Derivation treestring

20. 20

21. 21 Repeated Part

22. 22 Another possible derivation

23. 23

24. 24

25. 25

26. 26 Therefore, the string is also generated by the grammar

27. 27 We know: We also know this string is generated:

28. 28 We know: Therefore, this string is also generated:

29. 29 We know: Therefore, this string is also generated:

30. 30 Therefore, this string is also generated: We know:

31. 31 Therefore, knowing that is generated by grammar, we also know that is generated by

32. 32 In general: We are given an infinite context-free grammar Assume has no unit-productions no -productions

33. 33 Take a string with length bigger than Some variable must be repeated in the derivation of (Number of productions) x (Largest right side of a production) > Consequence:

34. 34 Last repeated variable String repeated

35. 35 Possible derivations:

36. 36 We know: This string is also generated:

37. 37 This string is also generated: The original We know:

38. 38 This string is also generated: We know:

39. 39 This string is also generated: We know:

40. 40 This string is also generated: We know:

41. 41 Therefore, any string of the form is generated by the grammar

42. 42 knowing that we also know that Therefore,

43. 43 Observation: Since is the last repeated variable

44. 44 Observation: Since there are no unit or productions

45. 45 The Pumping Lemma: there exists an integer such that for any string we can write For infinite context-free language with lengths and it must be:

46. 46 Applications of The Pumping Lemma

47. 47 Context-free languages Non-context free languages

48. 48 Theorem: The language is not context free Proof: Use the Pumping Lemma for context-free languages

49. 49 Assume for contradiction that is context-free Since is context-free and infinite we can apply the pumping lemma

50. 50 Pumping Lemma gives a magic number such that: Pick any string with length We pick:

51. 51 We can write: with lengths and

52. 52 Pumping Lemma says: for all

53. 53 We examine all the possible locations of string in

54. 54 Case 1: is within

55. 55 Case 1: and consist from only

56. 56 Case 1: Repeating and

57. 57 Case 1: From Pumping Lemma:

58. 58 Case 1: From Pumping Lemma: However: Contradiction!!!

59. 59 Case 2: is within

60. 60 Case 2: Similar analysis with case 1

61. 61 Case 3: is within

62. 62 Case 3: Similar analysis with case 1

63. 63 Case 4: overlaps and

64. 64 Case 4: Possibility 1:contains only

65. 65 Case 4: Possibility 1:contains only

66. 66 Case 4: From Pumping Lemma:

67. 67 Case 4: From Pumping Lemma: However: Contradiction!!!

68. 68 Case 4: Possibility 2:contains and contains only

69. 69 Case 4: Possibility 2:contains and contains only

70. 70 Case 4: From Pumping Lemma:

71. 71 Case 4: From Pumping Lemma: However: Contradiction!!!

72. 72 Case 4: Possibility 3:contains only contains and

73. 73 Case 4: Possibility 3:contains only contains and Similar analysis with Possibility 2

74. 74 Case 5: overlaps and

75. 75 Case 5: Similar analysis with case 4

76. 76 There are no other cases to consider (since, string cannot overlap, and at the same time)

77. 77 In all cases we obtained a contradiction Therefore: The original assumption that is context-free must be wrong Conclusion:is not context-free