1. Fall 2006Costas Busch - RPI1 Non-Deterministic Finite Automata

2. Fall 2006Costas Busch - RPI2 Alphabet = Nondeterministic Finite Automaton (NFA)

3. Fall 2006Costas Busch - RPI3 Two choices Alphabet =

4. Fall 2006Costas Busch - RPI4 No transition Two choices No transition Alphabet =

5. Fall 2006Costas Busch - RPI5 First Choice

6. Fall 2006Costas Busch - RPI6 First Choice

7. Fall 2006Costas Busch - RPI7 “accept” First Choice All input is consumed

8. Fall 2006Costas Busch - RPI8 Second Choice

9. Fall 2006Costas Busch - RPI9 Second Choice “reject” Input cannot be consumed Automaton Halts

10. Fall 2006Costas Busch - RPI10 An NFA accepts a string: if there is a computation of the NFA that accepts the string i.e., all the input string is processed and the automaton is in an accepting state

11. Fall 2006Costas Busch - RPI11 is accepted by the NFA: “accept” “reject” because this computation accepts this computation is ignored

12. Fall 2006Costas Busch - RPI12 Rejection example

13. Fall 2006Costas Busch - RPI13 First Choice “reject”

14. Fall 2006Costas Busch - RPI14 Second Choice

15. Fall 2006Costas Busch - RPI15 Second Choice “reject”

16. Fall 2006Costas Busch - RPI16 Another Rejection example

17. Fall 2006Costas Busch - RPI17 First Choice

18. Fall 2006Costas Busch - RPI18 “reject” First Choice Input cannot be consumed Automaton halts

19. Fall 2006Costas Busch - RPI19 Second Choice

20. Fall 2006Costas Busch - RPI20 Second Choice “reject” Input cannot be consumed Automaton halts

21. Fall 2006Costas Busch - RPI21 An NFA rejects a string: if 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 For each computation:

22. Fall 2006Costas Busch - RPI22 is rejected by the NFA: “reject” All possible computations lead to rejection

23. Fall 2006Costas Busch - RPI23 is rejected by the NFA: “reject” All possible computations lead to rejection

24. Fall 2006Costas Busch - RPI24 Language accepted:

25. Fall 2006Costas Busch - RPI25 Lambda Transitions

26. Fall 2006Costas Busch - RPI26

27. Fall 2006Costas Busch - RPI27

28. Fall 2006Costas Busch - RPI28 input tape head does not move

29. Fall 2006Costas Busch - RPI29 “accept” String is accepted all input is consumed

30. Fall 2006Costas Busch - RPI30 Rejection Example

31. Fall 2006Costas Busch - RPI31

32. Fall 2006Costas Busch - RPI32 (read head doesn’t move)

33. Fall 2006Costas Busch - RPI33 “reject” String is rejected Input cannot be consumed Automaton halts

34. Fall 2006Costas Busch - RPI34 Language accepted:

35. Fall 2006Costas Busch - RPI35 Another NFA Example

36. Fall 2006Costas Busch - RPI36

37. Fall 2006Costas Busch - RPI37

38. Fall 2006Costas Busch - RPI38 “accept”

39. Fall 2006Costas Busch - RPI39 Another String

40. Fall 2006Costas Busch - RPI40

41. Fall 2006Costas Busch - RPI41

42. Fall 2006Costas Busch - RPI42

43. Fall 2006Costas Busch - RPI43

44. Fall 2006Costas Busch - RPI44

45. Fall 2006Costas Busch - RPI45 “accept”

46. Fall 2006Costas Busch - RPI46 Language accepted

47. Fall 2006Costas Busch - RPI47 Another NFA Example

48. Fall 2006Costas Busch - RPI48 Language accepted (redundant state)

49. Fall 2006Costas Busch - RPI49 Remarks: The symbol never appears on the input tape Simple automata:

50. Fall 2006Costas Busch - RPI50 NFADFA NFAs are interesting because we can express languages easier than DFAs

51. Fall 2006Costas Busch - RPI51 Formal Definition of NFAs Set of states, i.e. Input aplhabet, i.e. Transition function Initial state Accepting states

52. Fall 2006Costas Busch - RPI52 Transition Function resulting states with following one transition with symbol

53. Fall 2006Costas Busch - RPI53

54. Fall 2006Costas Busch - RPI54

55. Fall 2006Costas Busch - RPI55

56. Fall 2006Costas Busch - RPI56

57. Fall 2006Costas Busch - RPI57 Extended Transition Function Same with but applied on strings

58. Fall 2006Costas Busch - RPI58

59. Fall 2006Costas Busch - RPI59

60. Fall 2006Costas Busch - RPI60 Special case: for any state

61. Fall 2006Costas Busch - RPI61 : there is a walk from to with label In general

62. Fall 2006Costas Busch - RPI62 The Language of an NFA The language accepted by is: where and there is some (accepting state)

