three models for the description of language, noam chomsky

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THREE MODELS FOB THE DESCRI PTI ON OF LANGUAGE*

Noam ChomskyDepart ment of Modern Languages and Research Laboratory of El ectroni cs

Massachusett s I nst i tute of Technol ogyCambri dge, Massachusetts

Abstr act

We i nvesti gate several concept i ons of l i ngui st i c st ructure to determi ne whether ornot they can provi de si mpl e and "r eveal i ng"grammars that generate al l of the sentencesof Engl i sh and onl y these. We f i nd that nof i ni te- state Markov process that producessymbol s wi t h t ransi t i on f romstate to statecan serve as an Engl i sh grammar. Furt hermore,the part i cul ar subcl ass of such processes t hatproduce n- order stat i st i cal approxi mat i ons toEngl i sh do not come cl oser, wi t h i ncreasi ng n,to matchi ng the output of an Engl i sh grammar.We f ormal i se- the not i ons of “phrase st ructure"and show that thi s gi ves us a method fordescr i bi ng l anguage whi ch i s essent i al l y morepowerf ul , though st i l l representabl e as a ratherelementary type of f i ni te- state process. Neverthel ess, i t i s successful onl y when l i mi ted to asmal l subset of si mple sentences. We study thef ormal propert i es of a set of grammati cal t ransf ormati ons t hat carry sentences wi t h phrasestructure i nto new sentences wi t h deri ved phrasest ructure, showi ng that t ransf ormati onal grammarsare processes of the same el ementary t ype asphrase-st ructure grammars; that the grammar of Engl i sh i s materi al l y si mpl i f i ed i f phrasestructure descri pt i on i s l i mi ted to a kernel of si mpl e sentences f romwhi ch al l other sentencesare constructed by repeated t ransf ormat i ons; andthat thi s vi ew of l i ngui sti c str ucture gi ves acertai n i nsi ght i nto the use and underst andi ng

of l anguage.

1. I nt roducti on

There are two cent ral probl ems i n thedescri pt i ve study of l anguage. One pri maryconcern of the l i ngui st i s to di scover si mpl eand "r eveal i ng* grammars f or natural l anguages.At the same ti me, by st udyi ng the propert i es of such successf ul grammars and cl ari f yi ng the basi cconcept i ons t hat underl i e them, he hopes toarr i ve at a general theory of l i ngui sti cst ructure. We shal l exami ne certai n f eatures of these rel ated i nqui ri es.

The grammar of a l anguage can be vi ewed asa theory of the st ructure of thi s l anguage. Any

sci ent i f i c theory i s based on a certai n f i ni teset of observat i ons and, by est abl i shi ng generall aws st ated i n terms of certai n hypotheti calconstructs, i t att empts to account f or these

•Thi s work was supported i n part by the Army(Si gnal Corps), the Ai r Force (Of f i ce of Sci ent i f i cResearch, Ai r Research and Devel opment Command) ,and the Navy (Of f i ce of Naval Besearch) , and i npart by a grant f romEast man Kodak Company.

observat i ons, to show how they are i nterrel ated,and to predi ct an i ndefi ni te number of newphenomena. A mathemat i cal theory has theaddi t i onal proper ty that predi cti ons f ol l owri gorousl y f romthe body of theory. Si mi l arl y,a grammar i s based on a fi ni te number of observedsentences ( the l i ngui st ' s corpus) and i t"proj ects" thi s set t o an i nf i ni te set of grammati cal sentences' by establ i shi ng general"l aws" (grammati cal rul es) f ramed i n terms of such hypothet i cal const ructs as the part i cul arphonemes, words, phrases, and so on, of thel anguage under analysi s. A proper l y f ormul atedgrammar shoul d determi ne unambi guousl y the setof grammati cal sentences.

General l i ngui st i c t heory can be vi ewed asa metatheory whi ch i s concerned wi t h the probl emof how to choose such a grammar i n the case of each par t i cul ar l anguage on the basi s of a f i ni tecorpus of sentences. I n part i cul ar, i t wi l lconsi der and att empt t o expl i cate the rel at i onbetween the set of grammati cal sentences and theset of observed sentences. I n other words,l i ngui sti c theory at tempts to expl ai n the abi l i t yof a speaker to produce and underst and newsentences, and to rej ect as ungrammati cal othernew sequences, on the basi s of hi s l i mi tedl i ngui st i c experi ence.

Suppose t hat f or many l anguages t here are

cert ai n cl ear cases of grammati cal sentences andcertai n cl ear cases of ungrammati cal sequences,e. g. , (1) and (2), respecti vel y, i n Engl i sh.

(1) J ohn ate a sandwi ch(2) Sandwi ch a ate J ohn.

I n thi s case, we can test the adequacy of aproposed l i ngui st i c t heory by determi ni ng, f oreach l anguage, whether or not the cl ear casesare handl ed properl y by the grammars const ructedi n accordance wi t h thi s theory. For exampl e, i f a l arge corpus of Engl i sh does not happen tocont ai n ei ther (1) or (2), we ask whether thegrammar t hat i s deter mi ned f or thi s corpus wi l lproj ect t he corpus to i ncl ude (1) and excl ude (2).Even though such cl ear cases may provi de only a

weak test of adequacy f or the grammar of a gi venl anguage taken i n i sol at i on, they provi de a veryst rong test f or any general l i ngui st i c t heory andf or the set of grammars to whi ch i t l eads, si ncewe i nsi st that i n the case of each l anguage thecl ear cases be handl ed proper l y i n a f i xed andpredetermi ned manner. We can take certai n st epstowards the const ructi on of an operat i onalcharacter i zati on of "grammati cal sentence" thatwi l l provi de us wi t h the cl ear cases r equi red toset the task of l i ngui sti cs si gni f i cantl y.

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Observe, f or exampl e, that ( 1) wi l l be r ead by anEngl i sh speaker wi t h the normal i ntonati on of asentence of the corpus, whi l e (2) wi l l be readwi t h a fal l i ng i ntonat i on on each word, as wi l lany sequence of unrel ated words. Ot her di st i ngui shi ng cri ter i a of the sane sort can bedescri bed.

Before we can hope to provi de a sati sf actoryaccount of the general rel ati on between observedsentences and grammat i cal sent ences, we mustl earn a great deal more about the f ormal proper t i es of each of these sets. Thi s paper i s concerned wi t h the f ormal st ructure of the set of grammati cal sentences. We shal l l i mi t oursel vesto Engl i sh, and shal l assume i ntui t i ve knowl edgeof Engl i sh sentences and nonsentences. We thenask what sort of l i ngui st i c theory i s requi red asa basi s f or an Engl i sh grammar that wi l l descri bethe set of Engl i sh sentences i n an i nterest i ngand sati sf actory manner.

The f i rst step i n the l i ngui st i c anal ysi s of a l anguage i s to provi de a f i ni te systemof

representat i on f or i ts sentences. We shal l assumethat thi s step has been carr i ed out, and weshal l deal wi t h l anguages onl y i n phonemi cor al phabet i c transcri pt i on. By a l anguage,then, we shal l mean a set ( f i ni te or i nf i ni te)of sentences, each of f i ni te l ength, al lconst ructed f roma f i ni te al phabet of symbol s.I f A i s an al phabet , we shal l say that anyt hi ngf ormed by concatenat i ng the symbol s of A i s ast r i ng i n A. By a grammar of the l anguage L wemean a devi ce of some sort that produces al l of the str i ngs that are sentences of L and onl ythese.

Ho matt er how we ul t i matel y deci de toconstruct l i ngui sti c theory, we shal l surel yrequi re t hat the grammar of any l anguage must he

f i ni te. I t f ol l ows that onl y a countabl e set of grammars i s made avai l abl e by any l i ngui st i ctheory; hence that uncountabl y many l anguages,i n our general sense, are l i t eral l y not descri babl ei n terms of the concept i on of l i ngui st i c structureprovi ded by any par t i cul ar +heory. Gi ven aproposed theory of l i ngui st i c str ucture, then, i ti s al ways appropr i ate to ask the f ol l owi ng question:

(3) Ar e there i nterest i ng l anguages t hat aresi mpl y outsi de the range of descri pti on of theproposed type?

I n part i cul ar, we shal l ask whet her Engl i sh i ssuch a l anguage. I f i t i s, then the proposedconcept i on of l i ngui st i c st ructure must be j udged

i nadequate. I f t he answer to (3) i s negati ve, wego on to ask such questi ons as the fol l owi ng:

(k) Can we const ruct reasonabl y si mpl egrammars f or al l I nteresti ng l anguages?

(5) Are such grammars "reveal i ng" i n thesense that the syntact i c structure that theyexhi bi t can support semant i c anal ysi s, can provi dei nsi ght i nto the use and underst andi ng of l anguage,etc. ?

We shal l f i rst exami ne vari ous concept i ons

of l i ngui sti c str ucture i n terms of the possi bi l i t y and compl exi ty of descri pt i on (quest i ons(3), (**■))• Then, i n § 6 , we shal l br i ef l yconsi der the same theori es i n terms of (ji) , andshal l see that we are i ndependent l y l ed to thesame concl usi ons as to rel at i ve adequacy f or t hpurposes of l i ngui sti cs.

