1. Courtesy Costas Busch - RPI1 Non Deterministic Automata

2. Courtesy Costas Busch - RPI2 Alphabet = Nondeterministic Finite Accepter (NFA)

3. Courtesy Costas Busch - RPI3 Two choices Alphabet = Nondeterministic Finite Accepter (NFA)

4. Courtesy Costas Busch - RPI4 No transition Two choices No transition Alphabet = Nondeterministic Finite Accepter (NFA)

5. Courtesy Costas Busch - RPI5 First Choice

6. Courtesy Costas Busch - RPI6 First Choice

7. Courtesy Costas Busch - RPI7 First Choice

8. Courtesy Costas Busch - RPI8 “accept” First Choice All input is consumed

9. Courtesy Costas Busch - RPI9 Second Choice

10. Courtesy Costas Busch - RPI10 Second Choice

11. Courtesy Costas Busch - RPI11 Second Choice No transition: the automaton hangs

12. Courtesy Costas Busch - RPI12 Second Choice “reject” Input cannot be consumed

13. Courtesy Costas Busch - RPI13 An NFA accepts a string: when there is a computation of the NFA that accepts the string all the input is consumed and the automaton is in a final state AND

14. Courtesy Costas Busch - RPI14 Example is accepted by the NFA: “accept” “reject” because this computation accepts

15. Courtesy Costas Busch - RPI15 Rejection example

16. Courtesy Costas Busch - RPI16 First Choice

17. Courtesy Costas Busch - RPI17 First Choice “reject”

18. Courtesy Costas Busch - RPI18 Second Choice

19. Courtesy Costas Busch - RPI19 Second Choice

20. Courtesy Costas Busch - RPI20 Second Choice “reject”

21. Courtesy Costas Busch - RPI21 An NFA rejects a string: when there is no computation of the NFA that accepts the string: All the input is consumed and the automaton is in a non final state The input cannot be consumed OR

22. Courtesy Costas Busch - RPI22 Example is rejected by the NFA: “reject” All possible computations lead to rejection

23. Courtesy Costas Busch - RPI23 Rejection example

24. Courtesy Costas Busch - RPI24 First Choice

25. Courtesy Costas Busch - RPI25 First Choice No transition: the automaton hangs

26. Courtesy Costas Busch - RPI26 “reject” First Choice Input cannot be consumed

27. Courtesy Costas Busch - RPI27 Second Choice

28. Courtesy Costas Busch - RPI28 Second Choice

29. Courtesy Costas Busch - RPI29 Second Choice No transition: the automaton hangs

30. Courtesy Costas Busch - RPI30 Second Choice “reject” Input cannot be consumed

31. Courtesy Costas Busch - RPI31 is rejected by the NFA: “reject” All possible computations lead to rejection

32. Courtesy Costas Busch - RPI32 Language accepted:

33. Courtesy Costas Busch - RPI33 Lambda Transitions

34. Courtesy Costas Busch - RPI34

35. Courtesy Costas Busch - RPI35

36. Courtesy Costas Busch - RPI36 (read head does not move)

37. Courtesy Costas Busch - RPI37

38. Courtesy Costas Busch - RPI38 “accept” String is accepted all input is consumed

39. Courtesy Costas Busch - RPI39 Rejection Example

40. Courtesy Costas Busch - RPI40

41. Courtesy Costas Busch - RPI41 (read head doesn’t move)

42. Courtesy Costas Busch - RPI42 No transition: the automaton hangs

43. Courtesy Costas Busch - RPI43 “reject” String is rejected Input cannot be consumed

44. Courtesy Costas Busch - RPI44 Language accepted:

45. Courtesy Costas Busch - RPI45 Another NFA Example

46. Courtesy Costas Busch - RPI46

47. Courtesy Costas Busch - RPI47

48. Courtesy Costas Busch - RPI48

49. Courtesy Costas Busch - RPI49 “accept”

50. Courtesy Costas Busch - RPI50 Another String

51. Courtesy Costas Busch - RPI51

52. Courtesy Costas Busch - RPI52

53. Courtesy Costas Busch - RPI53

54. Courtesy Costas Busch - RPI54

55. Courtesy Costas Busch - RPI55

56. Courtesy Costas Busch - RPI56

57. Courtesy Costas Busch - RPI57 “accept”

58. Courtesy Costas Busch - RPI58 Language accepted

59. Courtesy Costas Busch - RPI59 Another NFA Example

60. Courtesy Costas Busch - RPI60 Language accepted (redundant state)

61. Courtesy Costas Busch - RPI61 Remarks: The symbol never appears on the input tape Simple automata:

62. Courtesy Costas Busch - RPI62 NFA DFA NFAs are interesting because we can express languages easier than DFAs

63. Courtesy Costas Busch - RPI63 Formal Definition of NFAs Set of states, i.e. Input aplhabet, i.e. Transition function Initial state Final states

64. Courtesy Costas Busch - RPI64 Transition Function

65. Courtesy Costas Busch - RPI65

66. Courtesy Costas Busch - RPI66

67. Courtesy Costas Busch - RPI67

68. Courtesy Costas Busch - RPI68 Extended Transition Function

69. Courtesy Costas Busch - RPI69

70. Courtesy Costas Busch - RPI70

71. Courtesy Costas Busch - RPI71 Formally : there is a walk from to with label

72. Courtesy Costas Busch - RPI72

73. 73 Inductive Definition Basis: Induction: Suppose w = x a. Also Let Then

74. Courtesy Costas Busch - RPI74 The Language of an NFA

75. Courtesy Costas Busch - RPI75

76. Courtesy Costas Busch - RPI76

77. Courtesy Costas Busch - RPI77

78. Courtesy Costas Busch - RPI78

79. Courtesy Costas Busch - RPI79 Formally The language accepted by NFA is: where and there is some (final state)

80. Courtesy Costas Busch - RPI80

81. Courtesy Costas Busch - RPI81 NFAs accept the Regular Languages

82. Courtesy Costas Busch - RPI82 Equivalence of Machines Definition for Automata: Machine is equivalent to machine if

83. Courtesy Costas Busch - RPI83 Example of equivalent machines NFA DFA

84. Courtesy Costas Busch - RPI84 We will prove: Languages accepted by NFAs Regular Languages NFAs and DFAs have the same computation power Languages accepted by DFAs

85. Courtesy Costas Busch - RPI85 Languages accepted by NFAs Regular Languages Step 1 Proof: Every DFA is trivially an NFA Any language accepted by a DFA is also accepted by an NFA

86. Courtesy Costas Busch - RPI86 Languages accepted by NFAs Regular Languages Step 2 Proof: Any NFA can be converted to an equivalent DFA Any language accepted by an NFA is also accepted by a DFA

87. Courtesy Costas Busch - RPI87 Convert NFA to DFA NFA DFA

88. Courtesy Costas Busch - RPI88 Convert NFA to DFA NFA DFA

89. Courtesy Costas Busch - RPI89 Convert NFA to DFA NFA DFA

90. Courtesy Costas Busch - RPI90 Convert NFA to DFA NFA DFA

91. Courtesy Costas Busch - RPI91 Convert NFA to DFA NFA DFA

92. Courtesy Costas Busch - RPI92 Convert NFA to DFA NFA DFA

93. Courtesy Costas Busch - RPI93 Convert NFA to DFA NFA DFA

94. Courtesy Costas Busch - RPI94 NFA to DFA: Remarks We are given an NFA We want to convert it to an equivalent DFA With

95. Courtesy Costas Busch - RPI95 If the NFA has n states the DFA has 2 n states in the power set of Q N i.e.

96. Courtesy Costas Busch - RPI96 Procedure NFA to DFA 1. Initial state of NFA: Initial state of DFA:

97. Courtesy Costas Busch - RPI97 Example NFA DFA

98. Courtesy Costas Busch - RPI98 Procedure NFA to DFA 2. For every DFA’s state Compute in the NFA Add transition to DFA

99. Courtesy Costas Busch - RPI99 Exampe NFA DFA

100. Courtesy Costas Busch - RPI100 Procedure NFA to DFA Repeat Step 2 for all letters in alphabet, until no more transitions can be added.

101. Courtesy Costas Busch - RPI101 Example NFA DFA

102. Courtesy Costas Busch - RPI102 Procedure NFA to DFA 3. For any DFA state If some is a final state in the NFA Then, is a final state in the DFA

103. Courtesy Costas Busch - RPI103 Example NFA DFA

104. Courtesy Costas Busch - RPI104 Theorem Take NFA Apply procedure to obtain DFA Then and are equivalent :

105. Courtesy Costas Busch - RPI105 Proof AND

106. Courtesy Costas Busch - RPI106 First we show: Take arbitrary: We will prove:

107. Courtesy Costas Busch - RPI107

108. Courtesy Costas Busch - RPI108 We will show that if

109. Courtesy Costas Busch - RPI109 More generally, we will show that if in : (arbitrary string)

110. Courtesy Costas Busch - RPI110 Proof by induction on Induction Basis:

111. Courtesy Costas Busch - RPI111 Induction hypothesis:

112. Courtesy Costas Busch - RPI112 Induction Step:

113. Courtesy Costas Busch - RPI113 Induction Step:

114. Courtesy Costas Busch - RPI114 Therefore if

115. Courtesy Costas Busch - RPI115 We have shown: We also need to show: (proof is similar)