CS25 Lecture Presentation 5

8/20/2019 CS25 Lecture Presentation 5

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CS 25Automata Theory & Formal Languages

Lecture 5


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CONTEXT-FREE GRAMMARS(CFGs)

― a notation for describing languages

― !ore "o#erful t$an finite auto!ata or RE%s& but still cannot defineall "ossible languages

― can describe features #it$ recursi'e structure

―

First used in t$e stud of $u!an language ― !"ortant a""lication in s"ecification and co!"ilation of

"rogra!!ing languages


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CONTEXT-FREE GRAMMARS(CFGs)

E*a!"les+ CFG for , n.n / n 0 .1G1:

S -> 01

S -> 0S1

― Consists of 2RO34CTONS (collection of substitution rules)

― Rules $as 5ARA67E and STRNG ('ariables and ot$er s!bolscalled ter!inals) se"arated b arro#s


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FORMA7 3EFNTON OFCONTEXT-FREE GRAMMAR A conte*t-free gra!!ar is a 8-tu"le (V,

Σ

, R, S)& #$ere

1. V is a finite set called VARIABLES

Σ is a finite set& dis9oint fro! 5& called TERMINALS

3. R is a finite set of RULES/PRODUCTIONS

4. S is t$e START SYMBOL


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CFGs+ 2roductions

● A production $as t$e for! 'ariable -0 string of 'ariables and

ter!inals

●

Con'ention+ – A& 6& C&: are 'ariables

– a& b& c&: are ter!inals – α, β, γ ,…… are strings of ter!inals and;or 'ariables

– For convenience, we abbreviate rules with the same left-hand variable into a single line

● E*+ A→ A. / 6


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FORMA7 3EFNTON OF

CONTEXT-FREE GRAMMARExample !F" G for # 0n1n $ n > 1%&

G= (V, Σ, R, S)& #$ere

1. V < ,S1

Σ < ,& .1

3. R < ,S -0 .&

S -0 S.14. S is t$e start s!bol


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3ER5ATON fro! CFG=s

● 3ER5ATON is t$e se>uence of substitutions to obtain a string

● ?e derive strings in t$e language of a CFG b starting #it$ t$e

start s!bol& and re"eatedl re"lacing so!e 'ariable A b t$e

rig$t side of one of its "roductions – T$at is& t$e @"roductions for A are t$ose t$at $a'e A on t$e left

side of t$e -0

● All strings generated b t$e gra!!ar constitute t$e

7ANG4AGE OF TBE GRAMMAR – 7(G.) -language of gra!!ar G.


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3ER5ATON fro! CFG=s

● For!all& #e sa α Aβ


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3ER5ATON fro! CFG=s

7EFTMOST 3ER5ATON at e'er ste"& t$e left!ost

re!aining 'ariable is t$e one re"laced

RGBTMOST 3ER5ATON - at e'er ste"& t$e rig$t!ost

re!aining 'ariable is t$e one re"laced


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3ER5ATON fro! CFG=sG < (5& Σ& R& HEX2R0) #$ere

5 is ,HEX2R0& HTERM0& HFACTOR01&

Σ S ,a& I& X& (& )1&

R is HEX2R0 → HEX2R0 I HTERM0 / HTERM0

HTERM0 → HTERM0 X HFACTOR0 / HFACTOR0

HFACTOR0 → (HEX2R0) / a

7eft!ost 3eri'ation of a a ! a +

HEX2R0 ⇒ HEX2R0 I HTERM0

⇒HTERM0 I HTERM0

⇒HFACTOR0 I HTERM0

⇒a I HTERM0

⇒a I HTERM0 X HFACTOR0

⇒a I HFACTOR0 X HFACTOR0

⇒a I a X HFACTOR0

⇒a I a X a


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3ER5ATON fro! CFG=sG < (5& Σ& R& HEX2R0) #$ere

