NonDeterministic 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 if there is a
computation of the NFA that accepts the string
i.e., 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 if 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 DFAs

NFA
DFA
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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
Is the result of following exactly one
transition
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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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NFAs accept the Regular Languages

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Equivalence of Machines

• Definition
• Machine is equivalent to machine
• if

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Example of equivalent machines

NFA
DFA
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We will prove
Languages accepted by NFAs
Regular Languages
Languages accepted by DFAs
NFAs and DFAs have the same computation power
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We will show
Languages accepted by NFAs
Regular Languages
Languages accepted by NFAs
Regular Languages
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Proof-Step 1
Languages accepted by NFAs
Regular Languages
Proof
Every DFA is trivially an NFA
Any language accepted by a DFA is also
accepted by an NFA
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Proof-Step 2
Languages accepted by NFAs
Regular Languages
Any NFA can be converted to an equivalent DFA
Proof
Any language accepted by an NFA is also
accepted by a DFA
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Convert NFA to DFA

NFA
DFA
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Convert NFA to DFA

NFA
DFA
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Convert NFA to DFA

NFA
DFA
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Convert NFA to DFA

NFA
DFA
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Convert NFA to DFA

NFA
DFA
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Convert NFA to DFA

NFA
DFA
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Convert 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 DFAs 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 accepting state in NFA
• Then,
• is accepting state in DFA

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Example

NFA
DFA
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Theorem
Take NFA

Apply conversion procedure to obtain DFA
Then and are equivalent

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Proof
AND
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First we show
Take arbitrary
We will prove
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NFA
NFA
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denotes
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We will show that if
NFA
then
DFA
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More generally, we will show that if in
(arbitrary string)
NFA
then
DFA
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Proof by induction on
Induction Basis
NFA
DFA
Is true by construction of
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Induction hypothesis
NFA
DFA
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Induction Step
NFA
DFA
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Induction Step
NFA
DFA
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Therefore if
NFA
then
DFA
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We have shown
We also need to show
(proof is similar and omitted)