Chapter 04 - Context Free Language

8/9/2019 Chapter 04 - Context Free Language

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Chapter – 04

Context Free Language andGrammar

Context Free Languages – Parsing and ambiguity –Sentential forms – leftmost and rightmost derivation– Derivation tree or Parse tree – Denition andrelationship between parse tree and derivation -

Simpliation of ontext free grammar

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Context Free Language(CFL)

"

#o overome the problem in regular language$ %eneeded the solution is alled ontext free language$

Context free language is generated by the ontextfree grammar$ &egular grammar is simple butontext grammar is omplex ompared to theregular grammar$

From the diagram' regular language is a Sub set ofthe CFL$ CFL is omplex one and it an solve theproblems( faed in regular language


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Context Free Grammar

)

Denition*

+ , .( #( P( S/

. – Set of non terminals# – set of terminals

P – Prodution rules* this rules are used to derive theertain string

S – starting state


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Cont.,

0

Derivation and Languages:

+eneration of language using spei rules is alled derivation$

Exampe 4.!: Constru"t the CFG #or the anguage having an$num%er o# a&s over the set ' a *.

&$1 , a2 , 3 ( a( aa( aaa( aaaa( 4$5

Prodution rules for the same*

S aS rule !S rule "

.ow if you want to derive the string 6aaa7$ Start with the startsymbol$

+

a+ rule!

aa+ rule!

aaa+ rule!

aaa rule"

aaa


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Cont.,

8

Exampe 4.: Constru"t the CFG #or the givenanguage

L an

%n

- n 0 *L , 3 ab( aabb( aaabbb( 4$5 , 3 e9ual number of a andb5

Prodution rule for the same*

S aSb rule!

S rule"

.ow if you want to derive the string 6aabb7$ :ou anstart the derivation from the start symbol

+

a+b rule!

aa+bb rule!

aabb rule!

aabb rule"


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Le#tmost Derivation and /ightmost Derivation

;

+eneration of the string using prodution rulesis alled derivation$


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Derivation ree

=

Derivation tree is a graphial representation for thederivation of the given prodution rules for the given

CF+$


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Cont.,

>

Exampe 4.1: Let G %e the grammar

+ a2 - %3

3 a - a+ - %332 % - %+ - a22

For the string %aaa%%a%%a nd e#tmost, rightmostderivation and Derivation tree

First write the prodution rules separately li?e below

S a@ rule!

S bA rule"

A a rule)

A aS rule0

A bAA rule8

@ b rule;

@ bS rule=

@ a@@ rule>


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Cont.,

B

Le#tmost Derivation /ightmostDerivation

+ +b3 r! b3 r"

ba+ r0 ba+ r0

baa2 r! baa2 r!

baaa2@ r> baaa@2 r>

baaab+@ r= baaa@b+ r=

baaabb3@ r" baaa@bb3 r"

baaabba2 r) baaa2bba r)baaabbab+ r; baaab+bba r=

baaabbabb3 r! baaabb3bba r"

baaabbabba r) baaabbabba r)


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Cont.,

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5arsing

!!

Parsing is the proess of analying a string ofsymbols( either in natural or omputer language$


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3m%iguit$

!"

+rammar an be derived in either leftmost orrightmost derivation$ %e an draw the derivationtree alled as parse tree or syntax tree$ Parse tree isa uni9ue one though the derivation is leftmost orrightmost$


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Cont.,

!)

Exampe 4.4: he given CFG is G (6, , 5, +)7here

6 E * 5 E E 8 E id * E E 9 E

+ E * E id * s the given grammaram%iguous;

.ow if the string is id 9 id 8 id then we an draw theparse tree

So the above grammar is ambiguous grammar


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+impi"ation o# CFG

!0

All +rammars are not always optimied$ #hatmeans grammars are ontaining some unneessarysymbolsnon-terminals/ and this will inrease thelength of the grammar$ Simpliation of grammar

means redution of grammar by removing uselesssymbols$

i$&emoval of useless symbols

ii$1limination of prodution

iii$&emoval of unit prodution


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Cont.,

!8

/emova o# useess s$m%os:

remove the useless non-terminals from the produtionrules$


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Cont.,

!;

P, 3 S !!A E !!

A 5

Exampe 4.=: +impi#$ the given CFG

+ 32 - C3

2 2C - 32

3 a

C a2 - %


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Cont.,

!=

Eimination o# produ"tion rue:

&emove the symbol from the grammar withoutaffeting the meaning of the grammar$ #hus is notneessary for the grammar$

Exampe 4.>: /emove the produ"tion #rom the#oo7ing CFG

+ ?@A? 0? -

@ !@ -

#o remove the prodution( delete the prodution rule and

to preserve the meaning of the CF+( plaing the value in the

other prodution instead of the orresponding non-terminal$

+tep!: remove the produ"tion

&emove the rules and :


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Cont.,

!>

+tep: o preserve the meaning o# grammar:

Plae the to rule !* S : :

S :

: S : :

Plae the : to rule !* S

Plae the and : to rule !* S

Plae the to rule "*

Plae the : to rule 0* : ! !

Colletively an write the CF+*

+ ?@ - @? - ?? - ? -@

? 0

@ !


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Cont.,

!B

/emoving the Bnit produ"tion:

Gnit produtions are the produtions in whih one non-terminal gives another non-terminal only without terminal$

: and : H

#hen you an minimie the rule li?e H

Exampe 4.: /emove the unit produ"tions #rom given CFG

+ 03 - !2 - C

3 0+ - 002 ! - 3

C 0!

$S A E !@ E 0!

A S E

@ ! E A

C !


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Cont.,

"

Exampe 4.: ptimie the given CFG %$ redu"ingthe grammar

+ 3 - 0C!3 2 - 0! - !0

C - CD

2 and D are useess s$m%os %e"ause 2 and D is

not avaia%e+ 3 is a unit produ"tion

C - prodution

S ! E !

A ! E !

C


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End o# Chapter 04

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Chapter 04 - Context Free Language