2. Fi ni te State Harkov Processes.

2.1 The most el ementary grammars whi ch, wi tha f i ni te amount of apparatus, wi l l generate ani nf i ni te number of sentences, are those based oa f ami l i ar concept i on of l anguage as apart i cul arl y si mpl e type of i nf ormat i on source,namel y, a f i ni te-st ate Markov process. - Speci f i cal l y, we def i ne a f i ni t e- state grammar 8as a systemwi t h a f i ni te number of statess0, ~*sq. a set ^ j a ^ j 0<i , j <q; l ^k ^H^ fo

each i . j of t ransi t i on symbol s, and a setC={( Si , Sj jy of certai n pai rs of states of B t

are sai d to be connected. As the syst emmovesf romst ate S to S. , i t produces a symbol a. ..

At- , i «J I J JC

Suppose that

(6 )

i s a sequence of states of (J wi t h o^ kl O, and

(S ,S )eC f or each i <m. As tha systemmovai ai +l

f romS to S i t produces the symbolai ai +i

(7) a.aiai +l k

f or some k <N Usi ng the arch' ”"'' to si gnia i ai +l

concat enati on,2 we say that the sequence (6)generates al l sentences

( 8)al °2kl

a , a k 1ID—1 ID ID***!

f or al l appropri ate choi ces of k (i . e. , f ork, < K ). The l anguage contai ni ng al l ai _ a iai +l G

onl y such sentences i s cal l ed the l anguagegenerated by G.

Thus, to produce a sentence of we set thesystemG i n the i ni t i al state SQ and we run

through a sequence of connect ed states, endi ngagai n wi t h So, and produci ng one of the associ at ransi t i on symbol s of A wi t h each tr ansi t i on f rone st8te to the next. We say that a l anguage i s a f i ni te- state l anguage i f L i s the set of sentences generated by some f i ni te-st ate gramma

2. 2. Suppose that we take the set A of t ransi tsymbol s to be the set of Engl i sh phonemes. Wecan att empt to const ruct a f i ni te state grawtarwhi ch wi l l generate every st r i ng of Engl i shphonemes whi ch i s a grammati cal sentence of Engl i sh, and onl y such stri ngs. I t i s i mmedi atevi dent that the task of const ruct i ng a f i ni te-state grammar f or Engl i sh can be consi derabl ysi mpl i f i ed i f we take A as t he set of Engl i sh

l i l t



morphemes^ or words, and const r uct G bo that i twi l l generate exact l y t he grammati cal stri ngs of these uni ts. We can then compl ete the grammarby gi vi ng a f i ni te set of rul es t hat gi ve thephonemi c spel l i ng of each word or morpheme i neach context i n whi ch i t occurs. We shal lconsi der br i ef l y the status of such rul es i n

§ b.l and §5- 3.

Bef ore i nqui r i ng di rectl y i nto the probl emof const ruct i ng a f i ni te- state grammar f orEngl i sh morpheme or word sequences, l et usi nvest i gate the absol ute l i mi ts' of the set of f i ni te-state l anguages. Suppose that A i s theal phabet of a l anguage L, that a , . . , an are

symbol s of thi s al phabet, and that S=a /_''.. •/""'an

i s a sentence of L. We say that S has an (i , j ) -dependencr wi t h respect to X i f and onl y i f t hef ol l owi ng condi t i ons are met :

(9) ( i ) 1 < i < j < n

( i i ) - there are symbol s b . bj eAwi t h theproperty that i s not a sentence of L, and S£ i s a sentence of L, wherei s f ormed f rom S by repl aci ng the i ^symbol of S (namel y, aj J by bi , andS2i s formed f rom by repl aci ng the

j th symbol of (namel y, aj ) by bj .

I n other words, S has an ( i , j ) - dependency wi t hrespect to L I f repl acement of the 1 symbol a^

of S byb (b. a. ) requi res acor respondi ng

repl acement of the j symbol a. of S by b. J J

(bj aj ) f or the resul t i ng st r i ng to bel ong to L.

We say t hat D*

dependency set f or S i n L i f and onl y i f the

f ol l owi ng condi t i ons are met:

(10)( i ) For l <i <m, S has an ( o . Pi ) -dependency wi t h respect to L

(i i ) f or each i ,J , Pj

(i l i ) f or each l , j such that ifj,

and Pj / Pj .

Thus, i n a dependency set f or S i n L every twodependenci es are di st i nct i s both terms and each"determi ni ng" el ement i n S precedes al l "determi ned" el ements, where we pi ct ure a asdetermi ni ng the choi ce of a_ . ai

PiEvi dent l y, i f S has an m- t ermed dependency seti n L, at l east 2® st ates are necessary i n tbs

f i ni te-st ate gr ammar that generates thel anguage 1 .

Thi s observat i on enabl es us to state anecessary condi t i on f or f i ni te- state l anguages.

(11) Suppose that L i s a f i ni te- statel anguage. Then there I s an n such that nosentence S of L has a dependency set of morethan m terms i n L.

Wi t h thi s condi t i on i n mi nd, we can easi l yconstr uct many nonf i ni te-state l anguages. For

exampl e, the l anguages L. , I*,, L_ descri bed i n( 12) are not descri babl e' Sy any Ti ni te- stategrammar . ^ r\

( 12) (i ) L. contai ns a b, a a b b,r\ r\ r\. , ,

a a a o b d ,..., and in

general , al l sentences consi st i ngof n occurrences of a f ol l owed by

exact l y n occurrences of b, and onl ythese;

( i i ) I>2 contai ns a a, b b, a^b^b^a,.A A A, A A. A. A A

b a a b, a a b b a a,...,

and i n general , al l "mi rror- i mage"sentences consi st i ng of a str i ng Xf ol l owed by X i n reverse, and onl ythese; o r> r\ r\

(i l l ) L, contai ns a a, b b, a b a b,r\ r\.r\ r\ r\.

b a b a, a a b a a b • j

and i n general , al l sentences consi st i ng of a str i ng X fol l owed bythe i dent i cal st r i ng X, and onl ythese.

I n I v, , f or exampl e, f or any mwe can f i nd a

sentence wi t h a dependency set kDm={ ( 1 ,2m) , (2 ,2m- l ) , . . , (m, m+l ) j-

2. 3* Tur ni ng now to Engl i sh, we f i nd that thereare i nf i ni te sets of sentences that have dependencysets wi t h more than any f i xed number of terms.For example, l et be decl arat i ve sentences.

Then the f ol l owi ng are al l Engl i sh sentences:

( 13) U) I f Sl t then S2.(i i ) Ei ther S3 , or S4 .

( i l l ) The man who sai d that S5 , i sar r i vi ng today.

These sentences have dependenci es between ■i f -"t hen", Bei t herM- "or*, "man"- "i s". But we canchoose Sj_, S3 , Sc; whi ch appear between the i nterdependent words, as (131). (13i i )» or ( 13i i i ) t hemsel ves. Proceedi ng to const ruct sentences i n thi s

way we arr i ve at subparts of Engl i sh wi t h J ust themi r ror i mage propert i es of the l anguages Li and I 2of (12). Consequent l y, Engl i sh f ai l s condi t i on '(11). Engl i sh I s not a f i ni te-st ate l anguage, and1we are forced to rej ect the theory of l anguageunder di scussi on as f ai l i ng condi t i on (3).

We mi ght avoi d thi s consequence by anarbi t rary decree that there i s a f i ni te upperl i mi t t o sentence l engt h I n Engl i sh. Thi s woul dserve no usef ul purpose, however . The poi nt i sthat there are processes of sentence f ormati onthat thi s el ementary model f or l anguage i si nt ri nsi cal l y i ncapabl e of handl i ng. I f nof i ni te l i mi t 13 set for the operat i on of theseprocesses, we can prove the l i t eral i nappl i cabi l i t y of thi s model . I f t he processes have al i mi t, then the constr ucti on of a fi ni te- state

grammar wi l l not be l i t eral l y i mpossi bl e ( si ncea l i st i s a tri vi al f i ni te- state grammar ) , butthi s grammar wi l l be so compl ex as to be of l i tt l euse or i nterest. Bel ow, we shal l st udy a modelf or grammars t hat can handl e mi rror- i mage l anguages, The extra power of such a model i n thei nf i ni te case i s ref l ected I n the f act that i t i smuch more usef ul and reveal i ng i f an upper l i mi ti s set. I n general , the assumpt i on that l anguageare i nf i ni te i s made for the purpose of si mpl i f yi ng the descri pt i on. ^ I f a grammar has norecursi ve steps ( cl osed l oops, i n the model

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di scussed above) i t wi l l be prohi bi t i vel y compl ex-i t wi l l , i n f act, tarn out to be l i tt l e betterthan a l i st of st r i ngs or of morpheme cl asssequences i n the case of natural l anguages. I f i tdoes have recursi ve devi ces, i t wi l l producei nf i ni t el y many sentences.