5 is ,HEX2R0& HTERM0& HFACTOR01&

Σ S ,a& I& X& (& )1&

R is HEX2R0 → HEX2R0 I HTERM0 / HTERM0

HTERM0 → HTERM0 X HFACTOR0 / HFACTOR0

HFACTOR0 → (HEX2R0) / a

Rig$t!ost 3eri'ation of aIaXa +

HEX2R0 ⇒ HEX2R0 I HTERM0

⇒HEX2R0 I HTERM0 X HFACTOR0

⇒HEX2R0 I HTERM0 X a

⇒HEX2R0 I HFACTOR0 X a

⇒HEX2R0 I a X a

⇒HTERM0 I a X a

⇒HFACTOR0 I a X a

⇒a I a X a


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SENTENTA7 FORMS

● 'n( string of variables and)or terminals

derived from the start s!bol is called a

sentential form&● Formall(, α is a sentential form iff S *>+α&


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7ANG4AGE OF A GRAMMAR

● f " is a !F", then ."/, the language of G,

is #w $ S *>+ w%&

– ote w must be a terminal string, S is the starts(mbol&

● Example " has productions S -> ε andS -> 0S1&

● ."/ * #0n1n $ n > 0%&Note: ε is a legitimateright side.


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CONTEXT-FREE 7ANG4AGES

● ' language that is defined b( some !F" is

called a context-free language&

●

here are !F2s that are not regular languages,such as the example 3ust given&

● 4ut not all languages are !F2s&

● ntuitivel( !F2s can count two things, not three&

● 24SB3O?N A4TOMATA (23A) is a class of

!ac$ines recogniDing conte*t-free languages

(CF7s)


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EXERCSES+

. 4sing t$e gra!!ar G defined in t$e "re'ious slide& s$o# t$e left!ost and rig$t!ostderi'ation #it$ t$eir corres"onding "arse trees for t$e string (a + a) X a

Gi'en t$e follo#ing gra!!ar+

HSENTENCE0→ HNO4N-2BRASE0 H5ER6-2BRASE0

HNO4N-2BRASE0 → HCOM27X-NO4N0 / HCOM27X-NO4N0 H2RE2-2BRASE0

H5ER6-2BRASE0→ HCOM27X-5ER60 / HCOM27X-5ER60 H2RE2-2BRASE0

H2RE2-2BRASE0→ H2RE20 HCOM27X-NO4N0

HCOM27X-NO4N0 → HARTC7E0 HNO4N0

HCOM27X-5ER60→ H5ER60 / H5ER60 HNO4N-2BRASE0

HARTC7E0→ a / t$e

HNO4N0→ bo / girl / flo#er

H5ER60 → touc$es / liJes / sees

H2RE20 → #it$

a Gi'e t$e for!al definition

b S$o# t$e left!ost deri'ation for+

t"e #ir$ %ee% a &$o'er

a (o) 'it" a &$o'er $i*e% t"e #ir$


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2ARSE TREES

―

re"resents t$e (sntactic) structure of w ― an alternati'e re"resentation to deri'ations and

recursi'e inferences

― t$ere can be se'eral "arse trees for t$e sa!e

string ― ideall t$ere s$ould be onl one "arse tree (t$e

@true structure) for eac$ string& ie t$e language

s$ould be una!biguous


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AM6G4TK

●

So!eti!es& a gra!!ar generates a string in se'eral different#as (se'eral "arse trees and $ence& different !eanings) – E*+ HEX2R0 → HEX2R0 I HEX2R0 / HEX2R0 * HEX2R0 / (HEX2R0) / a

● 5ou should be able to prove that a a ! a is generated

ambiguousl(&

●

' grammar generates a string ambiguousl(, if the string has twodifferent parse trees, 6 two different derivations&

● ' string is derived ambiguously in CFG if it has two or more

different leftmost derivations&

● he grammar is ambiguous if it generates some string

ambiguousl(&

● INHERENTLY AMBIGUOUS - languages that can onl( be

generated b( ambiguous grammars&

E*+ , i.

