1 Non-Deterministic Finite Automata

2 Alphabet = Nondeterministic Finite Automaton (NFA)

3 Two choices Alphabet =

4 No transition Two choices No transition Alphabet =

5 First Choice

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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 an accepting state There is a computation:

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 OR For each computation:

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

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36 (read head does not move)

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38 “accept” String is accepted all input is consumed

39 Rejection Example

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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

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49 “accept”

50 Another String

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57 “accept”

58 Language accepted

59 Another NFA Example

60 Language accepted (redundant state)

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

62 NFA FA NFAs are interesting because we can express languages easier than FAs

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

64 Transition Function

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68 Extended Transition Function

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71 Formally : there is a walk from to with label

72 The Language of an NFA

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

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