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# Languages and Finite Automata_4_

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```									            Single Final State for NFA

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Any NFA can be converted

to an equivalent NFA

with a single final state

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Example
NFA

Equivalent NFA

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In General
NFA

Equivalent NFA

Single
final state
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Extreme Case

NFA without final state

Without transitions

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Properties of
Regular Languages

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For regular languages                 and
we will prove that:

Union:

Concatenation:

Star:                     Are regular
Languages
Reversal:

Complement:
Intersection:
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We say: Regular languages are closed under

Union:

Concatenation:

Star:

Reversal:

Complement:
Intersection:
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Regular language             Regular language

NFA                     NFA

Single final state              Single final state
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Example

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Union
NFA for

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Example
NFA for

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Concatenation

NFA for

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Example

NFA for

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Star Operation
NFA for

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Example

NFA for

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Reverse

NFA for

1. Reverse all transitions

2. Make initial state final state
and vice versa
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Example

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Complement

1. Take the DFA that accepts

2. Make final states non-final,
and vice-versa
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Example

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Intersection
DeMorgan’s Law:

regular

regular

regular

regular

regular
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Example

regular

regular          regular

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Regular Expressions

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Regular Expressions
Regular expressions
describe regular languages

Example:

describes the language

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Recursive Definition
Primitive regular expressions:

Given regular expressions        and

Are regular expressions

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Examples

A regular expression:

Not a regular expression:

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Languages of Regular Expressions

: language of regular expression

Example:

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Definition

For primitive regular expressions :

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Definition (continued)

For regular expressions         and

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Example
Regular expression:

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Example

Regular expression

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Example

Regular expression

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Example

Regular expression

= {all strings with at least
two consecutive 0}

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Example

Regular expression

= { all strings without
two consecutive 0 }

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Equivalent Regular Expressions

Definition:

Regular expressions            and

are equivalent if

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Example
= { all strings without
two consecutive 0 }

and
are equivalent
Reg. expressions
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Regular Expressions
and
Regular Languages

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Theorem

Languages
Regular
Generated by
Languages
Regular Expressions

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Theorem - Part 1

Languages
Regular
Generated by
Languages
Regular Expressions

1.       For any regular expression
the language         is regular

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Theorem - Part 2

Languages
Regular
Generated by
Languages
Regular Expressions

2.         For any regular language     , there is
a regular expression       with

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Proof - Part 1

1.     For any regular expression
the language        is regular

Proof by induction on the size of

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Induction Basis
Primitive Regular Expressions:
NFAs

regular
languages

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Inductive Hypothesis

Assume for regular expressions     and
that      and        are regular languages

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Inductive Step
We will prove:

are regular
Languages.

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By definition of regular expressions:

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By inductive hypothesis we know:
and        are regular languages

We also know:
Regular languages are closed under:
Union
Concatenation
Star

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Therefore:

Are regular
languages

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And trivially:

is a regular language

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Proof – Part 2

2.     For any regular language       there is
a regular expression        with

Proof by construction of regular expression

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Since        is regular, take an
NFA         that accepts it

Single final state
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From , construct an equivalent
Generalized Transition Graph in which
transition labels are regular expressions

Example:

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Another Example:

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Reducing the states:

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Resulting Regular Expression:

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In General
Removing states:

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The final transition graph:

The resulting regular expression:

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