2. 4 Al t hough we have f ound that no f i ni t e- state

Markov process t hat produces sentences f roml ef tto ri ght can serve as an Engl i sh grammar, wemi ght i nqui re i nto the possi bi l i t y of constr ucti nga sequence of such devi ces t hat. I n some nont r i vi alway, come cl oser and cl oser to cat chi ng the out putof a sat i sf actory Engl i sh grammar. Suppose, f orexampl e, that f or f i xed n we const ruct a f i ni te-st ate grammar i n the f ol l owi ng manner : one st ateof the grammar i s associ ated wi t h each sequence of Engl i sh words of l engt h n and the probabi l i t y thatthe word X wi l l be produced when the systemi s i nthe state S. i s equal to the condi t i onal probabi l i t y of X, gi ven the sequence of n words whi chdef i nes S. . The output of such grammar i scust omar i l y cal l ed an n+18* order approxi mati on toEngl i sh. Evi dent l y, as n i ncreases, the output of

such grammars wi l l come to l ook more and more l i keEngl i sh, si nce l onger and l onger sequences have ahi gh probabi l i t y of bei ng taken di rectl y f rom thesampl e of Engl i sh i n whi ch the probabi l i t i es weredetermi ned. Thi s f act has occasi onal l y l ed tothe suggest i on that a theory of l i ngui st i cst ructure mi ght be f ashi oned on such a model .

Whatever the other i nterest of stat i st i calapproxi mati on i n thi s sense may be, i t i s cl earthat i t can shed no l i ght on the probl ems of grammar. There i s no general rel ati on between thef requency of a str i ng (or i ts component parts) andi ts grammati cal ness. We can see thi s most cl ear l yby consi deri ng such str i ngs as

(14) col or l ess green i deas sl eep f ur i ousl y

whi ch i s a grammati cal sentence, even though i t i sf ai r to assume t hat no pai r of i ts words may everhave occur red together i n the past. Not i ce that aspeaker of Engl i sh wi l l read (14) wi t h theordi nary i ntonati on pat t ern of an Engl i sh sentence,whi l e he wi l l read the equal l y unf ami l i ar st r i ng

( 15) f ur i ousl y sl eep i deas green col or l ess

wi t h a fal l i ng i ntonat i on on each word, as i nthe case of any ungrammati cal st r i ng. Thus (14)di f f ers fr om (15) exact l y as (1) di f f ers from(2);our tentati ve operat i onal cri ter i on for grammat i cal ness support s our i ntui t i ve f eel i ng that(14) i s a grammati cal sentence and that (15) i s

not. We mi ght state the probl emof grammar, i npart , as that of expl ai ni ng and reconst ruct i ngthe abi l i t y of an Engl i sh speaker to recogni ze( 1), (14), etc. , as grammat i cal , whi l e rej ecti ng(2) , (15), etc. But no order of approxi mat i onmodel can di st i ngui sh (14) f rom (15) (or ani ndef i ni te number of si mi l ar pai r s) . As ni ncreases, an n*h order approxi mati on to Engl i shwi l l excl ude (as more and more i mprobabl e. ' ) anever- i ncreasi ng number of grammati cal sentenses ,whi l e i t st i l l contai ns vast numbers of compl etel yungrammati cal st r i ngs. ® We are f orced to concl ude

that there i s apparent l y no si gni f i cant appto the probl ems of grammar i n thi s di rect i on

Noti ce t hat al t hough f or every n, a pof n- order approxi mati on can be r epresent edf i ni te- state Markov process, the converse i

true. For exampl e, consi der the three- stateprocess wi t h (S Sj ) , ( S Sj ) , ( Sl tSo) ,

(Sq,S2) , ( S2 ,S2) , ( S2 , So) as i ts onl y connect

states, and wi t h a, b, a, c, b, c as t he rei ve t ransi t i on symbol s. Thi s process can berepresent ed by the f ol l owi ng st ate di agram:

- ‘ < x >

Thi s process can produce the sentences a a,a b a , a^b^b a, a ^b^b^b ^a . . . . c~c,

c^b c, c ~b~b~c , c ^b^b^b^c , . .. , but a' ' 'b'"' 'b'r' c, c b^b^a, etc. The generatedl anguage has sentences wi t h dependenci es of f i ni te l ength.

I n § 2. 4 we argued that there i s nosi gni f i cant cor rel at i on bet ween order of apmati on and grammati cal ness. I f we order thst r i ngs of a gi ven l ength i n terms of orderapproxi mati on to Engl i sh, we shal l f i nd botgrammati cal and ungrammati cal st r i ngs scattthroughout the l i st, f romtop to bot tom. Hthe noti on of stati st i cal approxi mati on appto be i r rel evant to grammar. I n § 2.3 we pout that a much broader cl ass of processes,namel y, al l f i ni te- state Markov processes t

produce transi t i on symbol s, does not i ncl udEngl i sh grammar. That I s, I f we const ruct f i ni te-st ate grammar t hat produces onl y Engsentences, we know that i t wi l l f al l to proan i nf i ni te number of these sentences; i n pti cul ar, i t wi l l f al l to produce an i nf i ni tnumber of true sentences, f al se sentences,reasonabl e quest i ons that coul d be i ntel l i gasked, and the l i ke. Bel ow, we shal l i nvesa sti l l br oader cl ass of processes that mi gprovi de us wi t h an Engl i sh grammar.

3. Phrase St ructure.

3. 1. Customar i l y, syntacti c descri pt i on i sgi ven i n terms of what i s cal l ed * i mmedi ateconsti tuent anal ysi s. 1* I n descri pt i on of tsort the words of a sentence are gr ouped i nphr ases, these are grouped i nto smal l er contuent phrases and so on, unt i l the ul t i matconst i t uent s (general l y mor phemes ) are rea These phrases are then cl assi f i ed as nounphrases (NP) , verb phrases ( VP) , etc. Torexampl e, the sentence (17) mi ght be anal yzei n the accompanyi ng di agram.

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1?) the man took the book

HP Verb HP

VP

Sentence

i dent l y, descr i pti on of sentences i n such t en*rmi ts consi derabl e si mpl i f i cat i on over t herd- by- word model , si nce the composi t i on of ampl ex cl ass of expressi ons such as HP pan beated j ust once i n the grammar, and thi s cl assn be used as a bui l di ng bl ock at var i ousi nt s i n the constructi on of sentences. We nowk what f ormof grammar cor responds t o thi sncept i on of l i ngui st i c structure.

2. A phrase-st ructure grammar i s def i ned by ani te vocabul ary (al phabet) V_, a fi ni te set 2i ni ti al str i ngs i n V , and S f i ni te set Fof es of the f orm: X- *f Y, wher e X and T are

ri ngs i n V_. Bach such rul e i s I nterpreted ase i nstr ucti on: rewr i te X as I . Tor r easonsat wi l l appear di rectl y, we requi re t hat i nch such [ 2 , T] grammar

(18) I : X r . . , X n

J :

X — Y_.

i s f ormed f r omX by the repl acement of a

ngl e symbol of Xi by soma st r i ng. Hei ther

e repl aced symbol nor the repl aci ng str i ng

y be the i dent i t y el ement TJ of f ootnote ^

Gi ven the [ 2 , F] grammar (18), we say that:

(19) (i ) a str i ng P f ol l ows f roma st ri ng ai f a=Z ' ' X^W and 0=Z 1 ^ W, f or

some 1 < m; ?(11) a der i vat i on of the st r i ng S. i s a

sequence D=(S , . . , St) of str i ngs,where Sj * S and for each l <t ,

f ol l ows f romS ;

( Hi ) a str i ng S i s der i vabl e f rom(18)i f there i s a deri vat i on of s i nterms of (18);

(l v) a der i vat i on of i s termi nated i f

there i s no str i ng that f ol l ows f rom

st ;(v) a str i ng i s a termi nal st r i ng i f

i t i s the l ast l i ne of a termi natedder i vat i on.

A deri vat i on i s t hus r oughl y anal ogous to aoof , wi t h Z taken as the axi om systemand I*

the rul es of I nf erence. Ve say that L I s ar i vabl e l anguage i f L i s the set of st r i ngs

that are deri vabl e f romsome £ 2 , T] grammar,and we say that L i s a termi nal l anguage i f i t i sthe set of termi nal st r i ngs fromsome systemC I , F]. *

I n every i nterest i ng case there wi l l be atermi nal vocabul ary V, ( V C Vp) that

exactl y characteri ses the termi nal str i ngs, i nthe sense that every termi nal st r i ng i s a st r i ngi n Yj and no symbol of V_ i s r ewr i t ten i n any of the rul es of F. I n such a caBe we can i nterpretthe termi nal st r i ngs as const i tut i ng the l anguageunder anal ysi s (wi th V_ as i ts vocabul ary) , andthe der i vat i ons of these st r i ngs as provi di ngthei r phrase structure.

