1. 1 CD5560 FABER Formal Languages, Automata and Models of Computation Lecture 9 Mälardalen University 2006

2. 2 Content Language Hierarchy Deterministic PDAs (DPDAs) Non-DPDA (NPDA) NPDAs Have More Power than DPDAs Positive Properties of Context Free Languages Negative Properties of Context Free Languages Intersection of CFL and RL (Regular Closure) The Pumping Lemma for CFL Applications of the Pumping Lemma for CFL

3. 3 Language Hierarchy

4. 4 Regular Languages Context-Free Languages Non-regular languages

5. 5 Deterministic PDAs (DPDAs)

6. 6 Allowed DPDAs

7. 7 Not allowed

8. 8 Allowed Something must be matched from the stack

9. 9 Not allowed

10. 10 DPDA example

11. 11 The language is deterministic context-free.

12. 12 Definition A language is deterministic context-free if some DPDA accepts it.

13. 13 Example of Non-DPDA (NPDA)

14. 14 Not allowed in DPDAs

15. 15 NPDAs Have More Power than DPDAs

16. 16 We will show: which is not deterministic context-free (not accepted by a DPDA). There is a context-free language (accepted by a NPDA)

17. 17 The language is:

18. 18 The language is context-free Context-free grammar for there is an NPDA that accepts

19. 19 is not deterministic context-free Theorem The language (i.e., there is no DPDA that accepts ). (Each a is to be matched by either one or two b. An initial choice must be made.)

20. 20 Proof (by contradiction) Assume the opposite, i.e. that is deterministic context free. Therefore: there is a DPDA that accepts

21. 21 accepts DPDA with

22. 22 The language is not context-free (we will prove it later on using Pumping Lemma for CFL) Fact 1

23. 23 The language is not context-free (we will prove later on that the union of two context-free languages is context-free) Fact 2

24. 24 We will construct a NPDA that accepts: Contradiction, as is not context-free!

25. 25 We modify Replace with Modified

26. 26 The NPDA that accepts Modified Original

27. 27 Since is accepted by a NPDA it is context-free. Contradiction!

28. 28 Therefore: There is no DPDA that accepts Not deterministic context free. END OF PROOF

29. 29 Positive Properties of Context-Free Languages

30. 30 Context-free languages are closed under Union is context free is context-free Union

31. 31 Example Union LanguageGrammar

32. 32 In general: For context-free languages with context-free grammars and start variables The grammar of the union has new start variable and additional production

33. 33 Context-free languages are closed under Concatenation is context free is context-free Concatenation

34. 34 Example Concatenation Language Grammar

35. 35 In general: For context-free languages with context-free grammars and start variables The grammar of the concatenation has new start variable and additional production

36. 36 Context-free languages are closed under star-operation is context free Star Operation is context free

37. 37 Example LanguageGrammar Star Operation

38. 38 In general: For context-free language with context-free grammar and start variable The grammar of the star operation has new start variable and additional production

39. 39 Negative Properties of Context-Free Languages

40. 40 Context-free languages are not closed under intersection is context free not necessarily context-free Intersection

41. 41 Example Context-free: NOT context-free Intersection

42. 42 Context-free languages are not closed under complement is context free not necessarily context-free Complement

43. 43 NOT context-free Example Context-free: Complement

44. 44 Intersection of Context-Free Languages and Regular Languages (Regular Closure)

45. 45 The intersection of a context-free language and a regular language is a context-free language context free regular context-free

46. 46 Construct a new NPDA machine that accepts forNPDA Machine context-free forDFA Machine regular simulates in parallel and

47. 47 transition NPDA transition DFA transition NPDA

48. 48 initial state NPDA DFA initial state NPDA

49. 49 final state NPDA final states DFA final states NPDA

50. 50 simulates in parallel and accepts string if and only if accepts string and

