(1982) Test-Score Semantics for Natural Languages [Zadeh]

8/12/2019 (1982) Test-Score Semantics for Natural Languages [Zadeh]

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COLING 82, J. Horeck.f, ed .)North-Holland Publishing Company Academia, 1982

TEST-SCORE SEMANTICSFOR NATURAL LANGUAGESL o t f i A . Za d eh

C o m p u t e r S c i e n c e D i v i s i o nU n i v e r s i t y o f C a l i f o r n i aB e r k e l e y , C a l i f o r n i aU . S . A .

T e s t - s c o r e s e m a n t i c s i s b a se d o n t h e p r e m i s e t h a ta l m o s t e v e r y t h i n g t h a t r e l a t e s t o n a t u r a l l a ng u a g esi s a m a t t e r o f d e g r e e . V ie w e d f ro m t h i s p e r s p e c t i v e ,a n y s e m a n t ic e n t i t y i n a n a t u r a l l a n g u a g e , e . g . , ap r e d i c a t e , p r e d i c a t e - m o d i f i e r , p r o p o s i t i o n , q u a n t i -f i e r , c om m a nd , q u e s t i o n , e t c . m ay b e r e p r e s e n t e d a sa s y st em o f e l a s t i c c o n s t r a i n t s o n a c o l l e c t i o n o fo b j e c t s o r d e r i v e d o b j e c t s i n a u n i v e r s e o f d i s -c o u r s e . I n t h i s s e n s e , t e s t - s c o r e s e m a n t i c s m ay b ev ie w e d a s a g e n e r a l i z a t i o n o f t r u t h - c o n d i t i o n a l ,p o s s i b l e - w o r l d an d m o d e l - t h e o r e t i c s e m a n t i c s , b u ti t s e x p r e s s i v e p ow er is s u b s t a n t i a l l y g r e a t e r .

INTRODUCTIONT e s t - s c o r e s e m a n t ic s r e p r e s e n t s a b r e ak w i t h t h e t r a d i t i o n a l a p p ro a c h es t o s em an -t i c s i n t h a t i t i s b as ed o n t h e p r e m is e t h a t a l m o s t e v e r y t h i n g t h a t r e l a t e s t on a t u r a l l a ng u a g es i s a m a t t e r o f d e g r e e . T he a c c e p ta n c e o f t h i s p r e m is e e n t a i l sa n a b an do n me nt o f b i v a l e n t l o g i c a l s y s t em s as a b a s i s f o r t h e a n a l y s i s o f n a t u r a ll a n g u a g e s a nd s u g g e s t s t h e a d o p t i o n o f f u z z y l o g i c Z ad e h 1 9 7 5 ) , B e l l m a n a ndZ ad e h 1 9 7 7 ) , Za de h 1 9 7 9 ) ) as t h e b a s i c c o n c e p t u a l f r a m e w o rk f o r d e a l i n g w i t hn a t u r a l l a n g u a g e s .I n f u z z y l o g i c , a s i n n a t u r a l l a n g u a g e s , a l m o s t e v e r y t h i n g i s a m a t t e r o f d e g r e e .T o p u t i t m e t a p h o r i c a l l y , t h e u se o f f u z T y l o g i c m ay b e l i k e n e d t o w r i t i n g w i t ha s p r a y - c a n , r a t h e r t h a n w i t h a b a l l - p o i n t p e n . T he s p r a y - c a n , h o w e ve r , h as a na d j u s t a b l e o r i f i c e , s o t h a t o n e m ay w r i t e , i f n ee d b e , as f i n e l y a s w i t h a b a l l -p o i n t p e n . T h u s , a c om m it me n t t o f u z z y l o g i c d o e s n o t p r e c l u d e t h e u se o f a b i v a -l e n t l o g i c w h e n i t i s a p p r o p r i a t e t o d o s o . I n e f f e c t , s uc h a c o n ~ i tm e n t m e re l yp r o v i d e s a l a n gu a g e t h e o r i s t w i t h a m u c h m ore f l e x i b l e f ra m e w o rk f o r d e a l i n g w i t hn a t u r a l l a ng u a g es a nd , e s p e c i a l l y , f o r r e p r e s e n t i n g m e a n in g , k n ow l ed g e a nd s t r e n g t ho f b e l i e f .A n a c i d t e s t o f t h e e f f e c t i v e n e s s o f a m e a n i n g - r e p r e s e n t a t i o n s y s te m i s i t s a b i l i t yt o p r o v i d e a b a s i s f o r i n f e r e n c e f r o m p re m i s e s e x p r e s s e d in a n a t u r a l l a n g u a g e .I n t h i s r e g a r d , an i n d i c a t i o n o f t h e c a p a b i l i t y o f t e s t - s c o r e s e m a n ti cs i s p r o -v i d e d b y t h e f o l l o w i n g e x a m p l e s , i n w h i c h t h e p r e m i s e s a p p e a r a b o v e t h e l i n e a n dt h e q u e s t i o n w h i c h ma y b e a n s w e re d i s s t a t e d b e l o w i t .