3-3. As a si mpl e exampl e of a systemof the f orm(18), consi der the f ol l owi ng smal l part of Engl i shgr ammar :

(20) £ : # Sent enc e°F: Sentence HP"VP

VP — Ver b~HPHP the' ' man, the' ’' book

Verb — t ookAmong the der i vat i ons f rom(20) we have, i npart i cul ar:

(21) D. : # '"‘Sentence' ' ##~HP' ' VP~##' ' HP' ' Ver b' ' HP~#

man' ' Verb' ' HP' ' ## ' ' the' ' man' ' Verb' ' t he' ' book' ' ##' ' t he' >man' ' took' ' the' ' book' ' #

D, : Sent ence ' ' #2 #~HP/ ' VP ' ' #

# ' ' the' ' man' ' VP' "' ##' Nt he' ' man' ' Ver b' ' HP~##' vthe' ' man' ' t ook' ' HP' ' #

#' ' t he' ' man"' took' ' the' ' book' ' #

These deri vat i ons are evi dentl y equi val ent ; theydi f f er onl y i n the order i n whi ch the rul es areappl i ed. We can represent thi s equi val encegr aphi cal l y by const ructi ng di agrams t hatcorrespond, i n an obvi ous way, to deri vat i ons.Bot h and Dg reduce to the di agram:

(22)

HP VP

/ \the man Verb HP

I / \t ook the book

The di agram( 22) gi ves the phrase st ructure of the termi nal sentence "t he man took the book, "

j ust as i n (1?). I n general , gi ven a deri vat i onD of a str i ng S, we say that a subst r i ng s of Si s an X i f i n the di agramcorrespondi ng to B, sI s t raceabl e back to a si ngl e node, and thi s nodei s l abel l ed X. Thus gi ven B or D, , correspond

i ng to (22), we say that l,the' ' man" i s an HP,l,t ook' ' the' ' book,1 i s a VP, " t he book11 i s anHP, 11the' ' man' ' took' ' the' ' book" i s a Sentence,"man' ' took, " however, i s not a phrase of thi s

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str i ng at al l , si nce i t i s not t raceabl e backto any node*

When we attempt to construct the si mpl estpossi bl e [ 2 , F] grammar f or Engl i sh we f i nd thatcert ai n sentences aut omati cal l y recei ve nonequi val ent der i vat i ons. Al ong wi t h (20), thegrammar of Engl i sh wi l l certai nl y have to contai n

such rules as

(23) Verb — are ~ f l yi ngVerb - « areHP — theyHP — pl anesHP — f l yi ng' "' pl anes

i n order to account f or such sentences as "theyare f l yi ng - a pl ane" ( HP-Verb- HP) , "( f l yi ng)pl anes - are - noi sy" (HP-Verb- Adj ect i ve) , etc.But thi s set of rul es provi des us wi t h two nonequi val ent deri vat i ons of the sentence "t hey aref l yi ng pl anes", reduci ng to the di agrams:

(24) Sentence # '‘' Sei tence qj fc

HP VP HP VP

I / \ I / 'they Verb HP they Verb

I / \ / \ Iare f l yi ng pl anes are f l yi ng pl anes

HP

Hence t hi s sentence wi l l have two phrasest ructures assi gned to i t; i t can be anal yzed as"t hey - are - f l yi ng pl anes" or "t hey - are f l yi ng- pl anes. " And i n f act, thi s sentence i s

ambi guous i n j ust thi s way; we can underst and i tas meani ng that "those specks on the hor i zon -are - f l yi ng pl anes* or "t hose pi l ots- are f l yi ng- pl anes. " When the si mpl est grammar aut omati c

al l y provi des nonequi val ent deri vat i ons f or somesentence, we say that we have a case of

const ruct i onal homonyml t y. and we can suggestthi s f ormal propert y as an expl anati on f or thesemant i c ambi gui ty of the sentence i n questi on.I n $1 we posed the requi rement that grammarsof f er I nsi ght i nto the use and underst andi ng of l anguage (cf . (5)) . One way to test t he adequacyof a grammar i s by det ermi ni ng whet her or notthe cases of const ructi onal homonyml ty areactual l y cases of semanti c ambi gui t y, as I n (24)We return to thi s i mportant probl em i n { 6.

I n (20) - (24) the el ement # I ndi catedsentence ( l ater, word) boundary. I t can betaken as an el ement of the termi nal vocabul aryVj , di scussed i n the f i nal paragraph of § 3-2.

3.4. These segments of Engl i sh grammar are muchoversi mpl i f i ed i n several respects. Tor onethi ng, each rul e of (20) and (23) has onl y asi ngl e symbol on the l eft , al t hough we pl aced nosuch l i mi t at i on on [ 2 , F] grammars i n { 3 -2.A rul e of the f or*

(25) Z^X^W — Z^T^W

i ndi cates that X can be rewr i t ten as T onl y i nthe cont ext Z—W. I t can easi l y be shown thatthe grammar wi l l be much si mpl i f i ed i f we permi t

such rules. I n J 3. 2 we requi red that i n sa rul e as (25), X must be a si ngl e symbol . ensures t hat a phrasest ructure di agramwi lconst ruct i bl e f romany deri vat i on. The gracan al so be si mpl i f i ed very great l y i f we othe rul es and requi re that they be appl i ed sequence ( begi nni ng agai n wi t h the f i rst rul e

appl yi ng the f i nal rul e of the sequence) , awe di st i ngui sh bet ween obl i gatory rul es whimust be appl i ed when we reach them i n the seand opti onal rul es whi ch may or may not beappl i ed. These revi si ons do not modi f y thegenerat i ve power of the grammar, al though tl ead to consi derabl e si mpl i f i cat i on.

I t seems reasonabl e to requi re f orsi gni f i cance some guarantee t hat the grammaactual l y generat e a l arge number of sentenca l i mi ted amount of t i me; more speci f i cal l yi t be i mpossi bl e to run through the sequencrul es vacuousl y ( appl yi ng no rul e) unl ess tl ast l i ne of the der i vat i on under const ructa termi nal st r i ng. We can meet thi s requi rby posi ng cert ai n condi t i ons on the occurre

obl i gatory rul es i n the sequence of rul es. def i ne a proper grammar as a syst em[ 2 ,Q.]where 2 I s a set of i ni t i al st r i ngs and Q,sequence of rul es X. - » T. as i n (18), wi t h addi t i onal condi t i on that f or each 1 there be at l east one J such that X =X. and X Tan obl i gat ory rule. Thus, each l eft - hand tthe rul es of (18) must appear i n at l east oobl i gatory rul e. Thi s i s the weakest si mplcondi t i on that guarantees that a nontermi nader i vati on must advance at l east one step et i me we run t hrough the rul es. I t provi desi f X. can be rewr i t t en as one of t. , . . , T.

n \

then at l east one of these rewr i t i ngs must pl ace. However, proper grammars are essent

di f f erent f rom[ 2 ,F] grammars. Let D(G) the set of der i vat i ons produci bl e f roma phst ructure grammar G, whet her proper or not. DJ , «-[D(0) | 8 » [ J J ] grammar j - and 0^0(

a proper grammar^ . Then

(26). Dj , and Dp are i ncomparabl e; i . e. ,

That i s, there are systems of phrase st ructuthat can be descr i bed by [ 2 , F] grammars bby proper grammars, and others that can bedescr i bed by proper grammars but not by [ Z grammars.

3. 5* We have def i ned three types of l anguag

f i ni t e- state l anguages ( i n §2. 1), der i vabl e termi nal l anguages (i n $3. 2) . These are rel ai n the f ol l owi ng way:

(27)( i ) every f i ni t e- state l anguage i s atermi nal l anguage, but not conversel y;

( i i ) every der i vabl e l anguage i s a termil anguage, but not conversel y;

(i i i ) t here are deri vabl e, nonf i ni te- statl anguages and f i ni te- state, nonderi vabl el anguages.

Suppose that Lg i s a f i ni te—st ate l angu

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t h the f i ni te- st ate grammar G as i n § 2. 1.We const ruct a [ 2 ,T] grammar i n thef ol l owi nganner: 2 ={s V S 7 contai ns arul e of the fora

281) f or each0 1, j , k such that ( S S t C, j O,

nd k < 7 contai ns a rul e of the f orm(28i i )

or each i , k such that ( Sj , S0)eC and k <

(28) (1) S j- a j j /' S j

(li) S j - a ^

earl y, the termi nal l anguage f rom thi s [ 2 , ?]rammar wi l l be exact l y Ii , establ i shi ng therst part of (271).

I n § 2. 2 we f ound that L , and 1 of

12) were not f i ni te-st ate l anguages. and I ,

owever, are termi nal l anguages. For L , e. g. ,

e have t he [ 2 , ?] grammar

(29) 2 : Z „y : Z — a b

Z — a^Z^b

hi s est abl i shes (2?i ).

Suppose that I s a deri vabl e l anguage wi t h

he vocabul ary V = {a. , . . , a } . Suppose that wedd to the grammar of a f i ni te set of rul es

. - b. , where the b^s are not i n Vp and are

al l di st i nct. Then thi s new grammar gi ves aermi nal l anguage whi ch i s si mpl y a notati onal

var i ant of V. Thus every deri vabl e l anguages al so termi nal .

