Assuming that representation is a crucial issue for Computational Semantics, I will show in this paper that equation solving is a very simple and elegant way or computing meaning representations or information structures conveyed by the use o

Lee Kiyong

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Lee, Kiyong

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Computing Information by Equation SolvingKiyong LeeKorea University0. IntroductionAssuming that representation is a crucial issue for Computational Semantics, I willshow in this paper that equation solving is a very simple and elegant way orcomputing meaning representations or information structures conveyed by the use ofnatural language. In a formal semantics like Montague's(1974), a type-theoretic A-calculus with /3-reduction is often used to handle with substitutions. But I arguein this paper that the substitution for equation solving is less constrained thatthe substitution by 13-reduction and that the former can treat the phenomena of freeword-order and partial information in natural language more conveniently that thelatter. To support my argument, I will analyze some fagments of Korean and English,particulary those involving scrambling in Korea, and quantification andNP-conjunction in English.1. Two Types of SubstitutionThere are two types of substitution in computing meaning representations: one by I?-reduction and the other, by equation solving. Consider the following substitutionsinstances:(1) /1-reductiona. Ax2yf(x,y)(b)(a)b. A yf(a)(b)c. f(a,b)(2) equation solvinga. f(x,y), x=a, y=b.b. f(a,y), y=b. OR b'. f(x,b , x=a.c. f(a,b).In (1), the functor A x A yf (x, y) takes two arguments a and b, yielding a closedformula A xAyf(x, y)(b)(a). This formula is reducible to f(a,b) by the repreatedapplication of fi -reduction: first, a substitutes for x and then b for y. Unlike theformula (1a), the formula AxAyf(x,y)(a)(b) reduces to a different formula f(b,a)by substituting b for x and a for y. But in the case (2) of equation solving, thereis no fixed order of substitution: whatever the order of application may be, aalways substitutes for x and b for y. (2a) is thus equivalent to f(x,y)fx=a, y=b},where there is no constraint over the order of substituting a for x and b for y.Language, Irtformation and ComputationChungmin Lee and Beom-mo Kang, editors, 1993, Seoul: ThaehaksaThere is another difference between /3-reduction and equation solving. Inthe /3-reduction approach, the meaning of a sentence is normally represented bya single formula, and this formula may often consist of an n-ary functor andits n arguments. This complex formula is then reduced by /3-reduction to asimpler or atomic formula. In the equation-solving approach, on the other hand,the meaning of a sentence may be represented by a set of open formulas and theseformulas may reduce to a single formula by the process of equation solving. Inthe former, a formula consisting of a functor and its arguments is notdecomposed into other formulas, whereas in the latter a set of formulas maydecompose to simpler formulas or simpler sets of formulas. If we accept the viewthat atomic formulas represent basic units of factual or non-factualinformation, then the equation-solving approach can represent any partialinformation associated with parts of a sentence in a more elegant way as will beillustrated in this paper.2. Scrambling in KoreanIn Korean, no-verbal expressions like NP's can be scrambled in a sentence, asin:(3) a. Kim-i	 Ann-eke ton-il cuassta.Kim-nom Ann-dat ton-acc gave'Kim gave Ann money.'b. Kim-i ton-il Ann-eke cuassta.c. Ann-eke Kim-i ton-il cuassta.d. Ann-eke ton-11 Kim-i cuassta.e. ton-il Kim-i Ann-eke cuassta.f. ton-il Ann-eke Kim-i cuassta.Despite a difference in word order, they all have the same semanticrepresentation.