a ) D u r i n g m u c h . o f t h e p a s t d e c a d e P a t e a r n e d f a r m o re t h a n a l l o f h i s c l o s ef r i e n d s p u t t o g e t h e rH ow m u c h d i d P a t e a r n d u r i n g t h e p a s t d e c a de ?

b ) M o s t t a l l me n a r e n o t f a tM any f a t m en a re ba ldB i g i s t a l l a nd f a t

4 5


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426 L A ZADEH

How ma n y b ig men a re b a ld ?( c ) I f X i s l a r g e th e n i t i s n o t l i k e l y t h a t Y i s s m a l lI f X i s n o t v e r y l a r ge t h e n i t i s v e r y l i k e l y t h a t Y i s l a r g eX i s n o t l a r g e

H ow l i k e l y i s i t t h a t Y i s m o re o r l e s s s m a l l ?I n fu z z y l o g i c , t h e an sw e r t o a q u e s t io n i s , i n g e n e r a l , a p o s s i b i l i t y d i s t r i b u t i o n( Za d eh ( 1 9 7 8 ) ) . F o r e x a m p l e , i n t h e c as e o f ( a ) t h e a n s w e r w o u l d b e a p o s s i b i l i t yd i s t r i b u t i o n i n t h e u n i v e r s e o f r e a l n um b e rs w h i c h a s s o c i a t e s w i t h e a ch nu m b er ut h e p o s s i b i l i t y , ~ ( u ) , o ~ ( u ) ~ I , t h a t u c o u l d b e t h e c u m u l a t i v e in co m e o fP at g i v e n ( i ) t h e p r e m i s e , a nd ( i i ) t h e i n f o r m a t i o n r e s i d e n t i n a d a t a b a s e .I n t e s t - s c o r e s e m a n t i c s, a s e m a n ti c e n t i t y s uc h a s a p r o p o s i t i o n , p r e d i c a t e , p r e d i -c a t e - m o d i f i e r , q u a n t i f i e r , q u a l i f i e r , c om m an d, q u e s t i o n , e t c . , i s r e p re s e n te d a s as y st em o f e l a s t i c c o n s t r a i n t s o n a c o l l e c t i o n o f o b j e c t s o r d e r i v e d o b j e c t s i n au n i v e r s e o f d i s c o u r s e . S i m p l e e x a m p l es o f s e m a n t i c e n t i t i e s w h os e m e a n in g c an ber e p r e s e n t e d in t h i s m a nn er a r e t h e f o l l o w i n g :I . A n ca h as a y o u n g s o n . ( P r o p o s i t i o n . )2 . W hen D an i s t i r e d o r t e n s e , he s m ok es a l o t . ( C o n d i t i o n a l p r o p o s i t i o n . )3 . I t i s n o t q u i t e tr u e t h a t J oh n h as v e r y f ew c l o s e f r i e n d s . ( T r u t h - q u a l i f i e dp r o p o s i t i o n . )4 . I t is v e r y l i k e l y t h a t M a r ie w i l l b ec om e w e l l -k n o w n . ( P r o b a b i l i t y - q u a l i f i e dp r o p o s i t i o n . )5 . I t i s a l m o s t i m p o s s i b l e f o r M a nu el t o b e u n k i n d . ( P o s s i b i l i t y - q u a l i f i e dp r o p o s i t i o n . )6 . E x p e n si v e c a r . ( F u zz y p r e d i c a t e . )7 . V e r y . ( M o d i f i e r )8 . S e v e r a l l a r g e a p p l e s . ( S e c o n d -o r d e r f u z z y p r e d i c a t e . )9 . M ore o r l e s s . ( M o d i f i e r / F u z z i f i e r . )I 0 . N o t v e r y t r u e . ( Q u a l i f i e r . )I I . V e ry u n l i k e l y . ( Q u a l i f i e r )1 2 . M u c h a l l e r t h a n m o s t . ( F u zz y p r e