As an exampl e of a termi nal , nonderi vabl eanguage cohsi der the l anguage cont ai ni ng j usthe str i ngs

( 30 ) a ~ b , c ^ a ^ b ^ d , c ~ c ~ a ' 'b ' " ' d ' 'd ,

c c c ^a ^b^ d d' ' d, .. .

An i nf i ni te der i vabl e l anguage must cont ai n annf i ni te set of str i ngs t hat can be arranged i n a

sequence S Sg, . . . i n such a way that f or seme

rul e X — T, St fol l ows f romSi _1 by appl i cat i on

of thi s rul e, f or each 1 >1. And T i n thi s rul emust be formed f romX by repl acement of a si ngl esymbol of X by a st r i ng (cf. (18)) . Thi s i sevi dentl y i mpossi bl e I n the case of L . Thi sanguage i s, however, the termi nal l anguage gi ven

by the fol l owi ng grammar:

(31) 2 : Z ^F : Z- »a bZ — c ^Z^d

An exampl e of a f i ni te- state, nonder i vabl eanguage i s the l anguage cont ai ni ng al l and

onl y the str i ngs consi st i ng of 2n or 3noccurrences of a, f or n=l , 2 , . . . . l anguage

of (12) i s a deri vabl e, nonf i ni te- state l anguage,wi t h the i ni t i al str i ng a~b and the rul e:a~b — a" a~b~b.

The maj or i mport of Theorem(27) i s thatdescri pt i on i n terms of phrase structure i sessent i al l y more powerf ul (not j ust si mpl er) thandescri pt i on i n terms of the f i ni te- st ate grammarsthat produce sentences f rom l ef t to ri ght. I n§ 2. 3 we f ound that Engl i sh i s l i t eral l y beyondthe bounds of these grammars because of mi r ror-i mage propert i es that i t shares wi t h and Lg

of ( 12). We have j ust seen, however, that

i s a termi nal l anguage and the same i s t rue of Lg. Hence, the consi derati ons that l ed us torej ect the f i ni te- etate model do not si mi l ar l yl ead us to rej ect the more powerf ul phrase-st ructure model .

Note that the l att er i s more abst ract thanthe f i ni te-st ate model i n the sense that symbol sthat are not i ncl uded i n the vocabul ary of al anguage enter i nto the descri pti on of t hi sl anguage. I n the terms of § 3. 2, Vp properl y

i ncl udes 7,. Thus i n the case of ( 29), we

descr i be L. i n terms of an el ement Z whi ch i s noti n L, ; and i n the case of (20)- (24), we i ntroducesuch symbol s as Sentence, HP, VP, etc. , whi ch arenot words of Engl i sh, i nto the descri pt i on of Engl i sh structure.

3. 6. We can i nterpret a [ 2 ,F] grammar of thef orm (18) as a rather el ementary fi ni te- stateprocess in the f ol l owi ng way. Consi der a systemthat has a f i ni te number of states S . - . S .

When i n state So, i t can produce any cf the

st ri ngs of 2 , thereby movi ng i nto a new state.I ts st ate at any poi nt i s determi ned by the subset of el ements of X. , . . ,X contai ned as sub-

1 ID

st ri ngs i n the l ast produced str i ng, and i t movesto a new state by appl yi ng one of the rul es t othi s st r i ng, thus produci ng a. new str i ng. The

systemr eturns to state S wi t h the product i onof a termi nal st r i ng. Thi s systemthus producesderi vat i ons, i n the sense of § 3»2. The processi s determi ned at any poi nt by i ts present stateand by the l ast st r i ng that has been produced,and there i s a f i ni te upper bound on the amountof i nspecti on of t hi s str i ng that i s necessarybef ore the process can cont i nue, produci ng a newst r i ng that di f f ers i n one of a f i ni te number of ways fromi ts l ast output.

I t i s not di f f i cul t to construct l anguagesthat are beyond the range of descr i pti on of [ 2 ,F] grammars. I n f act , the l anguage of

( 12i i i ) i s evi dent l y not a termi nal l anguage. I

do not know whether Engl i sh i s actual l y a termi nall anguage or whet her there are other actuall anguages that are l i t eral l y beyond the bounds of phrase st ructure descr i pti on. Hence I see no wayto di squal i f y thi s theory of l i ngui sti c str uctureon the basi s of consi derati on (3). When we turnto the questi on of the compl exi ty of descri pt i on(cf . (*»■)), however, we f i nd that there are ampl egrounds f or the concl usi on that thi s theory of l i ngui st i c st ructure i s f undamental l y i nadequate.We shal l now i nvest i gate a f ew of the probl ems

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that ari se when we attempt to extend (20) to af ul l - scal e grammar of Engl i sh.

4. I nadequaci es of Phrasa-Structure Grammar

4.1. I n (20) we consi dered onl y one way of devel opi ng the el ement Verb, namel y, ae "took*.

But even wi t h the verb st emf i xed there area great many other f orms that coul d appear i nthe context "the man — the book, * e. g. , "takes,*"has t aken, " "has been taki ng, " "i s t aking, "■has been taken, " "wi l l be taki ng, " and so on.A di rect descri pt i on of t hi s set of el ementswoul d be fai r l y compl ex, because of the heavydependenci es among them( e. g. , "has taken" butnot "has t aking, " I s bei ng taken" but not "i sbei ng taki ng, " etc. ) . We can, i n fact, gi ve avery si mpl e anal ysi s of "Verb" as a sequence of i ndependent elements, but onl y by sel ect i ng asel ements certai n di scont i nuous stri ngs. Forexampl e, i n the phrase "has been taki ng" we canseparate out t he di sconti nuous el ements "hasten, ""be. . i ng, " and "take", and we can then say thatthese elements combine f reel y. Fol l owi ng thi s

course systemat i cal l y, we repl ace the l ast rul ei n (20) by

(32) (1) Verb - * Auxi l i ary' ' V(i i ) V- t ake, eat, . . .

(i l l ) Auxi l i ary — C(M)(have ' ' en) (be' ' i ng)( be en)

(i v) M- *wi l l ' , can, shal l , may, must(v) C -*■past , present

The notat i ons i n (32111) are to be i nterpreted as fol l ows: i n devel opi ng "Auxi l i ary"i n a deri vati on we must choose t he unparenthesi zed el ement C, and we may choose cero or moreof the parenthesi zed el ements, i n the ordergi ven. Thus, i n cont i nui ng the deri vat i onD- of (21) bel ow l i ne f i ve, we mi ght proceed

as f ol l ows:

(33 ) #' ' the/"man' -' Verb' " t he' ' book #[ f rom of (21)]

#' "' the' ' man' ' Auxi l i ary' ' ' V~ the' ' book #[ (321)3

# the'"' man' "' Auxi l i ary take the bo ok' ' #

[ (3211)]

# ^ the' ' man' ' C' ' have' ' en' ' be' ' l ng' ' take' 'the' ' book' ' #

[ (32111), choosi ng the el ements C,have en, and be' ' l ng]

# ^ the' ' man' " past' ' have' ' en ' be' ' i ng' ' take' 'the' ' book' ' #

t(32v)]

Suppose that we defi ne the cl ass Af as contai ni ngthe aff i xes "en" , "i ng", and the C' s; and thecl ass v as i ncl udi ng al l V' s, M' s, "have", and "be. "We can then convert the l ast l i ne of (33) i nto aproper l y or dered sequence of morphemes by thef ol l owi ng rul e:

Appl yi ng thi s rul e to each of the three Af ' 'sequences i n the l ast l i ne of (33 ) • we deri v

(35) # ^ the' ' man' ' have past' ' #' ' be' ' *en' ' take ~ i ng'"' # ' ' the' ' book' ' #.

I n the fi rst paragraph of § 2.2 we ment i

that a grammar wi l l contai n a set of rul es ( cmorphophonemi c r ul es) whi ch convert st r i ngs omorphemes i nto st ri ngs of phonemes. I n themorphophonemi cs of Engl i sh, we shal l have sucrul es as t he fol l owi ng (we use convent i onal ,rather than phonemi c ort hography):

(36) have' ' past -*■ hadbe^en —*■ beentake ~ ing —► taki ngwi l l ' ' past - ® woul dcan ' ' past -*■ coul dM' -' present — M wal k' ' past —► wal kedtake' ' past - » tooketc.

(34) Af ' ' v • #

Appl yi ng the morphophonemi c rul es to (35der i ve the sentence:

(37) the man had been taki ng the book.

Si mi l ar l y, wi t h one maj or except i on to bdi scussed bel ow (and s' everal mi nor ones that shal l over l ook here), the rul es (32), (34) wigi ve al l the other f orms of the verb i ndecl arat i ve sentences, and onl y t hese forms.

