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1 Non-Deterministic Automata Regular Expressions

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2 NFA Example

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

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5 Transition Function

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6 Extended Transition Function

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

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8 The Language of an NFA

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

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13 Equivalence of NFAs and DFAs For DFAs or NFAs: Machine is equivalent to if

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14 Example NFA DFA

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15 Since machines and are equivalent DFA NFA

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16 Equivalence of NFAs and DFAs Every DFA is also an NFA A language accepted by a DFA will be accepted by an NFA An NFA is as least as powerful as a DFA Is an DFA as powerful as an NFA?

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17 Is a DFA as Powerful as an NFA? Answer: YES! A language accepted by an NFA will be accepted by some DFA For every NFA there is an equivalent DFA that accepts the same language

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18 NFAs Accept the Regular Languages For every NFA there is an equivalent DFA The language accepted by a DFA is regular The language accepted by an NFA is regular

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19 NFA to DFA NFA DFA

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20 NFA to DFA NFA DFA

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21 NFA to DFA NFA DFA

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22 NFA to DFA NFA DFA

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23 NFA to DFA NFA DFA

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24 NFA to DFA NFA DFA

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25 NFA to DFA NFA DFA

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26 NFA to DFA NFA DFA

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27 NFA to DFA Observations We are given an NFA We want to convert it to an equivalent DFA With

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28 If the NFA has states The DFA has states in the powerset

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29 Procedure NFA to DFA 1. Initial state of NFA: Initial state of DFA:

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30 Example NFA DFA

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31 Procedure NFA to DFA 2. For every DFA’s state Compute in the NFA Giving union Add a transition

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32 Exampe NFA DFA

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33 Procedure NFA to DFA Repeat Step 2 for all letters, until no more transitions can be added.

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34 Example NFA DFA

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35 Procedure NFA to DFA 3. For any DFA state If some is a final state for the NFA Then, is a final state for the DFA

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36 Example NFA DFA

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37 Theorem Take NFA Apply procedure to obtain DFA Then and are equivalent :

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