d i c a t e . )1 3 . B r i n g m e s e v e r a l l a r g e a p p l e s . ( F u z z y c om m a n d. )1 4 . / W h o a r e E d i e ' s c l o s e f r i e n d s . ( Q u e s t i o n . )A l t h o u g h t e s t - s c o r e s e m a n t i c s h as a m u ch g r e a t e r e x p r e s s i v e p o w e r t h a n t h e m e a n i n g -r e p r e s e n t a t i o n s y ste m s b as ed o n p r e d i c a t e , m o d al an d in t e n s i o n a l l o g i c s , i t s e x -p r e s s iv e n e s s is a t t a i n e d a t t h e c o s t o f d o w n p l a y i n g , i f n o t e n t i r e l y s e v e r i n g , t h ec o n n e c t i o n b et w ee n s y n t a x an d s e m a n t i c s . I n p a r t i c u l a r , t h e h om o m o rp hi c c o n n e c t i o nb e tw e e n s y n t a x a nd s e m a n t i c s w h i c h p l a y s a c e n t r a l r o l e i n M o nt ag ue s e m a n t i c s(Mo nta gu e (1 9 7 4 ) , P a r t e e (1 9 7 6 ) a n d a t t r i b u t e d g ramma rs f o r p ro g ra mmin g la n g u a g e s( K nu th ( 1 9 6 8 ) ) , p l a y s a m u c h l e s s e r r o l e i n t e s t - s c o r e s e m a n t i c s -a r o l e r e p r e s e n t edi n t h e m a in b y a c o l l e c t i o n o f l o c a l t r a n s l a t i o n r u l e s g o v e r n i n g t h e us e o f m o d i -f i e r s , q u a l i f i e r s , q u a n t i f i e r s a nd c o n n e c ti v e s . I n e f f e c t , t h e d o w n pl a yi n g o f th ec o n n e c t i o n b e tw e e n s y n t a x an d s e m a n ti c s i n t e s t - s c o r e s e m a n ti c s r e f l e c t s o u r b e l i e ft h a t , i n t h e c a s e o f n a t u r a l l a n g u a g e s , t h e c o n n e c t i o n i s f a r t o o c o m p l ex a nd f a rt o o f u z z y t o b e a m e na b le t o a n e l e g a n t m a t h e m a ti c a l f o r m u l a t i o n i n t h e s t y l e o fM o nt ag ue s e m a n t i c s , e x c e p t f o r v e r y s m a l l f ra g m e n t s o f n a t u r a l l a n g u a g e s i n w h i c ht h e c o n n e c t i o n c a n b e f o r m u l a t e d a n d e x p l o i t e d .T he c o n c e p tu a l f r am e w o r k o f t e s t - s c o r e s e m a n ti c s is c l o s e l y r e l a t e d t o t h a t o fP RU F ( Za de h ( 1 9 7 8 ) ) , w h i c h i s a m e a n i n g - r e p r e s e n t a t i o n s y s te m i n w h i c h a n e s s e n t i a lu se i s m a d e o f p o s s i b l i t y t h e o r y ( Za de h ( 1 9 7 8 ) ) - a t h e o r y w h ic h i s d i s t i n c t f ro mt h e b i v a l e n t t h e o r i e s o f p o s s i b i l i t y r e l a t e d t o m od al l o g i c a nd p o s s i b l e - w o r l ds e m a n t i c s ( C r e s s w e l l ( 1 9 7 3 ) , R e s c h e r ( 1 9 7 5 ) ) .I n e f f e c t , t h e b a s i c i d e a u n d e r l y i n g b o t h P RU F a nd t e s t - s c o r e s e m a n ti c s i s t h a tm o st o f t h e i m p r e c i s i o n an d l a c k o f s p e c i f i c i t y w h i c h i s i n t r i n s i c i n n a t u r a l l a n -g ua ge s i s p o s s i b i l i s t i c r a t h e r th a n p r o b a b i l i s t i c i n n a t u r e , a nd h en ce t h a t p o s s i -