Thi s very si mpl e anal ysi s, however, goesbeyond the bounds of [ 2 , F] grammars i n severespects. The rul e (34), al t hough i t I s qui tsi mpl e, cannot be I ncorporated wi t hi n a [ 2 ,grammar, whi ch has no pl ace f or di scont i nuousel ements. Furt hermore, to appl y the rul e (34

to the l ast l i ne of (33) we must know that "ti s a V, hence, a v. I n other words, i n orderaPPl y thi s rul e i t i s necessary to Inspect mothan J ust the st r i ng to whi ch the rul e appl i ei t i s necessary to know some of the const i tuestr ucture of thi s st ri ng, or equi val ent l y(cf. § 3. 3). to i nspect cert ai n earl i er l i nesi ts deri vat i on. Si nce (34) requi res knowl edgthe ’hi story of der i vat i on1 of a stri ng, i t vithe el ementary property of [ 2 , F] grammarsdi scussed I n § J.6.

4. 2. The f act t hat t hi s si mpl e anal ysi s of thverb phrase as a sequence of i ndependent l y chuni t s goes beyond the bounds of [ 2 ,F] grammsuggests that such grammars are too l i mi ted tgi ve a true pi cture of l i ngui sti c structure.Further st udy of the verb phrase l ends addi t isupport to thi s concl usi on. There i s one majl i mi tati on on the I ndependence of the el ementi nt roduced i n (32). I f we choose an i ntransiverb (e. g. , "come, " "occur, " etc. ) as V i n (3we cannot sel ect be^en as an auxi l i ary. We not have such phrases as "J ohn has been come,""J ohn i s occurred, " and the l i ke. Furthermorethe el ement be en cannot be chosen i ndependeof t he context of t he phrase "Verb. " I f we h

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he el ement "Verb" i n the context "the man — theood, " we are const rai ned not to sel ect be° en i nppl yi ng (32), al t hough we are f ree to choose anyher el ement of (32). That i s, we can have "the

an i s eati ng the food," "the man woul d have beenat i ng the f ood, " etc. , but not "t he man i s eatenhe f ood," “the man woul d have been eaten the food, "c. On the other hand, i f the context of the

hrase "Verb" i s, e. g. , "the f ood — by t he man, " are requi red to sel ect be en. We can havehe f ood i s eaten by the man, " but not "t he f oodeat i ng by the man, " etc. I n short , we fi nd that

he el ement be en enters i nto a detai l ed networkrestr i cti ons whi ch di sti ngui sh i t f romal l the

her el ements i ntr oduced i n the anal ysi s of "Verb"n (32). Thi s compl ex and uni que behavi or of e en suggests that i t woul d be- desi rabl e tocl ude i t f rom( 32) and to i nt roduce passi ves i ntoe grammar i n some ot her way.

There i s, i n f act, a very si mpl e way tocorporate sentences wi t h be en (i . e. , passi ves)to the grammar. Koti ce that f or every act i ventence such as " the man ate the f ood" we have arrespondi ng passi ve "t he f ood was eaten by then" and conversel y. Suppose then that we drop theement be en f rom(32i i i ) , and then add to theammar the f ol l owi ng rul e:

5. Transf ormati onal grammar.

5..1. Each grammati cal t ransf ormati on T wi l lessent i al l y be a rule that converts every sentencewi t h a gi ven const i tuent st ructure i nto a newsentence wi t h deri ved const i tuent structure. Thet ransf ormand i ts der i ved str ucture must be relatedi n a f i xed and const ant way to the st ructure of the t ransf ormed stri ng, f or each T. We can

characteri ze T by stat i ng, I n structural terms,the domai n of st r i ngs t o whi ch It appl i es and thechange that i t eff ect s on any such stri ng.

Let us suppose i n the f ol l owi ng di scussi onthat we have a [ Z , F] grammar wi t h a vocabul aryVp and a t ermi nal vocabul ary 7, C Vp , as i n ^ 3. 2.

I n § 3*3 ve showed that a [ 2 , F] grammarpermi t s t he der i vat i on of termi nal str i ngs, and wepoi nt ed out that i n general a gi ven termi nal str l i £wi l l have several equi val ent deri vat i ons. Twoderi vat i ons were sai d to be equi val ent i f they qreduce to the same di agramof the f orm(22), etc.Suppose that D , . . , Dn const i tute a maxi mal set

of equi val ent deri vat i ons of a termi nal st r i ngS. Then we defi ne a phrase marker of S as the set

of str i ngs t hat occur as l i nes i n the deri vat i onsDl ' ‘ ‘ ,Dn‘ A st r i ng wi l l have more than one phrase

marker i f and onl y i f i t has nonequi val entderi vat i ons (cf . (24)) .

(38) I f S i s a sentence of the f ormNP]_-uxi l i ary-V- HPg* then the correspondi ng st r i ng

the f ormHP2- Auxi l i ary' ' be<' en- V-by' HI ,2 I sso a sentence.

Tor exampl e, I f "t he man - past - eat theood" (HPj -Auxi l i ary-V- HFg) i s a sentence, then

he f ood - past be en - eat - by the man" (HP-r-uxi l i ar y be en- V- byPj _ ) i s al so a sentence.l es (34) and (36) woul d convert the f i rst of

hese i nto "the man ate the food" and the

econd i nto "t he f ood was eaten by the man. " The advantages of thi s anal ysi s Of passi ves

e unmi st akabl e. Si nce the el ement be en hasen dropped f rom(32) i t i s no l onger necessary toal i f y (32) wi t h the compl ex of rest r i cti onsscussed above. The fact that be ' "'en can occurl y wi t h t ransi t i ve verbs, that i t i s excl uded i ne context "the man — t he f ood" and that i t i squi red i n the context "t he f ood — by the man, "now, i n each case, an automati c consequence of

e anal ysi s we have j ust gi ven.

A rule of the f orm (38), however, i s wel lyond the l i mi ts of phrase- 3t rueture grammars.ke (3 ), i t rearranges the el ements of the st r i ngwhi ch i t appl i es, and i t requi res consi derabl e

f ormat i on about the const i tuent st ructure of thi s

ri ng. When we carr y the det ai l ed study of Engl i shntax further, we f i nd that there are many otherses i n whi ch the grammar can be si mpl i f i ed i f theZ , F] systemi s suppl emented by rul es of the sameneral f ormas (38). Let us cal l each such rul e aammati cal t ransf ormati on. As our t hi rd model f ore descri pt i on of l i ngui st i c str ucture, we nownsi der bri ef l y the formal propert i es of a t ransrmati onal grammar that can be adj oi ned to theZ ,T] grammar of phrase st ructur e. 8

Suppose that K i s a phrase marker of S.say that

We

(39) (S,K) i s anal yzabl e i nto (X- , . . , X ) i f and onl y I f there are str i ngs s, , . . , s such ?hat

(i ) S=81' ' . . . ' ' 8n 1 11

(i i ) f or each i <n, K contai ns the str i ng

"i -1 Ai 1+1 ** n

(40) I n thi s case, s. I s an X, i n S wi threspect to X. ®

The rel at i on def i ned I n (40) i s exact l y therel at i on "i s a" def i ned i n § 3 -3 ; I . e. , s. i s anX. i n the sense of (40) i f and' onl y i f s. I s asubst r i ng of S whi ch i s t raceabl e back to a si ngl enode of the di agramof the f orm (22), etc. , andthi s node i s l abel l ed X .

The noti on of an&l ystabi l i ty def i ned aboveal l ows us t o speci f y preci sel y the domai n of appl i cati on of any tr ansformat i on. We associ atewi t h each tr ansf ormat i on a restr i cti ng cl ass Sdef i ned as f ol l ows:

(41) B I s a rest ri cti ng cl ass I f and onl y i f f or some r, m, E i s t he set of sequences:

X1 X1

x

where i s a st r i ng i n the vocabul ary Vp, f or each

1, j . We then say that a str i ng S wi th the phrasemarker X bel ongs to the domai n of the t ransf ormati on

121



T i f the restr i ct i ng cl ass R associ ated wi t h T

contai ns a sequence (X , . . , X ) i nto whi ch (S, K) i s

anal yzabl e. The domai n of a t ransf ormati on i stfi us a set of ordered pai rs (S, K) of a str i ng Sand a phrase marker I of S. A transf ormati on maybe appl i cabl e t o S wi t h one phrase marker, but notwi t h a second phrase marker, i n the case of a st r i ngS wi t h ambi guous const i tuent st ructure.

I n part i cul ar , the passi ve t ransf ormat i ondescr i bed i n (38) has associ ated wi t h i t arest r i ct i ng cl ass E contai ni ng j ust one sequence:

P

<*a) Bp= | (HP, Auxi l i ary, T, HP)} .

Thi s t ransf ormati on can thus be appl i ed to anyst r i ng that i s anal yzabl e i nto an BP f ol l owed byan Auxi l i ary f ol l owed by a V fol l owed by an HP.For exampl e, i t can be appl i ed to the str i ng (43)anal yzed i nto substr i ngs s , . . , s^ i n accordancewi t h the dashes.

(43) the man - past - eat - the f ood.

