context free grammars normal

8/9/2019 Context Free Grammars Normal

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Fall 2005 Costas Buch - RPI 1

Simplifications

ofContext-Free Grammars


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A Substitution Rule

bB

aAB

abBcA

aaAA

aBS

p

p

p

p

p

Substitute

Equivalent

grammar

aAB

abbcabBcA

aaAA

abaBS

p

p

p

p

|

|

bB p


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Equivalent

grammarabaAcabbcabBcA

aaAA

aaAabaBS

||

||

p

p

p

aAB

abbcabBcA

aaAA

abaBS

p

p

p

p

|

|

SubstituteaABp


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In general:

1yB

xBzA

p

p

Substitute

zxyxBzA 1|pequivalent

grammar

1yBp


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Nullable Variables

:productionP PpX

Nullable Variable: P-Y

Example:

Pp

p

p

M

aMbM

aMbS

Nullable variable productionP


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PpM

Pp

p

p

M

aMbM

aMbSSubstitute

abaMbM

abaMbS

|

|

p

p

Removing sproductionP

After we remove all theall the nullable variables disappear

(except for the start variable)

sproductionP


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Unit-Productions

YpU

nit Production:(a single variable in both sides)

bbB

ABBA

aA

aAS

p

pp

p

pExample:

Unit Productions


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bbB

AB

BA

aA

aAS

p

p

p

p

p

Substitute

BAp

bbB

BAB

aA

aBaAS

p

p

p

p

|

|

Removal of unit productions:


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Remove

bbB

BAB

aA

aBaAS

p

p

p

p

|

|

bbB

AB

aA

aBaAS

p

p

p

p |

BB p

XX p

can be removed immediately

Unit productions of form


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SubstituteABp

bbB

aA

aAaBaAS

p

p

p ||

bbB

AB

aA

aBaAS

p

p

p

p |


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Remove repeated productions

bbB

aA

aBaAS

p

p

p |

bbB

aA

aAaBaAS

p

p

p ||

Final grammar


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Useless Productions

aAA

AS

S

aSbS

pp

p

p

P

--- aAaaaaAaAAS

Some derivations never terminate...

Useless Production


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bABA

aAA

AS

pp

p

p

P

Another grammar:

Not reachable from S

Useless Production


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In general:

If there is a derivation)(GLwxAyS --

Then variable is usefulA

Otherwise, variable is uselessA

consists ofterminals


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A production is useless

if any of its variables is useless

xAp

DC

CB

aAA

AS

S

aSbS

p

p

p

p

p

p

PProductionsuseless

useless

useless

useless

Variables

useless

useless

useless


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Example Grammar:

aCbC

aaB

aACAaSS

p

p

pp ||

Removing Useless Variables and Productions


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First: find all variables that can produce

strings with only terminals or

aCbC

aaB

aA

CAaSS

p

p

p

p || },{ BA

},,{ SBA

Round 1:

Round 2:

(these are possible useful variables)

P

(the right hand side of a production

has only terminals)

(the right hand side of a productionhas terminals and

variables of previous round)

T

his process can be generalized


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Then, remove productions that use variables

other than

aCbC

aaB

aA

CAaSS

p

p

p

p ||

},,{ SBA

aaB

aA

AaSS

p

p

p |


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Second: Find all variablesreachable from

aaB

aA

AaSS

p

p

p |

S A B

Use a Dependency Graph

not

reachable

S


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Keep only the variables

reachable from S

aaB

aA

AaSS

p

p

p |

aA

AaSS

p

p |

Final Grammar

Contains only

useful variables


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Removing All

Step 1: Remove Nullable Variables

Step 2: Remove Unit-Productions

Step 3: Remove Useless Variables

This sequence guarantees that

unwanted variables and production

are removed


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Normal Forms

forContext-free Grammars


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Chomsky Normal Form

Each productions has form:

BCAp

variable variable

aApor

terminal


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Examples:

bA

SAA

aS

ASS

p

p

p

p

Not ChomskyNormal Form

aaA

SAA

AASS

ASS

p

p

p

p

ChomskyNormal Form


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Convertion to Chomsky Normal Form

Example:

AcB

aabA

ABaS

p

p

pNot Chomsky

Normal Form

We will convert it to Chomsy Normal Form


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AcB

aabAABaS

p

p

p

Introduce new variables for the terminals:

cT

bT

aT

ATB

TTTA

ABTS

c

b

a

c

baa

a

p

p

p

p

p

pcba

TTT,,


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Introduce new intermediate variable

to break first production:

cT

bT

aT

ATB

TTTA

ABTS

c

b

a

c

baa

a

p

p

p

p

p

p

cT

bT

aT

ATB

TTTA

BTV

AVS

c

b

a

c

baa

a

p

p

p

p

p

p

p

1

1

1V


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Introduce intermediate variable:

cT

bT

aT

ATB

TTV

VTA

BTV

AVS

c

b

a

c

ba

a

a

p

p

p

p

p

p

p

p

2

2

11

2V

c

b

aT

ATB

TTTA

BTV

AVS

c

b

a

c

baa

a

p

p

p

p

p

p

p

1

1


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Final grammar in Chomsky Normal Form:

cT

bT

aT

ATB

TTV

VTA

BTV

AVS

c

b

a

c

ba

a

a

p

p

p

p

p

p

p

p

2

2

1

1

AcB

aabA

ABaS

p

p

p

Initial grammar


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From any context-free grammar

(which doesnt produce )

not in Chomsky Normal Form

we can obtain:

an equivalent grammar

in Chomsky Normal Form

In general:

P


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The Procedure

First remove:

Nullable variables

Unit productions


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Then, for every symbol :a

In productions: replace witha aT

Add production aTa p

New variable: aT


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Replace any productionnCCCA .21p

with

nnn CCV

VCV

VCA

12

221

11

p

p

p

-

New intermediate variables: 221 ,,, nVVV-


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Observations

Chomsky normal forms are good

for parsing and proving theorems

It is easy to find the Chomsky normal

form for any context-free grammar


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Greinbach Normal Form

All productions have form:

kVVVaA .21p

symbol variables

0uk


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Examples:

bB

bbBaAA

cABS

p

p

p

||

GreinbachNormal Form

aaS

abSbS

p

p

Not Greinbach

Normal Form


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aaS

abSbS

p

p

Conversion to Greinbach Normal Form:

bT

aT

aTS

STaTS

b

a

a

bb

p

p

p

p

Greinbach

Normal Form


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Observations

Greinbach normal forms are very good

for parsing strings (better than Chomsky Normal Forms)

However, it is hard to find the

Greinbach normal of a grammar

context free grammars normal

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