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TEST-SCORE SEMANTICS FOR NATURAL LANGUAGES 421

bil ity theory and fuzzy logic provide a more appropriate framework for dealin g withnatural language s than the tradit ional logical systems in which there are no gra-dations for truth, membership and belief, and no tools for coming to grips withvagueness, fuzziness and randomness.In what follows, we shall sketch some of the main ideas underly ing test-score seman-tics and il lustrate them with s imple examples. A more detailed exposit ion and ad-dit iona l examples may be found in Zadeh (1981).BASIC ASPECTS OF TEST-SCORE SEMANTICSAs was stated earlier, the point of departure in test-score semantics is the assump-tion that any semantic entity may be represented as a system of elastic constraintson a collection of objects or derived objects in a universe of discourse.Assuming that each object may be characterized by one or more fuzzy relations, thecollection of objects in a universe of discourse may be identif ied with a collectionof relations which constitute a fuzzv relational database or. equivalently, a statedescription (Carnap (1952)). In this database, then, a derived'object would be char-acterized by one or more fuzzy relations which are derived from other relations inthe database by operations expressed in an appropriate relation-man ipulating lan-guage.In more concrete terms, let SE denote a semantic entity , e.g., the proposit ion

p 4 During much of the past decade Pat earned far more thanall of his c lose fr iends put together,

whose mean ing we wish to represent. To this end, we must (a) identify the con-straints which are implic it or explic it in SE; (b) describe the tests which mustbe performed to ascertain the degree to which each constraint is satisfied; and(c) specify the manner in which the degrees in question or, equivalently, the par-t ial test scores are to be aggregated to y ield an overall test score. In general,the overall test score wo uld be represented as a vector whose components are num-bers in the unit interval or, more generally , possibil i ty /probabil ity distr ibutionsover this interva l.Spell ed out in greater detail, the process of meaning-representation in test-scoresemantics involves three distinct phases. In Phase 1, an explanatory databaseframe or EDF , for short, is constructed. EDF consists of a collec tion of rela-

tional frames each of which specif ies the name of a relation, the names of its at-tr ibutes and their respective domains,with the understanding that the mean ing ofeach relation in EDF is known to the addressee of the meaning-representation pro-cess. Thus, the choice of EDF is not unique and is strongly influenced by the de-sideratum of explanatory effectiveness as well as by the assumption made regardingthe knowledge profi le of the addressee of the meaning-representation process. Forexample, in the case of the proposit ion p 4 During much of the past decade Pat-- --earned far more than all of his c lose fr iends put together, the EDF might consist---7----- -----of the relational framesFRIEND [Namel; Name2; p], where p is the degree.to which Name1 is a fr iend ofName2; INCOME [Name; Income; Year], where Income is the income of Name in yearYear, coun ting backward from the present; MUCH [Proportion; ~1, where in is the de-gree to which a numerical value o f Proportion f its the mean ing of much in the con-text of p; and FAR.MORE [Numberl; Number2; P]. in which p is the degree to whichNumber1 fits the description far more in relation to NumberP. In effect, the com-posit ion of EDF is determined by the information that is neede d for an assessmentof the compatibil i ty of the given S E with any designated object or, more generally ,a specif ied state of affairs in the universe of discourse.In Phase 2, a test procedure is constructed which upo n application to an explan-atory database -that is , an instantiation of EDF - y ields the test scores,