9

J / i


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EXERCSES+

Gi'en t$e follo#ing CFG&

E → E E + E , E + -E. + id

a. S$o# t$at t$e gi'en gra!!ar is a!biguous b s$o#ing t$eleft!ost deri'ations and "arse trees for t$e string id I id id

b) Construct an e>ui'alent una!biguous gra!!ar


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3ESGNNG CONTEXT-FREEGRAMMARS

● Many CFGs are union of simpler CFGs. Construt CFGs

for simpler CFGs an! t"en om#ine S → S1 $ S% $ & $ S'

― E*+ , n.n / n ≥ 1 4 , .nn / n ≥ 1

● f reular, onstrut *F+ first an! t"en onert to CFG

. MaJe 'ariable Ri → aR 9 if δ(>i& a) < > 9

Add Ri→ L if >i is t$e acce"t state

MaJe R t$e start 'ariable #$ere > is t$e start state

● -o lin'e! su#strins. /se R→

uRv.

― E*+ , n.n / n≥ 1


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3ESGNNG CONTEXT-FREEGRAMMARS

● Strins may ontain ertain strutures t"at appearreursiely as part of t"e ot"er struture.

― 2lace 'ariable s!bol generating t$e structure in t$e location ofrules corres"onding to #$ere t$at structure !a recursi'el a""ear

– E*a!"le+ G < (5& Σ& R& HEX2R0) #$ere

– 5 is ,HEX2R0& HTERM0& HFACTOR01&

– Σ S ,a& I& X& (& )1&

– R is HEX2R0 → HEX2R0 I HTERM0 / HTERM0

– HTERM0 → HTERM0 X HFACTOR0 / HFACTOR0

– HFACTOR0 → (HEX2R0) / a


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CBOMSK NORMA7 FORM (CNF)

● Simplifie! form

● Form: A → BC

A → a

S → L

#$ere A, B and C are an 'ariables e*ce"t t$at B and C !anot be start 'ariables a is an ter!inal S is t$e start 'ariable


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CBOMSK NORMA7 FORM (CNF)

-"eorem:

An conte*t-free language is generated b a conte*t-free gra!!ar in C$o!sJ nor!al for!


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CON5ERSON OF ANK GRAMMARTO CNF

1. +!! ne start sym#ol. (S → S)

%. 0liminate all rules of form +→ .● (f R → uA' and A → L& R → uA' / u'● f R → uA'A# and A → L& R → uA'A# / u'A# / uA'# /

u'#● f R → A and A → L& do t$e sa!e for R)

3. 0liminate all unit rules of form + → 2.● (f 6 → u and A → 6& A → u)

4. Conert remainin rules into t"e proper form.● (Re"lace A → u0u1 2u* #$ere J ≥ and eac$ ui is a

'ariable or a ter!inal s!bol #it$ A → u0 A03 A0 → u1 A1 3

A1 → u4 A4323 A*51 → u*50u* f J ≥ & re"lace ui #it$ U i

and add U i → ui )


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CON5ERSON OF ANK GRAMMARTO CNF

0ample:

S → ASA / a6

A→ 6 / S

6 → b / L


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EXERCSES

. Find CFGs t$at generate t$ese regular languages o'er t$e

al"$abet ∑ < ,a& b1+

a) T$e language defined b (aaa I b)

b) All strings #it$ an e'en nu!ber of a%sc) All strings #it$ an odd nu!ber of a%s or an e'en nu!ber of

b%s

Gi'e a CFG t$at generates t$e non-regular language2A7N3ROME (strings t$at read t$e sa!e for#ard andbacJ#ard) o'er al"$abet ,a& b1


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EXERCSES

Gi'en t$e follo#ing CFG&G + S → aM

S → bS

M → aF

M → bS

F → bF

F → L

?$at is 7(G) P

8 Find an e>ui'alent gra!!ar in CNF for t$e follo#ing CFGs+i S → SS / A

A → SS / AS / a

ii S → aA / b6

A → aAA / bS / b

6 → b66 / aS / a

CS25 Lecture Presentation 5

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