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1 Non Deterministic Automata

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2 Alphabet = Nondeterministic Finite Accepter (NFA)

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3 Two choices Alphabet = Nondeterministic Finite Accepter (NFA)

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

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5 First Choice

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8 “accept” First Choice All input is consumed

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9 Second Choice

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10 Second Choice

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11 Second Choice No transition: the automaton hangs

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12 Second Choice “reject” Input cannot be consumed

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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

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

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15 Rejection example

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16 First Choice

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17 First Choice “reject”

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18 Second Choice

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19 Second Choice

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20 Second Choice “reject”

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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

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

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23 Rejection example

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24 First Choice

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25 First Choice No transition: the automaton hangs

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26 “reject” First Choice Input cannot be consumed

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27 Second Choice

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28 Second Choice

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29 Second Choice No transition: the automaton hangs

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30 Second Choice “reject” Input cannot be consumed

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

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32 Language accepted:

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33 Lambda Transitions

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36 (read head doesn’t move)

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38 “accept” String is accepted all input is consumed

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39 Rejection Example

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41 (read head doesn’t move)

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42 No transition: the automaton hangs

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43 “reject” String is rejected Input cannot be consumed

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44 Language accepted:

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45 Another NFA Example

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49 “accept”

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50 Another String

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57 “accept”

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58 Language accepted

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59 Another NFA Example

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60 Language accepted

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

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

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

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64 Transition Function

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68 Extended Transition Function

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

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72 The Language of an NFA

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

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79 Equivalence of NFAs and DFAs

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

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81 Example NFA DFA

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82 Since machines and are equivalent DFA NFA

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

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

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

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Slide 1Fig 25-CO, p.762. Slide 2Fig 25-1, p.765 Slide 3Fig 25-2, p.765.

Slide 1Fig 25-CO, p.762. Slide 2Fig 25-1, p.765 Slide 3Fig 25-2, p.765.

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