5. 2. I n thi s way, we can descri be i n st ructuralterns t he set of st r i ngs (wi th phrase markers) towhi ch eay t ransf ormati on appl i es. We must nowspeci f y the st ructural change that a tr ansf ormati oneff ect s on any str i ng i n i ts domai n. An el ementaryt ransf ormat i on t i s def i ned by the f ol l owi ngpropert y:

(44) f or each pai r of i ntegers n, r ( n<r ) ,there i s a uni que sequence of i ntegers (ao>a , . . , )

and a uni que sequence of st r i ngs i n Vp

such that (i)ao=0; k>0; l S a < r f or 1< j <k; Y0=*JU

(i i ) f or each .. , Tr ,

t ( T1 , . . , Tn{Tn, . . , Yr )=Ya^^ Y ^ Z2 Y . . Y ^ Z^.

Thus t can be underst ood as convert i ng t heoccurr ence of Y i n the context

TL

W ) V ' . / ' V j ' — Yn+1

i nto a certai n st r i ng Y_ Z, . - -ao 1 *k

whi ch i s uni que, gi ven the sequence of terms(Yj , . . , Y ) i nto whi ch Y, ' "' . Yf i s subdi vi ded,t carr i el the stri ng Y. - . . - Y_ i nto a

k+1

a new str i ngW. - . . - W whi ch i s rel at ed i n a f i xed way to

Yj - . . - Yr . More preci sel y, we associ ate wi t h t thederi ved1t ransf ormati on t*:

(46) t* is the der i ved t ransf ormat i on of t i f

and onl y i f f or al l Y1( . . , Yr , t*(Y1 , . . , Yr) =W1 . / " %where W »t ( Y, ,. . ,Y :Y .. . ,Y J f or each nSr .

ii x & n r

We now associ ate wi t h each tr ansf ormati on Tan el ementary transf ormati on t. For exampl e, wi ththe passi ve transf ormati on (38) we associ ate theel ementary t ransf ormati on t^ def i ned as f ol l ows:

(4?) = Yfr

V T1’T2;T2*T3' V =T2" t6 ' ' en

V Tr Y2*T3; Y3 ' V “ T3

W ' - ' W = by' ' Yl

tp(Yl *- *' Yn:Yn*- *>Yr ) * Yn f or a11 n~

The deri ved t ransf ormati on t* thus has the f oi ng ef f ect: **

( W) ( i ) t *(Y1( .. , Y ) =T1 - Y^e^en - -

(* 1) t*( the man, past , eat , the'"' f ood) *

the f ood - past' "' be^ en - eat - by' "man.

The rul es (34), (36) carry the r i ght - hand si de(4811) i nto *t he f ood was eaten by the man, " as they carry (43) i nto the corr espondi ng act"the man ate the f ood."

The pai r ( E , t ) as i n (42) , (47) compl etecharacteri zes the passi ve tr ansf ormati on asdescri bed i n (38). E tel l s us to whi ch str ithi s t ransf ormat i on af pl i es ( gi ven the phrasemarkers of these str i ngs) and how to subdi vi dthese str i ngs i n order to appl y the transformaand t tel l s us what st ructural change to effon thl subdi vi ded str i ng.

A grammat i cal t ransf ormat i on i s speci f i ecompl etel y by a restr i ct i ng cl ass B and anel ementary t ransf ormati on t, each of whi ch laf i ni tel y character l zabl e, as i n the case of tpassi ve. I t i s not di f f i cul t to def i ne ri gorothe manner of thi s speci f i cat i on, al ong the lsket ched above. To compl ete the devel opment t ransf ormati onal grammar i t i s necessary to show a t ransf ormat i on automat i cal l y assi gns ader i ved phrase marker t o each transf ormand tgeneral i ze to t ransformat i ons on sets of stri(These and rel ated topi qs are t reated i n refe[3]. ) A t ransformat i on wi l l then carry a st rwi t h a phrase marker K (or a set of such pai ri nto a st r i ng S' wi t h a deri ved phrase marker

5. 3. Fromthese consi derat i ons we are l ed topi cture of grammars as possessi ng a t ri part i tst ructure. Corr espondi ng to the phrase st rucanal ysi s we have a sequence of rul es of the fX- **, ®»g. , (20), (23), (32). Fol l owi ng thi shave a sequence of t ransf ormati onal rul es suc(34) and (38). Fi nal l y, we have a sequence omorphophonemi c r ul es such as (36), agai n of tf ormX-»T. To generate a sentence f romsuch grammar we construct an extended der i vati onbegi nni ng wi t h an i ni t i al st r i ng of the phrasst ructure grammar, e. g. , #' ,' Sentence/’' #, as i(20). We t hen run through the rul es of phrasest ructure, produci ng a termi nal str i ng. We thappl y cert ai n tr ansformat i ons, gi vi ng a st r i nmorphemes i n the correct order, perhaps qui tedi f f erent st r i ng f r omthe or i gi nal termi nal s

Appl i cat i on of the morphophonemi c r ul es convethi s i nto a str i ng of phonemes. We mi ght runthrough the phrase st ructure grammar several and then appl y a general i zed t ransf ormati on tresul t i ng set of termi nal str i ngs.

I n § 3. 4 we noted that i t i s advantageouorder the rul es of phrase st ructure i nto asequence, and to di st i ngui sh obl i gatory f romopt i onal rul es. The same i s true of the t ran

f ormati onal part of the grammar. I n 84 we di scussed the transf ormati on (34), whi ch convert s

122



sequence affix-veri i nto the sequence verb- af f i x,and the passi ve tr ansf ormati on (38). Noti ce that(34 ) must he appl i ed i n every extended deri vat i on,or the resul t wi l l not he a grammati cal sentence.Bui e (3 , then, i s an obl i gatory tr ansf ormat i on. The passi ve t ransf ormati on, however, may or maynot be appl i ed; ei ther way we have a sentence. Thepassi ve i s thus an opt i onal t ransf ormat i on. Thi s

di st i nct i on between opt i onal and obl i gatory t ransf ormati ons l eads us to di st i ngui sh between twocl asses of sentences of t he l anguage. We have, onthe one hand, a kernel of basi c sentences that areder i ved f romthe termi nal st r i ngs of the phrase-st ructure grammar by appl i cati on of onl yobl i gatory t ransf ormati ons. We then have a set of der i ved sentences t hat are generat ed by appl yi ngopt i onal t ransf ormat i ons to the st r i ngs underl yi ngkernel sentences.

When we actual l y carry out a detai l ed st udyof Engl i sh structure, we f i nd that the grammar canbe great l y si mpl i f i ed i f we l i mi t the kernel to avery smal l set of si mpl e, acti ve, decl arat i vesentences ( i n f act, probabl y a f i ni te set) such as"t he man ate the f ood," etc. We then deri vequesti ons, passi ves, sentences wi t h conj unct i on,sentences wi t h compound noun phrases ( e. g. ,"provi ng that theoremwas di f f i cul t , " wi t h the NP"provi ng that t heorem*) , ^ etc. , by transformati ons.Si nce the resul t of a transf ormati on i s a sentencewi t h der i ved consti tuent st ructure, transformati onscan be compounded, and we can f ormquest i ons f rompassi ves (e.g. , "was the f ood eaten by the man") ,etc. The actual sentences of real l i f e areusual l y not kernel sentences, but r athercompl i cated t ransf orms of t hese. We f i nd, however,that the transf ormati ons are, by and l arge, meani ng-preservi ng, so that we can vi ew the kernelsentences underl yi ng a gi ven sentence as bei ng, i nsome sense, the el ement ary "content el ements" i nterms of whi ch the actual t ransf ormi s Underst ood. BWe di scuss t hi s probl embri ef l y i n § 6, mor e

extensi vel y i n ref erences [ 1], [2].I n §3- 6 we poi nted out that a grammar of

phrase structure i s a rather el ementary t ype of f i ni te- state process t hat i s det ermi ned at eachpoi nt by i ts present st ate and a bounded amount of i ts l ast output. We di scovered i n § 4 that thi sl i mi tati on i s too severe, and that the grammar canbe si mpl i f i ed by addi ng tr ansf ormat i onal rul esthat take i nto account a cer tai n amount of const i tuent str ucture ( i . e. , a cert ai n hi story of der i vat i on) . However , each t ransf ormat i on i s sti l lf i ni tel y characteri zabl e (cf. §§ 5 . 1- 2), and thef i ni te rest r i cti ng cl ass (41) associ at ed wi th at ransf ormati on i ndi cates now much i nf ormati onabout a st r i ng i s needed i n order to appl y thi st ransf ormat i on. The grammar can therefore st i l lbe r egarded as an el ementary f i ni t e-st ate process

of the type correspondi ng to phrase str ucture. There i s st i l l a bound, f or each grammar, on howmuch of the past output must be i nspected i n orderf or the process of der i vat i on to cont i nue, eventhough more than j ust the l ast output ( the l astl i ne of the deri vat i on) must be known.

