1 Non-Deterministic Automata Regular Expressions

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

5 Transition Function

6 Extended Transition Function

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

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

14 Example NFA DFA

15 Since machines and are equivalent DFA NFA

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?

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

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

19 NFA to DFA NFA DFA

20 NFA to DFA NFA DFA

21 NFA to DFA NFA DFA

22 NFA to DFA NFA DFA

23 NFA to DFA NFA DFA

24 NFA to DFA NFA DFA

25 NFA to DFA NFA DFA

26 NFA to DFA NFA DFA

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

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

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

30 Example NFA DFA

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

32 Exampe NFA DFA

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

34 Example NFA DFA

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

36 Example NFA DFA

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