48-747 Shape Grammars

HISTORICAL

WHATISAGRAMMAR?ALANGUAGE?

language

OxfordEnglishDictionary

Languageisthemethodof‘human’communication,eitherspokenorwritten,consistingoftheuseof‘words’inanagreedway.

Languageisthestyleorfacultyofexpression

Languageisthesystemofsymbolsandrulesforwritingalgorithms

underlyinglanguageisthenotionofasentence

The happy li*le boy ran quickly

<DET> <ADJ> <ADJ> <NOUN> <VERB> <ADVerb>

<ADJPHRASE> <NOUNPHRASE> <VERBPHRASE>

<NOUNPHRASE>

<SENTENCE>

parsetree

The happy li*le boy ran quickly

<DET> <ADJ> <ADJ> <NOUN> <NOUN> <ADVerb>

<ADJPHRASE> <NOUNPHRASE> <VERBPHRASE>

<NOUNPHRASE>

<SENTENCE>

werecognizepartsofasentence

relationshipsbetweenparts(ofthesentence):

a<SENTENCE>isa<NOUNPHRASE>followedbya<VERB

a<NOUNPHRASE>isa<ADJPHRASE>followedbya<NOUNPHRASE>

a<ADJPHRASE>isa<DET>followedbya<ADJective>

a<NOUNPHRASE>isan<ADJective>followedbya<NOUN>

a<VERBPHRASE>isa<VERB>followedbya<ADVerb>

boyisa<NOUN>andranisa<VERB>

entities:

thingslike<SENTENCE><NOUNPHRASE>

thingslikethe,li*le,boy,quickly

relationshipsspecifyrules

<SENTENCE> <NOUNPHRASE><VERB>

<NOUNPHRASE> <ADJPHRASE><NOUNPHRASE>

<ADJPHRASE> <DET><ADJective>

<NOUNPHRASE> <ADJective><NOUN>

<VERBPHRASE> <VERB><ADVerb>

<NOUN>boy

<VERB> ran

vocabularyoralphabetsymbols• TERMINAL

• NONTERMINAL

entities• stringsofsymbolsjuxtapositionedtogether

sentence• stringmadeupofterminalvocabularysymbols

production• RULESmadeupofstringsexpressingarelationship

betweentwosuchentitiesandexpressedasa→b

initialstringorseedmadeupofvocabularyelements

grammar

Buildingblocks

producedfrom

Thelanguage,L,ofagrammar,G,isthesetofallsentences(i.e.,withoutnon‐terminals)thatareproducedfromtheinitialstring

L(G)={sentence|initial‐string⇒*sentence}

language

ruleapplication

Aruleisapplicable

tothecurrentstringwhichiseithertheinitialstringorastringproducedfromtheinitialstring

whenever

thelefthandsideoftherule‘occurs’intheobject

inwhichcase

itcanbereplacedbytherighthandsideoftheruleunderruleapplication

formal definition

GrammarG=(N,T,P,S)

seed

production

terminal

terminal

seedbelongstotheuniverseofstringsmadeofsymbolsinNandT

P(roduction)containrulesoftheforma→bwhereaandbbelongtotheuniverseofstringsmadeofsymbolsinT(erminal)andN(onterminal)whereacannotbeempty.

vocabularyanduniverse

VOCABULARYisalimitedsetofsymbolsnotwoofwhicharesimilaroridentical

IfwearegivenasetofsymbolsV,thenwecancreateasetV*calledtheUNIVERSE(orLEASTSET)ofVinthefollowingmanner:

TheemptystringeisinV*.

EverysymbolinVisinV*.

IfaandbarestringsinV*,sotooistheirjuxtaposition(concatenation)abinV*

V*isclosedunderconcatenation

WedenotethesetV*–{e}byV+.

Foranyproductiona→b,wehavea∈V+andb∈V*

example‐binarysequences

V={0,1}

V*={e,0,1,00,01,10,11,000,001,…}

V+={0,1,00,01,10,11,000,001,…}

exampleofagrammar

G=(N={S},T={0,1},P={S→0S1,S→01},S)

Whatcanwedowiththisgrammar?

FromSbyapplyingruleS→0S1weget0S1Thatis,substitutetheoccurrenceofthelefthandsideofthecurrentruleinthecurrentstringbytherighthandside.Thatis,S⇒0S1

ByapplyingruleS→0S1againweget00S11.Thatis,S⇒0S1⇒00S11.Or,S⇒*00S11

IfweapplyruleS→01wegetS⇒0S1⇒00S11⇒000111

typesofgrammars

PHRASESTRUCTUREGRAMMARSproductionsoftheformα→β

CONTEXTSENSITIVEGRAMMARSproductionsoftheformα1Aα2→β1Bβ2equivalently,|α|≤|β|

CONTEXTFREEGRAMMARSproductionsoftheformA→βwhereAisasinglevariable

REGULARGRAMMARSproductionsoftheformA→aBorA→awhereAisasinglevariable

acfgexample

G=(

N={S,A,B}, non‐terminals

T={a,b}, terminals

P={S→aB,S→bA, productionsA→a,A→aS,A→bAA,B→b,B→bS,B→aBB},

S) seed

WhatisthelanguageL(G)?

Allstringswithequalnumbersofa’sandb’s.

grammars,proceduresandalgorithms

Notethatifwecanrecognizewhetherasentenceisinalanguagethenwecanalwaysgeneratethelanguagesimplybygeneratingeachelementintheuniverseandcheckingwhethertheelementisinthelanguage

example:

givenanintegeri>1,isitprime?

1. setjto2

2. ifj≥i,thenhalt.iisprime.

3. ifi/jisaninteger,thenhalt.iisnotprime.

4. setjtoj+1andgoto2.

anotherexample:

givenanintegeri>1,isthereaperfectintegergreaterthani?

1. setk=i

2. setk=k+1,sum=0,j=1

3. ifj<k,goto6

4. ifsum|k,goto2

5. Halt.Thereisaperfectinteger.

6. ifk/jisnotaninteger,goto8.

7. setsum=sum+j

8. setjtoj+1andgoto3.

recursivelyenumerableandrecursive

Asetisrecursivelyenumerable(r.e)ifitcanbegeneratedbyaprocedure

Anotherwayoflookingatthisisthatthereisa1‐1correspondencebetweenelementsofthesetandthenaturalnumbers.Inotherwords,wecan“index”theelementsoftheset

Asetisrecursiveifitcanbegeneratedbyanalgorithm

Languagesofgrammarsarerecursivelyenumerable

pslarerecursivelyenumerablecsl,cfl,rlarerecursive

Equivalently,asetthatisrecursivelyenumerableisthelanguageofsomegrammar

noamchomsky－syntacticstructures

“thesearchforrigorousformulationinlinguisticshasamuchmoreseriousmotivationthanmereconcernforlogicalnicetiesorthedesireto

purifywell‐establishedmethodsforlanguageanalysis

preciselyconstructedmodelsforlinguisticstructurecanplayanimportantrole,bothnegativeandpositive,intheprocessof

discoveryitself

bypushingaprecisebutinadequateformulationtoanunacceptableconclusion,wecanoftenexposetheexactsourceofthisinadequacy

and,consequently,gainadeeperunderstandingofthelinguisticdata”