reverse of regular grammer

8/13/2019 Reverse of Regular Grammer

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Reverse of a Regular Language


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Theorem:

The reverse of a regular languageis a regular language

R

L L

Proof idea:

Construct NFA that accepts :RL

invert the transitions of the NFA

that accepts L


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Proof

Since is regular,

there is NFA that accepts

L

Example:

baabL *

a

b

ba

L


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Invert Transitions

a

b

ba


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Make old initial state a final state

a

b

ba


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Add a new initial state

a

b

ba


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a

b

ba

Resulting machine accepts RL

baabL *

ababLR *

RL is regular


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Grammars


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Grammars

Grammars express languages

Example: the English language

verbpredicate

nounarticlephrasenoun

predicatephrasenounsentence

_

_


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runsverb

dognounboynoun

thearticle

aarticle


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A derivation of the boy walks:

walksboythe

verbboythe

verbnounthe

verbnounarticle

verbphrasenoun


_

_


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A derivation of a dog runs:

runsdoga

verbdoga

verbnouna

verbnounarticle

verbphrasenoun


_

_


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Language of the grammar:

L = { a boy runs,

a boy walks,

the boy runs,the boy walks,

a dog runs,

a dog walks,the dog runs,

the dog walks }


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Notation

dognoun

boynoun

Variable

orNon-terminal

TerminalProduction

rule


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Another Example

Grammar:

Derivation of sentence :

S

aSbS

abaSbS

ab

aSbS S


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aabbaaSbbaSbS

aSbS S

aabb

S

aSbSGrammar:

Derivation of sentence :


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Other derivations:

aaabbbaaaSbbbaaSbbaSbS

aaaabbbbaaaaSbbbb

aaaSbbbaaSbbaSbS


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Language of the grammar

S

aSbS

}0:{ nbaL nn


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More Notation

Grammar PSTVG ,,,

:V

:T

:S

:P

Set of variables

Set of terminal symbols

Start variable

Set of Production rules


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Example

Grammar :

S

aSbSG

PSTVG ,,,

}{SV },{ baT

},{ SaSbSP


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More Notation

Sentential Form:

A sentence that containsvariables and terminals

Example:


Sentential Forms sentence


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We write:

Instead of:

aaabbbS

*



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In general we write:

If:

nww

*

1

nwwww

321


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By default: ww*


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Example

S

aSbS

aaabbbS

aabbS

abS

S

*

*

*

*

Grammar Derivations


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baaaaaSbbbbaaSbb

aaSbbS

S

aSbS

Grammar

Example

Derivations


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Another Grammar Example

Grammar :

A

aAbA

AbS

Derivations:

aabbbaaAbbbaAbbS

abbaAbbAbSbAbS

G


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More Derivations

aaaabbbbbaaaaAbbbbb

aaaAbbbbaaAbbbaAbbAbS

bbaS

bbbaaaaaabbbbS

aaaabbbbbS

nn


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Language of a Grammar

For a grammarwith start variable :

GS

}:{)( wSwGL

String of terminals


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Example

For grammar :

A

aAbA

AbS

}0:{)( nbbaGL nn

Since: bbaS nn

G


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A Convenient Notation

A

aAbA|aAbA

thearticle

aarticle

theaarticle |


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Linear Grammars


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Linear Grammars

Grammars with

at most one variable at the right sideof a production

Examples:

AaAbA

AbS

S

aSbS

G


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A Non-Linear Grammar

bSaS

aSbS

SSSS

Grammar :G

)}()(:{)( wnwnwGL ba

h L G


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Another Linear Grammar

Grammar :

AbB

aBA

AS

|

}0:{)( nbaGL nn

G

Ri h Li G


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Right-Linear Grammars

All productions have form:

Example:

xBA

xA

or

aS

abSS

L f Li G


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Left-Linear Grammars

All productions have form:

Example:

BxA

aBBAabA

AabS

|

xA

or


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Regular Grammars

R l G


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Regular Grammars

A regular grammaris any

right-linear or left-linear grammar

Examples:

aS

abSS

aB

BAabA

AabS

|

1G 2G

Ob ti


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Observation

Regular grammars generate regular languages

Examples:

aS

abSS

aabGL *)()(1

aB

BAabA

AabS

|

*)()(2

abaabGL

1G

2G


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Regular Grammars

GenerateRegular Languages

Th


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Theorem

LanguagesGenerated by

Regular Grammars

Regular

Languages

Th P t 1


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Theorem - Part 1


Regular Grammars

Regular

Languages

Any regular grammar generates

a regular language

Th P t 2


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Theorem - Part 2


Regular Grammars

Regular

Languages

Any regular language is generated

by a regular grammar

P f P t 1


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Proof Part 1


Regular Grammars

Regular

Languages

The language generated by

any regular grammar is regular

)(GL

G

Th s f Ri ht Li G mm s


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The case of Right-Linear Grammars

Let be a right-linear grammar

We will prove: is regular

Proof idea: We will construct NFA

with

G

)(GL

)()( GLML


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Grammar is right-linearG

Example:

aBbBBaaA

BaAS

|

|


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Construct NFA such that

every state is a grammar variable:

aBbB

BaaABaAS

|

|

S FV

A

B

specialfinal state


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Add edges for each production:

S FV

A

B

a

aAS


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S FV

A

B

a

BaAS |


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S FV

A

B

a

BaaA

BaAS

|

a

a


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S FV

A

B

a

bBB

BaaABaAS

|

a

a

b


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S FV

A

B

a

abBB

BaaABaAS

|

|

a

a

b

a


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aaabaaabBaaaBaAS

S FV

A

B

a

a

a

b

a

GrammarNFA


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SFV

A

B

a

a

a

b

aabBB

BaaA

BaAS

|

|

GGrammarNFA

abaaaab

GLML

**

)()(

In General


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In General

A right-linear grammar

has variables:

and productions:

G

,,, 210 VVV

jmi VaaaV 21

mi aaaV 21

or


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We construct the NFA such that:

each variable corresponds to a node:iV

0V

F

V

1V

2V

3V

4V special

final state


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For each production:

we add transitions and intermediate nodes

jmi VaaaV 21

iV jV1a 2a ma


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For each production:

we add transitions and intermediate nodes

mi aaaV 21

iV FV1a 2a ma


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Resulting NFA looks like this:

0V

FV

1V

2

V

3V

4V

1a

3a3a

4a

8a

2a 4a

5a

9a5a

9a

)()( MLGL It holds that:

The case of Left-Linear Grammars


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The case of Left-Linear Grammars

Let be a left-linear grammar

We will prove: is regular

Proof idea:

We will construct a right-linear

grammar with

G

)(GL

G RGLGL )()(


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Since is left-linear grammar

the productions look like:

G

kaaBaA

21

kaaaA 21


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Construct right-linear grammar G

In :G kaaBaA 21

In :G BaaaA k 12

vBA

BvA R


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Construct right-linear grammar G

In :G kaaaA 21

In :G 12aaaA k

vA

RvA


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It is easy to see that:

Since is right-linear, we have:

RGLGL )()(

)(GL RGL )(

G

)(GL

Regular

Language RegularLanguage RegularLanguage

Proof - Part 2


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Proof Part 2


Regular Grammars

Regular

Languages


by some regular grammar

L

G


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Proof idea:

Let be the NFA with .

Construct from a regular grammar

such that


by some regular grammar

L

G

)(MLL

G)()( GLML


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Since is regular

there is an NFA such that

L

)(MLL

Example: a

b

a

b*)*(* abbababL

)(MLL

1q 2q

3q

0q


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Convert to a right-linear grammar

a

b

a

b

0q 1q 2q

3q

10 aqq


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a

b

a

b

0q 1q 2q

3q

21

11

10

aqq

bqq

aqq


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a

b

a

b

0q 1q 2q

3q

32

21

11

10

bqq

aqq

bqq

aqq

LMLGL )()(


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a

b

a

b

0q 1q 2q

3q

3

13

32

21

11

10

q

qq

bqq

aqq

bqq

aqq

G

)()(

In General


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In General

For any transition:

a

q p

Add production: apq

variable terminal variable


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For any final state: fq

Add production: fq


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reverse of regular grammer

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