(4) cuasssta(kim,ann,ton) 'Kim gave Ann money'In the traditional Montague grammar based on A-calculus, we had to setseveral different categories of the di-transitive verb cu-ta 'give' and severaldifferent rules of combination. But in the proposed equation-solving approach,we need only a single recursive PS rule for the category V and an NP ruleintroducing a 04, Case Marker:(5) RulesV --> NP VNP --> N CMWe also need a Lexicon in which each lexical item is provided with appropriateinformation or necessary constraints. Factual information is mostly representedin an equational form, while a constraint is represented by a rule form p qread 'p if q', where p is a head and q, a body".15Before I discuss scrambled sentences in (3), I will analyze a simplersentence kia-i conta 'Kim dozes' for illustration's sake,(6) Lexiconconta: V, v(SEM)=[conta(X)	 member(X,Agent)], Role=Agent	 Case=nom.Kim: N, Role=[kim],-i: CM, Case=nom.Here, conta 'doze' belongs to a category V and its semantic content is aconstraint or rule consisting of a head conta(X) and a body member(X,Agent),where Agent is a list of agents. 2 This verb contains a constraint that relatesthe nom(inative) Case to the argument Role Agent. The noun kis denotes a list[kim] with an uninstantiated Role: this Role will be instantiated when the nouncombines with an appropriate verb. The Case Marker(CM) -i carries informationabout the Case-value nom.Given the Lexicon, the Case-Marking rule licenses the following tree:(7) NPRole=[kim]Case=nomN	 CMRole=[kim] Case=nomKimNow, by unifying pieces of information given in the daughter nodes N and CM, 1 weobtain the meaning or information structure associated with the mother node NPas a list of equations [Role=[kim],Case=nom].The V Rule will then combine the NP structure given above with V, yieldingthe following:1 In this paper, I will freely use terminology or notations commonly used inProlog.2 Here, I will introduce an infix	as defined: X:L	 member(X,L). By usingthis infix, we can treat a formula f(X:L) as equivalent to f(X) & member(X,L).Henceforth, I will abbreviate a rule like conta(X) 	 member(X,Agent) asconta(X:Agent).3 I here assume the Principle of Unification, which can be roughly statedas: The information associated with the Mother node is a result of unifying allthe relevant pieces of information given in its Daughter nodes. Unification is aseries of complex operations involving appending, matching or substitution.16(8)NPRole=[kim]	 v(SEM)=[conta(X:Agent)]Case=nom	 Role=Agent	 Case=nomN	 CMRole=[kim] Case=nom	 contaKim	 -iAgain, by unifying the pieces of information given in NP and V, we obtain theinformation for the Mother node V:(9) v(SEM)=[conta(X:Agent)]Role=Agent	 Case=nomRole=[kim]Case=nomHere, we first get Role=Agent by resolving the constraint Role=Agent Case=nomwith an equation Cas(=nom and then obtain an equation Agent=[kim] on the basisof two equations Role=Agent and Role=[kim]. Finally, we substitute [kim] forAgent in the top formula, thus obtaining:(10) v(SEM)=[conta(X:[kim])]This is equivalent to:(11) v(SEM)=[conta(kim)],for X:L is defined as 'X:L	 member(X,L)' and X should be a member of the list[kin].By simply extending the Lexicon, we can treat the free word orderillustrated by sentences (3a-f).(12) Extended Lexiconconta: V, v(SEM)=[conta(X:Agent)], Role=Agent 	 Case=nom.cuassta: V, v(SEM)=[cuassta(X:Agent,Y:Recipient,Z:Object)],Role=Agent	 Case=nom,Role=Recipient	 Case=dat,Role=Object	 Case=acc.Kim: N, Role=[kim].Ann: N, Role=[ann].ton: N, Role=[ton].-i: CM, Case=nom.11: CM, Case=acc.eke: CM, Case=dat.17Based on the extended Lexicon, the Case-Marking Rule derives the followingtrees:c.(13) a.	 NP	 b.	 NPRole=[kim]	 Role=[ton]Case=nom	 Case=acc	Kim-i	 ton-ilNPRole=[ann]Case=datAnn-ekeThe V-Rule will then derive the following trees:(14)Vv(SEM)=[cuassta(X:[kim],Y:[ann],Z:[ton])],1Vv(SEM)=[cuassta(X:Agt,Y:[ann],Z:[ton])],Role=Agt	 Case=nom.	 1Vv(SEM)=[cuassta(X:Agt,Y:Rec,Z:[ton])],Role=Agt	 Case=nom,Role=Rec	 Case=dat.1	 1NP	 NP	 NP	 VRole=[kim] Role=[ann] Role=[ton] 	 v(SEM)=[cuassta(X:Agt,Y:Rec,Z:Obj)],Case=nom Case=dat Case=acc 	 Role=Agt	 Case=nom,	Role=Rec	 Case=dat,	Role=Obj	 Case=acc.Kim-i	 Ann-eke	 ton-il	 cuassta(15) Vv(SEM)=[cuassta(X:jkimLY:[ann],Z:[ton])],1Vv(SEM)=[cuassta(X:Agt,Y:[ann],Z:[ton])],	Role=Agt	 Case=nom.	 1Vv(SEM)=[cuassta(X:Agt,Y:[ann],Z:Obj)],Role=Agt	 Case=nom,Role=Obj	 Case=acc.1	 1NP	 NP	 NP	 VRole=[kim] Role=[ton] Role=[ann]] 	v(SEM)=[cuassta(X:Agt,Y:Rec,Z:Obj)],Case=nom Case=acc Case=dat 	 Role=Agt	 Case=nom,	Role=Rec	 Case=dat,	Role=Obj	 Case=acc.Kim-i	 ton-11	 Ann-eke	 cuassta18These trees show how sentences (3a) and (3b) can be analyzed. The othertrees can be analyzed in the same manner, yielding the same semanticrepresentation. For deriving these trees in which NP's are scrambled, we haveonly utilized two PS rules and the Lexicon in which each lexical item isprovided with appropriate syntactic and semantic information and constraints.The computation of meaning or information required unification and equationsolving: the meanings of Daughter nodes were unified and their representationswere simpified by the repeated application of a substitution.3. Qunatifiers in EnglishThe use of A-abstraction easily represents the meaning of a quantified NP asa functor:(16) every man: APN/x[man'(x) -> P(x)]Taking as argument snores'which represents the meaning of a verb 'snores' andundergoing /3-reduction, the given functor yields a formula:(17) a. ANNI[man'(x) -> P(x)](snores')b. Vx[man'(x) -> snores'(x)]In this section, the proposed equation solving approach can treat quantifiersequally well. For this, the following annotated PS rules and the Lexicon will beset up.(18) Annotated RulesS --> NP	 VPGRel=subjNP --> NNP --> DET NVP --> VVP --> V	 NPGRel=obj(19) LexiconAnn: N, Role=[ann].snores: V, v(SEM)=[snores(X:Agent)],X=[var, Agent],Role=Agent	 GRel=subj.loves: V, v(SEM)=[loves(X:Agent,Y:Patient)],X=[var,Agent],Y4var,Patient],Role=Agent	 GRel=subj,Role=Patient	 GRel=obj.man: N, n(SEM)=[man].every: DET, q(SEM)=every([var,Role],P1,P2),P1=n(SEM),P2=v(SEM).,Role=avar,Rolefl.19This small grammar generates sentences like:(20) a. Ann snores.b. Every man snores.c. Ann loves every man.First, (20a) is analyzed as follows:(21)GRel=subjRole=[ann]v(SEM)=[snores(X:Agent)],X4var,Agent],Role=Agent	 GRel=subj.NP	 VPGRel=subjN	 VRole=[ann]]	 v(SEM)=[snores ( X: Agent )],X=[var,Agent],Role=Agent	 GRel=subj.snoresThe information structure associated with S simplifies by equation solving to:(22) v(SEM)=[snores(X:[ann])], X4var,[ann]].Since X denotes a variable X such that it ranges over the list ann], (22)further simplifies to:(23) v(SEM)=[snores(ann)].The analysis of sentence (20b) undergoes a similar process of unificationand equation solving:(24)NPGRel=subjDETq(SEM)=every([var,Role],P1,P2)Pl=n(SEM)P2=v(SEM)Role=[[var,Role]]everyn(SEM)=Cman]Vv(SEM )4snores( X: Agent )]X= [var, Agent]Role=Agent	 GRel=subj    snoresBy unification and substitution, we first obtain:(25) NPq(SEM)=everyavar,Role],[man],P2)P2=v(SEM)Role=[[var,Role]]GRel=subjWe then obtain the information structure for S:(26)q(SEM)=everyavar,Role],[man][man], snores(X:Agent)])Role=[[var,Role]]X=[var, Agent]	Role=Agent	 GRel=subjGRel=subjSince NP's GRel(ation) is subj(ect), its Role will be Agent. Then bysubstituting Agent for Role and replacing [var,Agent] with X, we obtain:(27)q(SEM)=every(X,[man],[snores(X:[X])])Furthermore, since f(X:[X]) is equivalent to (f(X) member(X,[X]) andmember(XJX]) always holds true, f(X:[X]) simplifies to f(X). Thus, from (27)we get:(28)q(SEM)=every(X,[man],[snores(x)])41A quantifier expression may occur in VP as in sentence (20c). But it canalso be treated nicely in the proposed grammar. The following is an analysis ofthe VP 'loves every man':4 In Prolog, the quantifier every may be treated as a functor defined as:every(X,[f],[g(X)])	 all(X,f(X) -> g(X)).21(29) VPV	 NPv(SEM)=[loves(X:Agent,Y:Patient)] 	 GRel=objX4var,Agent]	q(SEM)=everyffvar,Role],[man],P2)Y4var,Patient]	P2=v(SEM)Role=Agent	 GRel=subj	 Role=[[var,Role]]Role=Patient	GRel=objloves	 every manTo obtain the information structure for VP, we first unify the two informationstructures of V and NP and then deduce the fact Role=Patient by resolving theconstraint Role=Patient	GRel=obj and GRel=obj. We now substitute Patient forRole, and Y for [var,Role], [loves(X:Agent,Y:Patient)] for P2 	 ineveryffvar,Role], [man],P2). As a result of all these operations, we now have:(30) VPq(SEM)=[every(Y,[man],[loves(X:Agent,Y:Patient)])],Xgvar,Agent]	Role=Agent	 GRel=subjMen this combines with an NP 'Ann', we obtain:(31)q(SEM)=[every(Y,[man],[loves(ann,Y)])]Translated in Prolog, q(SEM) will be:(32) all(Y,man(Y) -> loves(ann,Y))We have thus captured the meaning of a universally quantified sentence`Ann loves every man' through a step-by-step process. The meaning structure ofthe entire sentence is obtained by unifying parts of the information assignedto each of its phrasal parts. As in Montague Semantics, it is possible to assigna semantic structure to the quantifier 'every' as well as to the quantified NP'every man'.4. Conjoined Noun PhrasesIt has been pointed out by Moore (1989) and Halvorsen (1991) that thetreatment of conjoined NP's like (33) causes a problem for unification-basedapproaches.22(33) Ann and Bob snore.But I will show that this conjunction can also be treated nicely in theequation-solving approach. For such an treatment, we just need an NP-ConjunctionRule with the specification of the meaning of 'and' in the Lexicon:(34) NP	 NP and NP(35) Lexiconand: CONJ, MRole=X	 append(DRolel,DRole2,X).Ann:	 Role=[ann].Bob: N, Role=[bob].snore: V, v( SEM ) = [snore ( X: Agent )],X=[var, Agent] ,Role=Agent	 GRel=subj.Here in the Lexicon, the constraint assigned to CONJ may be formulated as ageneral principle on conjunction, stating that MRole, the Role, or semanticcontent, of the Mother node is the result of appending a DRole, the Role of oneof its Daugheter nodes to the other DRole.(36)v(SEM)=[snore(X:[ann,bob])]NP	 VPGRel=subj	 v(SEM)=[snore(X:Agent)]Role=[ann,bob]	 X=[var,Agent]I	 Role=Agent	 GRel=subjNP	 CONJ	 NPRole=[ann]	 C	 Role=[bob]Ann	 and	 Bob	 snoreBy the contraint C on and-Conjunction, we first obtain (37) as representing themeaning of the conjoined NP 'Ann and Bob':(37) Role=[ann,bob]And then, by unification, we obtain the semantic structure for S:(38) v(SEM)=[snore(X:[ann,bob])]23The formula given in the list is equivalent to:(39) a. snore(X)	member(X, [ann,bob]).b, snore(ann), snore(bob).c. snore(ann) & snore(bob)It has thus be shown that a conjoined NP like 'Ann and Bob' can be treatedadequately in an equation-solving approach without /3-reduction.I will also show that the proposed analysis of a conjoined NP can applyto the case of a group interpretation,(40) Ann and Bob met yesterday.Here, the verb 'met' requires a group as its Subject. By just encoding thisinformation into the semantic structure of the verb, we can obtain the groupinterpretation of sentence (40).(41) Addition to the Lexiconmet: V, v(SEM) = (met(L:[Agent])],L=[var,[Agent]],Role=Agent	 GRel=subject.yesterday: ADV, s(SEM) = [yesterday(v(SEM))].On the basis of this expanded Lexicon, we can construct the followingtree:(42)s(SEM)=[yesterday(met(L:[[ann,bob]]))]NP	 VPGRel=subj	 s(SEM)=(yesterday(met(L:[Agent]))]	Role=[ann,bob]	 L=[var, Agent]Role=Agent	 GRel=subject.V	 ADVv(SEM)=[met(L:[Agent])] 	 s(SEM)=[yesterday(v(SEM))]L=Nar,[Agent]],Role=Agent	 GRel=subject.Ann and Bob	 met	 yesterday24The above tree is obtained by a routine process of unification andsubstitutions, yielding the semantic structure (43) for S.(43) a. yesterday(met(L:Uann,bobfl))b. yesterday(met([ann,bob]))Note in (43a) that L is a list [ann,bob]. Hence, (43a) reduces to (43b), asbefore. (43b) is then interpreted as representing a factual or non-factualinformation that Ann and Bob as a unit was involved in the meeting of yesterday.5. Concluding RemarksOne of the motivations for adopting an equation solving approach is to make theprocess of computing meaning representation more transparent and implementablein a computer language like Prolog. Each information structure is represented bya set of Horn clauses or equations and the composition of informaton structuresis carried out by the repeated application of appending, matching orsubstituion. This process we call equation solving.There are many problems left unresolved. One of the residual problems hasto do with a conjoined NP like:(44) a. Ann and every man snore. s)b. Bob and every other man snore.But note here that the set of objects referred to by one conjunct 'Ann' must bedisjoint from that referred to by the other conjunct 'every man'. Hence, thefollowing expressions sound inconsistent.(45) a.*Ann and every female snore.b.*The man and every man snore.c. *I and I snore.5 According to the present formulation of the information structure of 'and',sentences like (44a) fail to be treated properly. What is obtained is a wrongrepresentation:[i] every(X, [man], [snore(X:[ann,X]),which is equivalent to:[ii]all(X,man(X) -> (member(X,[ann,X]) -> snore(X))).This does not entail snore(ann). Hence, the analysis is wrong.25On the other hand, the Ibilowing expressions 'became 'accept-Able, when 1/10y1MND usedto refer to two different persons:(46) a, This man and this man snored.b. You and you snored.A problem like this cannot be resolved by just employing Type Raising or g-reduction. But when this type of constraint is clearly understood, anequation-solving approach can easily encode it into the information structure of aConjuction 'and'. This I hope to do in the next paper,REFERENCESClocsin, W. F. and Mellish, C. S. (1981) Programing in Prolog Springer-Verlag,Berlin.Dahl, V. and Patrick, S, (1985) Natural Language Understading and Logic ProgrammingNorth-Hol land, Amsterdam.DeGroot, D. and Gary L.(1986) Logic Progamming : Fuctions, Relations, and EquationsPrentice-Hall, Englewood Cliffs, New Jersey.Fenstad, J. et al. (1987) Situation, Language and Logic, D. Reidel, Dordrechet.Fenstad, J. et al. (1990) "Computational Semantics: stpes towards "intelligent" textprocessing," Computational Semantics, COSMOS-Report No. 15.Gogeun, J.A. (1988) "What is Unification? A Categorical View of Substitution,Equation and Solution," Report No. CSLI-88-124, Stanford.Halvorsen, P. and Jan T.L. (1991) " Computational Semantics Lectures Slides,"LSA Linguistic Institute, Snata Cruz,Kowalski, R. (1979) Logic for Problem Solving, North Holland, New York.Lee, K. (1989) "The problem of meaning in natural language processing [written inKoreanl," Language Research (Seoul Nat'l Univ. ), 25,1: 225-238.Lee, K. (1992) "An analysis of some fragments of Korean for computationalsemantics," Festschrift for Prof. Tae-Oak Kim, (in preparation).Montague, R. (1973) "The proper treatment of quantification in ordinary English."in Hintikka, et al. eds. (1986) Approaches to Natural Language, 221-242,D. Reidel, Dordrecht.Moore, R.C.(1989)"Unification-based semantic interpretation," 27th Annual Meeting-ofthe Association for Computational Linguistics, 33-41.Pereira, F e C. N. and Stuart M.S. (1987) Prolog and	 Natural-Language Analysis,Center for the Study of Language and Information, Stanford.Yoo, S. (1989) Situation Semantics and Extended Categorial Grammar, HanshinPublishing Co. Seoul.

Computing Information by Equation Solving

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Computing Information by Equation Solving

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