51. 51 Therefore: (since is NPDA) is context-free

52. 52 Applications of Regular Closure

53. 53 Prove that is context-free An Application of Regular Closure

54. 54 is regular We know is context-free

55. 55 regularcontext-free is context-free context-free (regular closure) END OF PROOF

56. 56 Prove that is not context-free Another Application of Regular Closure

57. 57 context-free regular context-free Ifis context-free Then impossible! Therefore, is not context free (regular closure) END OF PROOF

58. 58 The Pumping Lemma for Context-Free Languages

59. 59 Take an infinite context-free language Example: Generates an infinite number of different strings

60. 60 A derivation

61. 61 Derivation tree string

62. 62 Derivation tree string repeated

63. 63

64. 64 Repeated Part

65. 65 Another possible derivation

66. 66

67. 67

68. 68

69. 69 Therefore, the string is also generated by the grammar

70. 70 We know: We also know following string is generated:

71. 71 We know: Therefore, following string is also generated:

72. 72 We know: Therefore, following string is also generated:

73. 73 Therefore, following string is also generated: We know:

74. 74 Therefore, knowing that is generated by grammar is generated by We also know that

75. 75 In general We are given an infinite context-free grammar Assume has no unit-productions and no -productions

76. 76 Take a string with length bigger than (Number of productions) x (Largest right side of a production) > Some variable must be repeated in the derivation of Consequence:

77. 77 Last repeated variable String repeated strings of terminals

78. 78 Possible derivations

79. 79 We know: Following string is also generated:

80. 80 This string is also generated: The original We know:

81. 81 This string is also generated: We know:

82. 82 This string is also generated: We know:

83. 83 This string is also generated: We know:

84. 84 Therefore, any string of the form is generated by the grammar

85. 85 knowing that we also know that Therefore

86. 86 Observation Since is the last repeated variable

87. 87 Observation Since there are no unit or productions

88. 88 The Pumping Lemma for CFL there exists an integer such that for any string we can write For infinite context-free language with lengths and

89. 89 Applications of The Pumping Lemma for CFL

90. 90 Regular Languages Context-Free Languages Non-regular languages Unrestricted grammar languages

91. 91 Theorem The language is not context free Proof Use the Pumping Lemma for context-free languages

92. 92 Assume the contrary, that is context-free. Since is context-free and infinite we can apply the pumping lemma.

93. 93 Pumping Lemma gives a number such that: For any string with length We can choose e.g.

94. 94 We can write: with lengths and

95. 95 Pumping Lemma says: for all

96. 96 We examine all the possible locations of string in

97. 97 Case 1: is within

98. 98 Case 1: and consist from only

99. 99 Case 1: Repeating and

100. 100 Case 1: From Pumping Lemma:

101. 101 Case 1: From Pumping Lemma: However: Contradiction!

102. 102 Case 2: is within

103. 103 Case 2: Similar analysis to case 1

104. 104 Case 3: is within

105. 105 Case 3: Similar analysis to case 1

106. 106 Case 4: overlaps and

107. 107 Case 4: Possibility 1: contains only

108. 108 Case 4: Possibility 1:contains only

109. 109 Case 4: From Pumping Lemma:

110. 110 Case 4: From Pumping Lemma: However: Contradiction!

111. 111 Case 4: Possibility 2:contains and contains only

112. 112 Case 4: Possibility 2: contains and contains only

113. 113 Case 4: From Pumping Lemma:

114. 114 Case 4: From Pumping Lemma: However: Contradiction!

115. 115 Case 4: Possibility 3:contains only contains and

116. 116 Case 4: Possibility 3: contains only contains and Similar analysis with Possibility 2

117. 117 Case 5: overlaps and

118. 118 Case 5: Similar analysis to case 4

119. 119 There are no other cases to consider (since, string cannot overlap, and at the same time)

120. 120 In all cases we obtained a contradiction Therefore: The original assumption that is context-free must be wrong Conclusion:is not context-free END OF PROOF