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Tl,...bv the

3Tn, which represent the degrees to which the elastic constraints inducedconstituents of SE are satisfied. For example, in the case of p', the test

ppocedure would y ield the test scores for the constraints induced by c lose fr iend,@, far more, etc.--In Phase 3, the partial test scores obtained in Phase 2 are aggregated into anoverall test score, T, which serves as a measure of the compatibil i ty of SE withED, the explanatory database. As was stated earlier, the components of T are num-bers in the unit interval or, more generally , possibil i ty /probabil ity distr ibutionsover this interval. In particular, when the semantic entity is a proposit ion, p,and the overall test score, T, is a scalar, ' I may be interpreted as the truth ofp relative to ED or, equivalently, as the possibil i ty of ED given p. In this in-terpretation, then, the c lassical truth-condit ional semantics may be v iewed as aspecial case of test-score semantics which results whe n the constraints indu cedby p are inelastic and the overall test score is allowe d to be only pass or fail--The test procedure which y ields the overall test score T is interpreted as themeaning of SE.To il lustrate the phases in question, we shall consider a few simple examples

(a) SE 4 Ellen resides in a'small c ity near Oslo.In this case, EDF is assumed to comprise the followin g relationalframes ( + stands for unio n ):EDF 4 RESIDENCE [Name; C i ty .Name]+

POPULATION [City.Name; Popu lation]+SMALL [Populat ion; PI+NEAR [City.Namel; City.Name2; lo]

In RESIDE NCE, City.Name is the name of the c ity in which Name resides; in POPULA -TION, Popula tion is the number of residents in City.Name; in SMA LL, P is the de-gree to which a c ity with a popula tion equal to the value of Popu lation is small;and in NEAR , P is the degree to which City.Namel is near City.NameZ.The test procedure which leads to the overall test score T -- and thus representsthe mean ing of SE - is described below. In this procedure, Steps 1 and 2 involvethe determina tion of the value of an attr ibute given the values of other attr i-butes; Steps 3 and 4 involve the testing of constraints; and Step 5 involves anaggrega tion of the partial test scores into the overall test score T.

1. Find the name of the residence of Ellen :RE c , ty RameRESIDENCEIName=El len]

which means that th e value of Name is set to Ellen and the value of City.Name isread, y ielding RE, the residence of Ellen .

2. Find the popula tion of the residence of Ellen :n

PRE = Populat ion PDPULATION[City.Name=RE]3. Test the constraint induced by SMAL L:

r ,guSMALLIPopulat ion=RE]where ~~ denotes the result ing test score.

4. Test the constraint induced by NEAR :T2=uNEAR[City.Name=Oslo; City.Name2=RE]

5. Aggregate ~~ and T2:T = T1 , . T2

where A stands for min in infix posit ion, and T is the overall test score. This


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'TEST-SCORESEMANTICSFORNATURALLANGUAGES 429

mode of aggregation implies that, in SE, the denotation of conjunction is taken tobe the Cartesian product of the denotations of the conjuncts (Zadeh (1981)).

(b) SEA During much of the past decade Pat earned far more than all ofhis c lose fr iends put together.

In this case, we shall employ the EDF described earlier, that is :EDF 2 INCOME[Name; Year ; Amount ]+

FRIEND[Namel ; Name2; u]+FAR.MORE [Numberl; NumberP; n]+MUCH[Proportion; u]

The test procedure comprises the followin g steps:1. Find the fuzzy set of Pat's friends:

FP4 NamelxnFRIEND [Name2 = Pat ]in which the left subscript Namelxu s ignif ies that the relation FRIEND [NameP= Pat]is projected on the doma in of the attr ibutes Name1 and u, y ielding the fuzzy set offr iends of Pat.

