context free grammars_simplification

Costas Busch - LSU 1

Simplifications of

Context-Free Grammars


A Substitution Rule

bBaABabBcAaaAAaBS

Substitute

Equivalentgrammar

aABabbcabBcA

aaAAabaBS

|

|

bB


EquivalentgrammarabaAcabbcabBcA

aaAAaaAabaBS

||

||

aABabbcabBcA

aaAAabaBS

|

|

Substitute aAB


In general:

1yB

xBzA

Substitute

zxyxBzA 1|equivalentgrammar

1yB


Nullable Variables:production X

Nullable Variable: Y

Example:

MaMbMaMbS

Nullable variable production


M

MaMbMaMbS

SubstituteabaMbMabaMbS|

|

Removing sproduction

After we remove all the all the nullable variables disappear(except for the start variable)

sproduction


Unit-ProductionsYX Unit Production:

(a single variable in both sides)

bbBABBAaAaAS

Example:

Unit Productions


bbBABBAaAaAS

SubstituteBA

bbBBAB

aAaBaAS

|

|

Removal of unit productions:


Remove

bbBBAB

aAaBaAS

|

|

bbBABaA

aBaAS

|

BB

XX can be removed immediatelyUnit productions of form


SubstituteAB

bbBaA

aAaBaAS

||

bbBABaA

aBaAS

|


Remove repeated productions

bbBaA

aBaAS

|

bbBaA

aAaBaAS

||

Final grammar


Useless Productions

aAAAS

SaSbS

aAaaaaAaAAS

Some derivations never terminate...

Useless Production


bABA

aAAAS

Another grammar:

Not reachable from S

Useless Production


In general:

If there is a derivation)(GLwxAyS

Then variable is usefulA

Otherwise, variable is uselessA

consists of terminals


A production is useless if any of its variables is useless

xA

DCCBaAAAS

SaSbS

Productionsuselessuselessuselessuseless

Variablesuselessuselessuseless


Example Grammar:

aCbCaaBaA

CAaSS

||

Removing Useless Variables and Productions


First:find all variables that can producestrings with only terminals or

aCbCaaBaA

CAaSS

|| },{ BA

},,{ SBA

Round 1:

Round 2:

(possible useful variables)

(the right hand side of production that has only terminals)

(the right hand side of a production has terminals and variables of previous round)

This process can be generalized


Then, remove productions that use variablesother than

aCbCaaBaA

CAaSS

||

},,{ SBA

aaBaA

AaSS

|


Second:Find all variablesreachable from

aaBaA

AaSS

|

S A B

Use a Dependency Graphwhere nodes are variables

unreachable

S


Keep only the variablesreachable from S

aaBaA

AaSS

|

aAAaSS

|

Final Grammar

Contains onlyuseful variables


Removing AllStep 1: Remove Nullable Variables

Step 2: Remove Unit-Productions

Step 3: Remove Useless Variables

This sequence guarantees thatunwanted variables and productionsare removed

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