Non-Deterministic Finite Automata

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Non-Deterministic Finite Automata
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Nondeterministic Finite Automaton (NFA)
Alphabet
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Alphabet
Two choices
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Alphabet
Two choices
No transition
No transition
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First Choice
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First Choice
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First Choice
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First Choice
All input is consumed
accept
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Second Choice
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Second Choice
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Second Choice
No transition the automaton hangs
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Second Choice
Input cannot be consumed
reject
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An NFA accepts a string when there is a
computation of the NFA that accepts the string
There is a computation
all the input is consumed and the automaton is in
an accepting state
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Example
is accepted by the NFA
accept
reject
because this computation accepts
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Rejection example
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First Choice
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First Choice
reject
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Second Choice
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Second Choice
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Second Choice
reject
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An NFA rejects a string when there is no
computation of the NFA that accepts the string.
For each computation

• All the input is consumed and the
• automaton is in a non final state

OR

• The input cannot be consumed

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Example
is rejected by the NFA
reject
reject
All possible computations lead to rejection
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Rejection example
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First Choice
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First Choice
No transition the automaton hangs
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First Choice
Input cannot be consumed
reject
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Second Choice
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Second Choice
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Second Choice
No transition the automaton hangs
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Second Choice
Input cannot be consumed
reject
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is rejected by the NFA
reject
reject
All possible computations lead to rejection
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Language accepted
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Lambda Transitions
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(No Transcript)
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(No Transcript)
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(read head does not move)
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(No Transcript)
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all input is consumed
accept
String is accepted
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Rejection Example
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(No Transcript)
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(read head doesnt move)
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No transition the automaton hangs
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Input cannot be consumed
reject
String is rejected
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Language accepted
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Another NFA Example
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(No Transcript)
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(No Transcript)
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(No Transcript)
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accept
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Another String
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(No Transcript)
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(No Transcript)
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(No Transcript)
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(No Transcript)
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(No Transcript)
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(No Transcript)
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accept
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Language accepted
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Another NFA Example
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Language accepted
(redundant state)
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Remarks

• The symbol never appears on the
• input tape

• Simple automata

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• NFAs are interesting because we can
• express languages easier than FAs

NFA
FA
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Formal Definition of NFAs

Set of states, i.e.
Input aplhabet, i.e.
Transition function
Initial state
Accepting states
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Transition Function
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(No Transcript)
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(No Transcript)
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(No Transcript)
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Extended Transition Function

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(No Transcript)
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(No Transcript)
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Formally
there is a walk from to with label
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The Language of an NFA

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Formally

• The language accepted by NFA is
• where
• and there is some

(accepting state)
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