2. Intensify FP to account for the modifier.*:CFP 4 FP2

in which FP* denotes the fuzzy set which results from squaring the grade of member-ship of each component of FP. The assumption underly ing this step is that thefuzzy set of c lose fr iends of Pat may be derived from that of fr iends of Pat by in-tensif ication.

3. Find the fuzzy multiset of incomes of c lose fr iends of Pat in yearYear; , i=l,..., lO:

ICFP. 4 Amount INCOMEIName = CFP; Year=Yeari ]In stipulating that the r ight-hand member be treated as a fuzzy multiset, we implythat the identical elements should not be combined, as they would be in the case ofa fuzzy set. With this understanding, ICFPi wil l be of the general form

ICFPi = 6,/e1+6,/e+...+6,/em .

where el,..., e m are the incomes of Name ..., Name1' m,respectively, in Year., andsl,...,,m 6 are the grades of membership of Name,;.-,Namem in the fuzzy se t of c lose fr iends of Pat.

4. Find the total income of c lose fr iends of Pat in Yeari , i=l;.., 10:TICFPi =6,el+...+ 6 emm

which represents a weighted arithmeticin Year i .

5. Find Pat's income in Yeari:IP i 4Amount INCOME[Name=Pat ;

6. Test the constraint induced

sum of the incomes of c lose fr iends of Pat

Year=Yeari].by FAR.MORE:

ri$pFAR.MOR EINumberl=IPi; Nurnber2= TICFPi]7. Find the s igma-count (Zadeh (1981)) of years d uring which Pat's income

was far.greater than the total income of all of his c lose fr iends:c iq'.;


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430 L.A. ZADEH

a. Test the constraint induced by MUCH:TePMUCH[Proportion=C]

'16where T represents the overa ll test score.The two examples described above are intende d merely to provide a rough outl ineof the meaning-representation process in test-score semantics. A more detailed ex-posit ion of some of the related issues may be found in Zadeh (1978) and Zadeh(1981)'Research supported in part by the NSF Grants MCS79 -06543 and IST-801896.REFERENCES AND RELATED LITERATURE1.

2.3.4.

5.

6.

7.8.9.

Bellma n, R. E. and Zadeh, L. A., Local and Fuzzy Logics, in: Modern Uses ofMultiple -Value d Logic Epstein, G.,(ed.). Dordrecht: (D. Reide l 103-165, 1977).Carnap, R., Mean ing and Necessity (University of Chicago Press, 1952).Cresswell, M. J., Logics and Languag es (London: Methuen , 1973).Knuth, D., Semantics of context-free languages, Mathem atical Systems Theory 2(1968) 127-145..Lambert, K. and van Fraassen, 8. C., Mean ing Relation s, Possible Objects andPossible World&Philosophical Problems in Logic (1970) l-19.Montagu e, R., Formal Philosophy, in: Selected Papers,Thomason, R.,(ed.).New Haven: (Ya le University Press, '974).Partee, B., Montag ue Grammar (New York: Academic Press, 197 6).Rescher, N., Theory of Possibil i tyRieger, B., Feasible fuzzy semantics,Words, Worlds and Contexts (1981) 193-209.

10. Zadeh, L. A., Fuzzy logic and approximate reasoning, Synthese 30 (1975) 407-428.

11. Zadeh, L. A., A theory o f approximate reasoning, in: Electronics ResearchLaboratory Memora ndum M77/58, University of California, Berkeley, 1977. AlsoMachine Intell igence 9, Hayes, J. E., Michie, M. and Kulich, L. I., (eds.).

k& York: (Wiley, 149-194, 1979).12. Zade h, L. A., Fuzzy sets as a basis for a theory of possib ility, Fuzzy Sets

and Systems 1 (1978) 3-28.13. Zadeh, L. A., PRUF--a mean ing representation langua ge for natural languages,

Int. J. Man-Machine Studies 10 (1978) 395-460.14. Zadeh, L. A., Test-score semantics for natural languages and meaning-represen-

tation v ia PRUF, Tech. Note 247, AI Center, SRI International, Menlo Park,CA. , 1981.

Documents

(1982) Test-Score Semantics for Natural Languages [Zadeh]