# Languages and Finite Automata_1_

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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 aphabet, 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
: 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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NFA accept Regular Languages

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Equivalence of FA

Definition:

An FA         is equivalent to FA

if
that is if both accept the same language.

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Example of equivalent FA
NFA

DFA

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We will prove:

Languages
Regular
accepted
Languages
by NFA
Languages
accepted
by DFA
That is, NFA and DFA have
the same computation power
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Step 1

Languages
Regular
accepted
Languages
by NFA

Proof: Every DFA is trivially an NFA

Any language     accepted by a DFA
is also accepted by an NFA
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Step 2

Languages
Regular
accepted
Languages
by NFA

Proof: Any NFA can be converted into an
equivalent DFA

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
into an equivalent DFA

That is,

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If the NFA has states

Then 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 the following transition to the DFA

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Example
NFA

DFA

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Procedure NFA to DFA

Repeat step 2 for all symbols in the 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 the procedure to obtain DFA

Then,     and   are equivalent:

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Proof

AND

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First we show:

Take arbitrary string :

We will prove:

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We will show that if

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More generally, we will show that if in   :

(arbitrary string)

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Proof by induction on

The basis case:

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Induction hypothesis:

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Induction Step:

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Induction Step:

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Therefore if

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We have shown:

We also need to show:

(proof is similar)

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```
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