4 regular properties

7/28/2019 4 Regular Properties

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1

Properties of

Regular Languages


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2

1L 2L

21LLConcatenation:

*1LStar:

21 LL Union:

Are regular

Languages

For regular languages and

we will prove that:

1L

21 LL

Complement:

Intersection:

R

L1Reversal:


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We say: Regular languages are closed under

21LLConcatenation:

*1LStar:

21 LL Union:

1L

21 LL

Complement:

Intersection:

R

L1Reversal:


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a

b

b

aNFA

Equivalent

NFA

a

b

b

a

A useful transformation: use one accept state

2 accept states

1 accept state


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NFA

Equivalent NFA

Singleaccepting

state

In General


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NFA without accepting state

Add an accepting state

without transitions

Extreme case


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1LRegular language

11 LML

1

Single accepting state

NFA 2

2L

Single accepting state

22 LML

Regular language

NFA

Take two languages


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}{1 baLn

a

b

1

baL 2ab

2

0n

Example


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Union

NFA for

1

2

21 LL


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a

b

ab

}{1 baLn

}{2 baL

}{}{21 babaLLn

NFA for

Example


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Concatenation

NFA for 21LL

1 2


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NFA for

a

b ab

}{1 baLn

}{2 baL

}{}}{{21 bbaababaLL nn

Example


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Star Operation

NFA for *1L

1

*1L

1

21

Lw

wwww

i

k


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NFA for *}{*1 baL n

a

b

}{1 baLn

Example


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Reverse

R

L1

1

NFA for1

1. Reverse all transitions

2. Make initial state accepting state

and vice versa

1L

E l


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}{1 baLn

a

b

1

}{1nR

baL a

b

1

Example


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Complement

1. Take the DFAthat accepts 1L

11L

11L

2. Make accepting states non-final,

and vice-versa


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}{1 baLn

a

b

1

ba,

ba,

}{*},{1 babaL n a

b

1

ba,

ba,

Example


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Intersection

1L regular

2L regularWe show 21

LL

regular


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DeMorgans Law: 2121 LLLL

21 , LL regular

21 , LL regular

21 LL regular

21 LL regular

21 LL regular


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Example

}{1 baLn

},{2 baabL

regular

regular

}{21 abLL

regular


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Regular Expressions


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Regular Expressions

Regular expressions

describe regular languages

Example:

describes the language

*)( cba

,...,,,,,*, bcaabcaabcabca


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Recursive Definition

1

1

21

21

*

r

r

rr

rr

Are regular expressions

2r1rGiven regular expressions and


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Examples

)(* ccbaA regular expression:

baNot a regular expression:


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Languages of Regular Expressions

: language of regular expression

Example

rL r

,...,,,,,*)( bcaabcaabcacbaL


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Definition

For primitive regular expressions:

aaL

L

L


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Definition (continued)

For regular expressions and1r 2r

2121 rLrLrrL

2121 rLrLrrL

** 11 rLrL

11 rLrL


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Example

Regular expression: *aba

*abaL *aLbaL

*aLbaL

*aLbLaL

*aba

,...,,,, aaaaaaba

,...,,,...,,, baababaaaaaa


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Example

Regular expression bbabar *

,...,,,,, bbbbaabbaabbarL


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Example

Regular expression bbbaar **

}0,:{ 22 mnbbarL mn


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Example

Regular expression *)10(00*)10( r

)(rL = { all strings containing substring 00 }

E l


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Example

Regular expression )0(*)011( r

)(rL = { all strings without substring 00 }

E l l E


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Equivalent Regular Expressions

Definition:

Regular expressions and

are equivalent if

1r 2r

)()( 21 rLrL

E l


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Example

L = { all strings without substring 00 }

)0(*)011(1 r

)0(*1)0(**)011*1(2 r

LrLrL )()( 211r 2rand

are equivalent

regular expressions


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Regular Expressionsand

Regular Languages

Th


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Theorem

LanguagesGenerated by

Regular Expressions

Regular

Languages

P f


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Regular Expressions

RegularLanguages

Languages

Generated by

Regular Expressions

RegularLanguages

Proof:

P f P t 1


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Proof - Part 1

r

)(rL

For any regular expression

the language is regular


Regular Expressions

Regular

Languages

Proof by induction on the size of r

I d ti B i


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Induction Basis

Primitive Regular Expressions: ,,

CorrespondingNFAs

)()( 1 LML

)(}{)( 2 LML

)(}{)( 3 aLaML

regular

languages

a

I d ti H th i


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Inductive Hypothesis

Supposethat for regular expressions and ,

and are regular languages1r 2r

)( 1rL )( 2rL

I d ti St


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Inductive Step

We will prove:

1

1

21

21

*

rL

rL

rrL

rrL

Are regular

Languages

d f f l


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By definition of regular expressions:

11

11

2121

2121

**

rLrL

rLrL

rLrLrrL

rLrLrrL

B d h h k


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)( 1rL )( 2rL

By inductive hypothesis we know:

and are regular languages

Regular languages are closed under:

*1

21

21

rL

rLrL

rLrL Union

Concatenation

Star

We also know:

Th f :


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Therefore:

** 11

2121

2121

rLrL

rLrLrrL

rLrLrrL

Are regular

languages

)())(( 11 rLrL is trivially a regular language(by induction hypothesis)

End of Proof-Part 1

U i h l l f h i


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Using the regular closure of these operations,

we can construct recursively the NFA

that accepts

M

)()( rLML

Example:21 rrr

)()( 11rLML

)()( 22 rLML

)()( rLML

Pr f P rt 2


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For any regular language there is

a regular expression with

Proof - Part 2


Regular Expressions

Regular

Languages

L

rLrL )(

We will convert an NFA that accepts

to a regular expression

L


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Since is regular, there is a

NFA that accepts it

L

LML )(

Take it with a single final state


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From construct the equivalent

Generalized Transition Graphin which transition labels are regular expressions

Example:

a

ba,

c a

ba

c

Corresponding

Generalized transition graph


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Another Example:

ba a

b

b

0q 1q 2q

ba,a

b

b

0q 1q 2q

b

bTransition labels

are regularexpressions


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Reducing the states:

ba

ab

b

0q 1q 2q

b

0q 2q

babb

*

)(* babb

Transition labels

are regularexpressions


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Resulting Regular Expression:

0q 2q

babb*

)(* babb

*)(**)*( bbabbabbr

LMLrL )()(

In General


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In General

Removing a state:

iq q jq

a b

cd

e

iq jq

dae* bce*

dce*

bae*

By repeating the process until


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0q fq

1r

2r

3r4r

*)*(* 213421 rrrrrrr

LMLrL )()(

The resulting regular expression:

y r p at ng th proc ss unt

two states are left, the resulting graph is

Initial graph Resulting graph

End of Proof-Part 2

Stan ar epresentations


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Stan ar epresentationsof Regular Languages

Regular Languages

DFAs

NFAs RegularExpressions


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When we say: We are given

a Regular Language

We mean:

L

Language is in a standard

representation

L

(DFA, NFA, or Regular Expression)

4 regular properties

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