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Non Deterministic Automata
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Nondeterministic Finite Accepter (NFA)
Alphabet =
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Nondeterministic Finite Accepter (NFA)
Alphabet = Two choices
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Nondeterministic Finite Accepter (NFA)
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 All the input is consumed and the automaton is in a final 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 All the input is consumed and the automaton is in a non final state 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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(read head doesn’t move)
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all input is consumed “accept” String is accepted
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Rejection Example
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(read head doesn’t 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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“accept”
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Another String
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“accept”
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Language accepted
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Another NFA Example
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Language accepted
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Remarks: The symbol never appears on the input tape Extreme automata:
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NFAs are interesting because we can
express languages easier than DFAs NFA DFA
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Formal Definition of NFAs
Set of states, i.e. Input aplhabet, i.e. Transition function Initial state Final states
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Transition Function
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Extended Transition Function
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Formally It holds if and only if 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
(final state)
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Equivalence of NFAs and DFAs
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Equivalence of Machines
For DFAs or NFAs: Machine is equivalent to machine if
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Example NFA DFA
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Since machines and are equivalent NFA DFA
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Equivalence of NFAs and DFAs
Question: NFAs = DFAs ? Same power? Accept the same languages?
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Equivalence of NFAs and DFAs
Question: NFAs = DFAs ? YES! Same power? Accept the same languages?
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We will prove: Languages accepted by NFAs Languages accepted by DFAs NFAs and DFAs have the same computation power
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Step 1 Languages accepted by NFAs Languages accepted by DFAs Proof: Every DFA is trivially an NFA A language accepted by a DFA is also accepted by an NFA
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Step 2 Languages accepted by NFAs Languages accepted by DFAs Proof: Any NFA can be converted to an equivalent DFA A language accepted by an NFA is also accepted by a DFA
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NFA to DFA NFA DFA
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NFA to DFA NFA DFA
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NFA to DFA NFA DFA
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NFA to DFA NFA DFA
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NFA to DFA NFA DFA
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NFA to DFA NFA DFA
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NFA to DFA NFA DFA
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NFA to DFA: Remarks We are given an NFA We want to convert it
to an equivalent DFA With
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If the NFA has states the DFA has states in the powerset
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Procedure NFA to DFA 1. Initial state of NFA: Initial state of DFA:
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Example NFA DFA
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Procedure NFA to DFA 2. For every DFA’s state Compute in the NFA
Add transition to DFA
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Exampe NFA DFA
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Procedure NFA to DFA Repeat Step 2 for all letters in alphabet, until
no more transitions can be added.
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Example NFA DFA
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Procedure NFA to DFA 3. For any DFA state
If some is a final state in the NFA Then, is a final state in the DFA
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Example NFA DFA
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Theorem Take NFA Apply procedure to obtain DFA
Then and are equivalent :
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Finally We have proven Languages accepted by NFAs Languages accepted
by DFAs
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We have proven Languages accepted by NFAs Languages accepted by DFAs Regular Languages
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We have proven Languages accepted by NFAs Languages accepted by DFAs Regular Languages Regular Languages
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We have proven Languages accepted by NFAs Languages accepted by DFAs Regular Languages Regular Languages Thus, NFAs accept the regular languages
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