6. Explanatory Power of Linguistic Theories

We have thus f ar consi dered the rel ati veadequacy of theor i es of l i ngui st i c str ucture onl y

i n terms of such essent i al l y f ormal cri ter i a assi mpl i ci t y. I n 5 1 we suggest ed that t here areother rel evant consi derat i ons of adequacy f orsuch theori es. We can ask (cf . (5)) whether ornot t he syntact i c st ructure reveal ed by thesetheor i es provi des i nsi ght i nto the use and under st andi ng of l anguage. We can bar el y touch onthi s probl em here, but even thi s br i ef di scussi on

wi l l suggest that t hi s cr i teri on provi des thesame order of rel ati ve adequacy f or the threemodel s we have consi dered.

I f the grammar of a l anguage i s to provi dei nsi ght i nto the way the l anguage i s understood,i t must be true, i n part i cul ar , that i f a sentencei s ambi guous (underst ood i n more t han one way),then thi s sentence i s provi ded wi t h al ternat i veanal yses by the grammar. I n other words, i f acer tai n sentence S is ambi guous, we can test theadequacy of a gi ven l i ngui st i c t heory by aski ngwhet her or not the si mpl est grammar const ructl bl ei n terms of thi s t heory for the l anguage i nquesti on aut omat i cal l y provi des di st i nct ways of generati ng the sentence S. I t i s i nst ructi ve tocompare the Markov process, phrasest ructure, and

t ransf ormat i onal model s i n the l i ght of thi s test.

I n § 3- 3 we poi nted out that the si mpl estC 2 ,F] grammar f or Engl i sh happens to provi denonequi val ent der i vat i ons f or t he sentence "t heyare f l yi ng pl anes, " whi ch i s, i n f act, ambi guous. Thi s reasoni ng does not appear to carry over f orf i ni te-st ate grammars, however. That i s, therei s no obvi ous moti vat i on f or assi gni ng twodi f f erent paths t o thi s ambi guous sentence i n anyf i ni te-st ate grammar t hat mi ght be proposed f ora part of Engl i sh. Such exampl es of const ruct i onal homonyml t y ( there are many others) const i tutei ndependent evi dence f or the superi ori t y of thephrase- st ructure model over f i ni t e-state grammars.

Further i nvest i gat i on of Engl i sh br i ngs tol i ght exampl es that are not easi l y expl ai ned i nterms of phrase st ructure. Consi der the phrase

(49) the shooti ng of t he hunters.

We can underst and thi s phrase wi t h "hunters" asthe subj ect, anal ogousl y to (50), or as theobj ect, anal ogousl y to (51 ).

(50) the growl i ng of l i ons

(51) the rai si ng of f l owers.

Phrases (50) and (51 ), however, are not si mi l ar l y

ambi guous. Yet i n terms of phrase st ructure, eachof these phrases i s represented as: the - V i ng-of' ' ' NP.

Caref ul anal ysi s of Engl i sh shows that we cansi mpl i f y the grammar i f we st ri ke the phrases(49)-(51) out of the kernel and rei ntr oduce themtransf ormati onal l y by a t ransf ormati on that

carr i es such sentences as "l i ons growl " i nto (50),and a t ransf ormat i on T2 that carr i es such sentences

123



as "t hey rai se f l owers" i nto (51 ). T^ and Tg wi l l

be si mi l ar to the nomi nal i zi ng t ransf ormat i ondescr i bed i n f n.12, when they are correct l yconst ructed. But both "hunters shoot" and "theyshoot the hunters" are kernel sentences; andappl i cat i on of T-^to the f ormer and to the

l at ter yi el ds the resul t ( 9) • Hence (49) hastwo di sti nct tr ansformati onal ori gi ns. I t i s acase of const ructi onal homonymi t y on the t ransf ormati onal l evel . The ambi gui ty of the grammati cal rel ati on i n ( 9) i s a consequence of the f actthat t he rel ati on of "shoot" to "hunters"di f f ers i n the two underl yi ng kernel sentences.We do not have thi s ambi gui t y i n the case of (50),(51 ), si nce nei t her "t hey growl l i ons" nor"f l owers rai se" i s a grammati cal kernel sentence.

There are many other exampl es of the samegeneral ki nd (cf. [ l ] , [2] ) , and to my mi nd, theyprovi de qui te convi nci ng evi dence not onl y f orthe greater adequacy of the transf ormati onalconcept i on of l i ngui sti c str ucture, but al so f orthe vi ew expressed i n §5. ^ that t ransf ormati onalanal ysi s enabl es us t o reduce part i al l y the

probl emof expl ai ni ng howve understand a sentenceto that of expl ai ni ng how we underst and a kernelsentence.

I n summary, then, we pi ct ure a l anguage ashavi ng a smal l , possi bl y f i ni te kernel of basi csentences wi t h phrase st ructure i n the sense of f}3« al ong wi t h a set of t ransf ormati ons whi ch

can be appl i ed to kernel sentences or t o earl i ert ransf orms to produce new and more compl i catedsentences fromel ementary components. We haveseen cer tai n i ndi cat i ons that thi s approach mayenabl e us to reduce the i mmense compl exi ty of actual l anguage to manageabl e proport i ons and-, i naddi t i on, that i t may provi de consi derabl e i nsi ghti nto t he actual use and underst andi ng of l anguage.Footnotes

1. Cf . [7] . Fi ni t e- state grammars can berepresented graphi cal l y by state di agrams, as

in [?]. P-15f.

2. See C<S] , Appendi x 2, f or an axi omati zat i on of concatenati on algebras.

3. By ' morphemes' we refer to the smal l estgrammati cal l y f unct i oni ng el ements of thel anguage, e. g. , "boy", "run" , "i ng" i n"r unni ng", "s" i n "books", etc.

k. I n the case of L , bj of (9i l ) can be taken

as an i dent i ty el ement U whi ch has thepropert y that f or al l X, U' "' X=X' SU=X. Then

wi l l al so be a dependency set f or a

sentence of l ength 2m i n L .

5. Note that a grammar must ref l ect and expl ai nthe abi l i t y of a speaker to produce and under st and new sentences whi ch may be much l ongerthan any he has previ ousl y heard.

6. Thus we can al ways f i nd sequences of n+1words whose f i rst n words and l ast n wordsmay occur, but not i n the same sentence (e.g.

repl ace "i s" by "are" i n ( 13i i i ) , and chooSj. of any requi red l engt h) .

?. Z or W may be t he i dent i t y el ement U (cf. i n thi s case. Note that si nce we l i mi ted so as t o excl ude U f romf i gur i ng si gni f i caon ei ther the r i ght - or the l ef t - hand si deof a rul e of and si nce we requi red thatonl y a si ngl e symbol of the l ef t - hand si demay be-repl aced i n any rule, i t f ol l ows th

Y^ must be at l east as l ong as X . Thus we

have a si mpl e deci si on procedure f or der i vabi l i t y and termi nal i ty i n the sense of (19i i i ). (19v).

8. See £3] a detai l ed devel opment of anal gebra of t ransformat i ons f or l i ngui stidescr i pt i on and an account of t ransf ormat i onal grammar. For f ur ther appl i cat i onthi s type of descri pt i on to l i ngui sti cmater i al , see [1], [2], and f roma somewdi f f erent poi nt of vi ew, [4].

9 . I t i s not di f f i cul t to gi ve a ri gorousdef i ni t i on of the equi val ence rel at i on iquest i on, though thi s i s f ai r l y tedi ous.

10. The not i on "i s a" shoul d actual l y berel at i vi zed f ur ther to a gi ven occurrencof i n S. We can def i ne an occurrence

si i n S as an ordered pai r (s . X) , where

i s an i ni t i al substr i ng of -S, and s^ i s

f i nal substr i ng of X. Cf . [5], p. 297.11. Where U i s the i dent i ty, as i n fn.

12. Noti ce that thi s sentence requi res general i zed t ransf ormati on that operatesa pai r of st ri ngs wi t h thei r phrase mark Thus we have a t ransf ormat i on that conveSpSg of the f orms NP-VP , i t -VPg, resp

i vel y, i nto the str i ng; i ng VP. ^ - TPg.

Convert s S- ="they -■prove that theorem"

S2= "i t - was di f f i cul t" i nto "i ng prove

that theoremwas di f f i cul t , " whi ch by becomes "provi ng that theoremwas di f f i cOf . C1]. C3] f °r detai l s.

Bi bl i ography[1] Chomsky, N. , The Logi cal Structure of

Li ngui st i c Theory (mi meographed) .[2] Chomsky, N. , Syntacti c Str uctures, to be

publ i shed by Mouton & Co., 1S~GravenhageNetherl ands.

[ 3] Chomsky, N. , Transf ormati onal Anal ysi s, PhDi ssertat i on, Uni versi t y of Pennsyl vani a,

J une, 1955.[4] Harr i s , Z. S. , Di scourse Anal ysi s, Languag28, 1 (1952).

[5] Q,ui ne, Vf. V. , Mathemati cal Logi c, revi sededi t i on, Harvard Uni versi t y Press. , Cambri1951.

[6] Rosenbl oom, P. , El ement s of Mathemati calLogi c, Dover, New York, 1950

[ 7] Shannon & Weaver, The Mathemati cal TheoryCommuni cat i on, Uni versi t y of I l l i noi s PrUrbana, 19 9.

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three models for the description of language, noam chomsky

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