1 Non Deterministic Automata

2 Alphabet = Nondeterministic Finite Accepter (NFA)

3 Two choices Alphabet = Nondeterministic Finite Accepter (NFA)

4 No transition Two choices No transition Alphabet = Nondeterministic Finite Accepter (NFA)

5 First Choice

6

7

8 “accept” First Choice All input is consumed

9 Second Choice

10 Second Choice

11 Second Choice No transition: the automaton hangs

12 Second Choice “reject” Input cannot be consumed

13 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

14 Example is accepted by the NFA: “accept” “reject” because this computation accepts

15 Rejection example

16 First Choice

17 First Choice “reject”

18 Second Choice

19 Second Choice

20 Second Choice “reject”

21 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

22 Example is rejected by the NFA: “reject” All possible computations lead to rejection

23 Rejection example

24 First Choice

25 First Choice No transition: the automaton hangs

26 “reject” First Choice Input cannot be consumed

27 Second Choice

28 Second Choice

29 Second Choice No transition: the automaton hangs

30 Second Choice “reject” Input cannot be consumed

31 is rejected by the NFA: “reject” All possible computations lead to rejection

32 Language accepted:

33 Lambda Transitions

34

35

36 (read head doesn’t move)

37

38 “accept” String is accepted all input is consumed

39 Rejection Example

40

41 (read head doesn’t move)

42 No transition: the automaton hangs

43 “reject” String is rejected Input cannot be consumed

44 Language accepted:

45 Another NFA Example

46

47

48

49 “accept”

50 Another String

51

52

53

54

55

56

57 “accept”

58 Language accepted

59 Another NFA Example

60 Language accepted

61 Remarks: The symbol never appears on the input tape Extreme automata:

62 NFA DFA NFAs are interesting because we can express languages easier than DFAs

63 Formal Definition of NFAs Set of states, i.e. Input aplhabet, i.e. Transition function Initial state Final states

64 Transition Function

65

66

67

68 Extended Transition Function

69

70

71 Formally It holds if and only if there is a walk from to with label

72 The Language of an NFA

73

74

75

76

77 Formally The language accepted by NFA is: where and there is some (final state)

78

79 Equivalence of NFAs and DFAs

80 Equivalence of Machines For DFAs or NFAs: Machine is equivalent to machine if

81 Example NFA DFA

82 Since machines and are equivalent DFA NFA

83 Equivalence of NFAs and DFAs Question: NFAs = DFAs ? Same power? Accept the same languages?

84 Equivalence of NFAs and DFAs Question: NFAs = DFAs ? Same power? Accept the same languages? YES!

85 Languages accepted by NFAs Languages accepted by DFAs Regular Languages Thus, NFAs accept the regular languages