63. Fall 2006Costas Busch - RPI63

64. Fall 2006Costas Busch - RPI64

65. Fall 2006Costas Busch - RPI65

66. Fall 2006Costas Busch - RPI66

67. Fall 2006Costas Busch - RPI67

68. Fall 2006Costas Busch - RPI68

69. Fall 2006Costas Busch - RPI69 NFAs accept the Regular Languages

70. Fall 2006Costas Busch - RPI70 Equivalence of Machines Definition: Machine is equivalent to machine if

71. Fall 2006Costas Busch - RPI71 NFA DFA Example of equivalent machines

72. Fall 2006Costas Busch - RPI72 Theorem: Languages accepted by NFAs Regular Languages NFAs and DFAs have the same computation power, accept the same set of languages Languages accepted by DFAs

73. Fall 2006Costas Busch - RPI73 Languages accepted by NFAs Regular Languages accepted by NFAs Regular Languages we only need to show Proof: AND

74. Fall 2006Costas Busch - RPI74 Languages accepted by NFAs Regular Languages Proof-Step 1 Every DFA is trivially an NFA Any language accepted by a DFA is also accepted by an NFA

75. Fall 2006Costas Busch - RPI75 Languages accepted by NFAs Regular Languages Proof-Step 2 Any NFA can be converted to an equivalent DFA Any language accepted by an NFA is also accepted by a DFA

76. Fall 2006Costas Busch - RPI76 Conversion NFA to DFA NFA DFA

77. Fall 2006Costas Busch - RPI77 NFA DFA

78. Fall 2006Costas Busch - RPI78 NFA DFA empty set trap state

79. Fall 2006Costas Busch - RPI79 NFA DFA union

80. Fall 2006Costas Busch - RPI80 NFA DFA union

81. Fall 2006Costas Busch - RPI81 NFA DFA trap state

82. Fall 2006Costas Busch - RPI82 NFA DFA END OF CONSTRUCTION

83. Fall 2006Costas Busch - RPI83 General Conversion Procedure Input: an NFA Output: an equivalent DFA with

84. Fall 2006Costas Busch - RPI84 The NFA has states The DFA has states from the power set

85. Fall 2006Costas Busch - RPI85 1. Initial state of NFA: Initial state of DFA: Conversion Procedure Steps step

86. Fall 2006Costas Busch - RPI86 NFA DFA Example

87. Fall 2006Costas Busch - RPI87 2. For every DFA’s state compute in the NFA add transition to DFA Union step

88. Fall 2006Costas Busch - RPI88 NFA DFA Example

89. Fall 2006Costas Busch - RPI89 3. Repeat Step 2 for every state in DFA and symbols in alphabet until no more states can be added in the DFA step

90. Fall 2006Costas Busch - RPI90 NFA DFA Example

91. Fall 2006Costas Busch - RPI91 4. For any DFA state if some is accepting state in NFA Then, is accepting state in DFA step

92. Fall 2006Costas Busch - RPI92 NFA DFA Example

93. Fall 2006Costas Busch - RPI93 If we convert NFA to DFA then the two automata are equivalent: Lemma: Proof: AND We only need to show:

94. Fall 2006Costas Busch - RPI94 First we show: We only need to prove:

95. Fall 2006Costas Busch - RPI95 symbols Consider NFA

96. Fall 2006Costas Busch - RPI96 denotes a possible sub-path like symbol

97. Fall 2006Costas Busch - RPI97 We will show that if then NFA DFA state label state label

98. Fall 2006Costas Busch - RPI98 More generally, we will show that if in : (arbitrary string) then NFA DFA

99. Fall 2006Costas Busch - RPI99 Proof by induction on Induction Basis: is true by construction of NFA DFA

100. Fall 2006Costas Busch - RPI100 Induction hypothesis: NFA DFA Suppose that the following hold

101. Fall 2006Costas Busch - RPI101 Induction Step: NFA DFA Then this is true by construction of

102. Fall 2006Costas Busch - RPI102 Therefore if NFA DFA then

103. Fall 2006Costas Busch - RPI103 We have shown: With a similar proof we can show: END OF LEMMA PROOF Therefore: