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Citation Fernando C. N. Pereira and Stuart M. Shieber. The semantics of grammar formalisms seen as computer languages. In Proceedings of the Tenth International Conference on Computational , pages 123-129, Stanford University, Stanford, California, July 2-6 1984.

Published Version http://dx.doi.org/10.3115/980491.980518

Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:2309658

Terms of Use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Other Posted Material, as set forth at http:// nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of- use#LAA The Semantics of Grammar Formalisms Seen as Computer Languages Fernando C. N. Pereira and Stuart M. Shieber Artificial Intelligence Center SRI International and Center for the Study of Language and Information Stanford University

Abstract a declarative semantics z extended in the natural way from the declarative semantics of context-free . The design, implementation, and use of grammar for- The last point deserves amplification. Context-free ma]isms for natural language have constituted a major grammars possess an obvious declarative semantics in branch of coml)utational linguistics throughout its devel- which nonterminals represent sets of strings and rules rep- opment. By viewing grammar formalisms as just a spe- resent n-ary relations over strings. This is brought out by cial ease of computer languages, we can take advantage of the reinterpretation familiar from theory the machinery of denotational semantics to provide a pre- of context-free grammars as polynomials over concatena- cise specification of their . Using Dana Scott's do- tion and set union. The grammar formalisms developed main theory, we elucidate the nature of the feature systems from the definite-clause subset of first order are the used in augmented phrase-structure grammar formalisms, only others used in natural-language analysis that have in particular those of recent versions of generalized phrase been accorded a rigorous declarative semantics--in this structure grammar, lexical functional grammar and PATR- case derived from the declarative semantics of logic - I1, and provide a (lcnotational semantics for a simple gram- grams [3,12,1 I]. mar formalism. We find that the mathematical structures Much confusion, wasted effort, and dissension have re- developed for this purpose contain an operation of feature sulted from this state of affairs. In the absence of a rigorous generalization, not available in those grammar formalisms, semantics for a given grammar formalism, the user, critic, that can be used to give a partial account of the effect of or implementer of the formalism risks misunderstanding the coordination on syntactic features. intended interpretation of a construct, and is in a poor posi- tion to compare it to alternatives. Likewise, the inventor of a new formalism can never be sure of how it compares with 1. Introduction I existing ones. As an example of these dillqculties, two sim- ple changes in the implementation of the ATN formalism, The design, implementation, and use of grammar for- the addition of a well-formed substring table and the use malisms for natural lang,age have constituted a major of a bottom-up strategy, required a rather subtle branch of computational linguistics throughout its devel- and unanticipated reinterpretation of the register-testing opment. Itowever, notwithstanding the obvious superfi- and -setting actions, thereby imparting a different meaning cial similarily between designing a grammar formalism and to grammars that had been developed for initial top-down designing a programming language, the design techniques backtrack implementation [22]. used for grammar formalisms have almost always fallen Rigorous of grammar formalisms can and short with respect to those now available for programming should be made available. Looking at grammar formalisms language design. as just a special case of computer languages, we can take advantage of the machinery of denotational semantics [20i Formal and computational linguists most often explain to provide a precise specification of their meaning. This the effect of a grammar formalism construct either by ex- approach can elucidate the structure of the data objects ample or through its actual operation in a particular im- manipulated by a formalism and the mathematical rela- plementation. Such practices are frowned upon by most tionships among various formalisms, suggest new possibil- programming-language designers; they become even more ities for linguistic analysis (the matter of the for- dubious if one considers that most grammar formalisms malisms), and establish connections between grammar for- in use are based either on a context-free skeleton with malisms and such other fields of research as programming- augmentations or on some closely related device (such as 2This use of the term "semantics" should not be confused with the ATNs), consequently making them obvious candidates for more common usage denoting that portion of a grammar concerned IThe research reported in this paper has been made possible by a gift with the meaning of sentences. Here we are concerned with the from the System Development Foundation. meaning of the metalanguage.

123 language design and theories of abstract data types. This eral may not terminate and may therefore never produce a last point is particularly interesting because it opens up fully defined output, although each individual step may be several possibilities--among them that of imposing a type adding more and more information to a partial discipline on the use of a formalism, with all the attendant of the undeliverable output. advantages of compile-time error checking, modularity, and Domain theory is a mathematical theory of consider- optimized compilation techniques for grammar rules, and able complexity. Potential nontermination and the use of that of relating grammar formalisms to other knowledge functions as "first-class citizens" in computer languages ac- representation languages [l]. count for a substantial fraction of that complexity. If, as is As a specific contribution of this study, we elucidate the case in the present work, neither of those two aspects the nature of the feature systems used in augmented phrase- comes into play, one may be justified in asking why such structure grammar formalisms, in particular those of recent a complex apparatus is used. Indeed, both the semantics versions of generalized (GPSG) of context-free grammars mentioned earlier and the seman- [5,15], lexical functional grammar (LFG) [2] and PATR-II tics of logic grammars in general can be formulated using [ 18,17]; we find that the mathematical structures developed elementary [7,21]. for this purpose contain an operation of feature generaliza- However, using the more complex machinery may be tion, not available in those grammar formalisms, that can beneficial for the following : be used to give a partial account of the effect of coordina- tion on syntactic features. • Inherent partiality:, many grammar formalisms oper- ate in terms of constraints between elements that do Just as studies in the semantics of programming lan- not fully specify all the possible features of an ele- guages start by giving semantics for simple languages, so ment. we will start with simple grammar formalisms that capture the essence of the method without an excess of obscuring • Technical economy, results that require laborious detail. The present enterprise should be contrasted with constructions without utilizing domain theory can be studies of the generative capacity of formalisms using the reached trivially by using standard results of the the- techniques of formal language theory. First, a precise defini- ory. !;ion of the semantics of a formalism is a prerequisite for such • Suggestiveness: domain theory brings with it a rich generative-capacity studies, and this is precisely what we mathematical structure that suggests useful opera- are trying to provide. Second, generative capacity is a very tions one might add to a grammar formalism. coarse gauge: in particular, it does not distinguish among different formalisms with the same generative capacity that • Eztensibilit~. unlike a domain-theoretic account, a may, however, have very different semantic accounts. Fi- specialized semantic account, say in terms of sets, nally, the tools of formal language theory are inadequate to may not be easily extended as new constructs are describe at a sufficiently abstract level formalisms that are added to the formalism. based on the simultaneous solution of sets of constraints [9,10]. An abstract analysis of those formalisms requires a 3. The Domain of Feature Struc- notion of partial information that is precisely captured by the constructs of denotationai semantics. tures

We will start with an abstract denotational description 2. Denotational Semantics of a simple feature system which bears a close resemblance to the feature systems of GPSG, LFG and PATR-II, al- In broad terms, denotational semantics is the study of though this similarity, because of its abstractness, may not the connection between programs and mathematical enti- be apparent at first glance. Such feature systems tend to ties that represent their input-output relations. For such use data structures or mathematical objects that are more an account to be useful, it must be compositional, in the or less isomorphic to directed graphs of one sort or an- sense that the meaning of a program is developed from the other, or, as they are sometimes described, partial func- meanings of its parts by a fixed set of mathematical oper- tions. Just what the relation is between these two ways ations that correspond directly to the ways in which the of viewing things will be explained later. In general, these parts participate in the whole. graph structures are used to encode linguistic information For the purposes of the present work, denotational se- in the form of attribute-vahm pairs. Most importantly, par- mantics will mean the semantic domain theory initiated tial information is critical to the use of such systems--for by Scott and Strachey [20]. In accordance with this ap- instance, in the variables of definite clause grammars [12] proach, the meanings of programming language constructs and in the GPSG analysis of coordination [15]. That is, the are certain partial mappings between objects that represent elements of the feature systems, called fealure struclures partially specified data objects or partially defined states of (alternatively, feature bundles, f-structures [2], or terms} computation. The essential idea is that the meaning of a can be partial in some sense. The partial , be- construct describes what information it adds to a partial ing in a domain of attributes and complex values, tend to be description of a data object or of a state of computation. equational in nature: some feature's value is equated with Partial descriptions are used because computations in gen- some other value. Partial descriptions can be understood

124 in one of two w:ays: either the descriptions represent sets between elements of Con and elements of D and A is a of fully specilied elements of an underlying domain or they special least informative element that gives no information are regarded as participating in a relationship of partiality at all. We say that a subset S of D is deductively closed with respect to each other. We will hold to the latter view if every proposition entailed by a consistent subset of S is here. in S. The deductive closure -S of S ___ /9 is the smallest What are feature structures from this perspective? deductively closed subset of/9 that contains S. They are repositories of information about linguistic enti- The descriptor equations discussed earlier are the ties. In domain-theoretic terms, the underlying domain of propositions of the information system for feature structure feature structures F is a recursive domain of partial func- descriptions. Equations express constraints among feature tions from a set of labels L (features, attribute , at- values in a feature structure and the entailment relation tributes) to complex values or primitive atomic values taken encodes the reflexivity, symmetry, transitivity and substi- from a set C of constants. Expressed formally, we have the tutivity of equality. More precisely, we say that a finite set domain equation of equations E entails an equation e if • Membership: e E E F=IL~F]+G • Reflezivit~. e is A or d = d for some descriptor d The solution of this domain equation can be understood as • Symmetry. e is dl = d2 and dz = dl is in E a set of trees (finite or infinite} with branches labeled by elements of L, and with other trees or constants as nodes. • Transitivity. e is da = dz and there is a descriptor d The branches la .... , Im from a node n point to the values such that dl = d and d = dz are in E n{lt),..., n(Im) for which the node, as a partial function, is • Substitutivit~r. e is dl = Pl • d2 and both pl = Pz and defined. dl = P2 • d.~ are in E • Iteration: there is E' C E such that E' b e and for all 4. The Domain of Descriptions e'E~ EF-e' With this notion of entailment, the most natural What the grammar formalism does is to talk about F, of the set Con is that a finite subset E of 19 is consistent if not in F. That is, the grammar formalism uses a domain of and only if it does not entail an inconsistent equation, which descriptions of elements of F. From an intuitive standpoint, has the form e~ = cz, with et and Cz as distinct constants. this is because, for any given phrase, we may know facts An arbitrary subset of/9 is consistent if and only if all about it that cannot be encoded in the partial function its finite subsets are consistent in the way defined above. associated with it.. The consistent and deductively closed subsets of D ordered A partial description of an element n of F will be a set by inclusion form a complete partial order or domain D, of equations that constrain the values of n on certain labels. our domain of descriptions of feature structures. In general, to describe an element z E F we have equations Deductive closure is used to define the elements of D of the following forms: so that elements defined by equivalent sets of equations are (... (xII. })-..)ll;.) = (..-(z(li,))...)(l;.) the same. In the rest of this paper, we will specify elements of D by convenient sets of equations, leaving the equations (".(x{li,))".)(li,~) = ck , in the closure implicit. which we prefer to write as The inclusion order K in D provides the notion of a description being more or less specific than another. (t~,...I;.) = (Ij,..-i;.) The least-upper-bound operation 12 combines two descrip- (li,"'li=) = ck tions into the least instantiated description that satisfies the equations in both descriptions, their unification. The with x implicit. The terms of such equations are constants greatest-lower-bound operation n gives the most instanti- c E C' or paths {ll, ". It=), which we identify in what follows ated description containing all the equations common to with strings in L*. Taken together, constants and paths two descriptions, their generalization. comprise the descriptors. The foregoing definition of consistency may seem very Using Scott's information systems approach to domain natural, but it has the technical disadvantage that, in gen- construction [16], we can now build directly a characteriza- eral, the union of two consistent sets is not itself a consistent tion of feature structures in terms of information-bearing set; therefore, the corresponding operation of unification elements, equations, that engender a system complete with may not be defined on certain pairs of inputs. Although notions of compatibility and partiality of information. this does not cause problems at this stage, it fails to deal The information system D describing the elements of with the fact that failure to unify is not the same as lack of F is defined, following Scott, as the tuple definition and causes technical difficulties when providing rule denotations. We therefore need a slightly less natural D = (/9,A, Con, ~-) , definition. where 19 is a set of propositions, Con is a set of finite subsets First we add another to the specification of of P, the consistent subsets, I- is an entailment relation the entailment relation:

125 • Falsitv. if e is inconsistent, {e} entails every element cycles when viewed as a directed graph--then the resulting of P. feature tree will be infinite2 - That is, falsity entails anything. Next we define Con to be Stated more precisely, an element f of a domain is fi- simply the set of all finite subsets of P. The set Con no nite, if for any ascending sequence {d~} such that f E_ U~ d~, longer corresponds to sets of equations that are consistent there is an i such that f C_ d~. Then the cyclic elements in the usual equational sense. of D are those finite elements that are mapped by 6 into With the new definitions of Con and I-, the deductive nonfinite elements of F. closure of a set containing an inconsistent equation is the whole of P. The partial order D is now a lattice with top 5. Providing a Denotation for a element T = P, and the unification operation t_l is always defined and returns T on unification failure. Grammar We can now define the description mapping 6 : D --* F that relates descriptions to the described feature structures. We now move on to the question of how the domain D The idea is that, in proceeding from a description d 6 D to is used to provide a denotational semantics for a grammar a feature structure f 6 F, we keep only definite informa- formalism. tion about values and discard information that only states We take a simple grammar formalism with rules con- value constraints, but does not specify the values them- sisting of a context-free part over a nonterminal vocabu- selves. More precisely, seeing d as a set of equations, we lary .t/= {Nt,..., Ark} and a set of equations over paths in consider only the subset LdJ of d with elements of the form ([0..c~]- L*)0C. A sample rule might be

(l~-..lm)=c~ . . S ~ NP VP (o s,,bj) = (I) Each e 6 [d] defines an element f(e) of F by the equations (o predicate) = (2) (1 agr) = (2 agr) f(e)(l,) = f, This is a simplification of the rule format used in the PATR- II formalism [18,17]. The rule can be read as "an S is an fi-,(li) ---- fl NP followed by a VP, where the subject of the S is the

f,._,(l,.) = ek , NP, its predicate the VP, and the agreement of the NP the same as the agreement of tile VP'. with each of the f~ undefined for all other labels. Then, we More formally, a grammar is a quintuple G = can define 6(d) as (//, S, L, C, R), where • ,t/is a finite, nonempty set of nonterminals Nt,..., Nk 6(d) = L] f(e) • S is the set of strings over some alphabet (a fiat do- ~eL~l main with an ancillary continuous function concate- This description mapping can be shown to be continu- nation, notated with the symbol .). ous in the sense of domain theory, that is, it has the prop- • R is a set of pairs r = (/~0 ~ N,, .. . N,., E~), erties that increasing information in a description leads where E. is a set of equations between elements of to nendecreasing information in the described structures ([0..m] - L') 0 C. {monotonieity) and that if a sequence of descriptions ap- As with context-free grammars, local ambiguity of a proximates another description, the same condition holds grammar means that in general there are several ways of for the described structures. assembling the same subphrases into phra.ses. Thus, the Note that 6 may map several elements of D on to one semantics of context-free grammars is given in terms of element of F. For example, the elements given by the two sets of strings. The situation is somewhat more compli- sets of equations cated in our sample formalism. The objects specified by the grammar are pairs of a string and a partial description. Because of partiality, the appropriate construction cannot (fh) = c (gi) = e be given in terms of sets of string-description pairs, but rather in terms of the related domain construction of pow- describe the same structure, because the description map- erdomains [14,19,16]. We will use the Hoare powerdomain ping ignores the link between (f h) and (g i) in the first P = PM(S x D) of the domain S x D of string-description description. Such links are useful only when unifying with pairs. Each element of P is an approximation of a transdue- further descriptive elements, not in the completed feature tion relation, which is an association between strings and structure, which merely provides feature-value assignments. their possible descriptions. Informally, we can think of elements of D as directed We can get a feeling for what the domain P is doing rooted graphs and of elements of F as their unfoldings as by examinin~ our notion of lexicon. A lexicon will be an trees, the unfolding being given by the mapping 6. It is SMote precisely a rational tree, that is, a tree with a finite number of worth noting that if a description is cyclic---that is, if it has distinct subtrees.

126 element of the domain pk, associating with each of the k rest of the work of applying a rule, extracting the result, is nonterminals N;, I < i < k a transduction relation from the done by the projection and deindcxing steps• corresponding coordinate of pk. Thus, for each nontermi- The four steps for applying a rule nal, the lexicon tells us what phrases are under that non- terminal and what possible descriptions each such phrase r = (N,, --* U,, . .. N,.., E,) has. llere is a sample lexicon: to string-description pairs (s,,d,} ..... (sk,dk} are as fol- {"Uther", } lows. First, we index each d,, into d~ by replacing every {(agr n,tm) = sg, (agr per) = 3}) • . . • . $ • NP: path p m any of tts equatmns with the path I " P. We ("many knights", then combine these indexed descriptions with the rule by {

127 DCG-IIa PATR-II LFG GPSGb features of the head children. Formally, we have: FEATURE SYSTEM full finite finite nonrec. CF SKELETON full full off-line full (0 head) ---- [7 (i head) aDCGs based on Prolog-lI which allows cyclic terms. i~heads of 0 bHPSG, the current Hewlett-Packard implementation derived from GPSG, would come more accurately under the PATR-II In the case of parents with one head child, this final HFC classification. reduces to the old HFC requiring identity; it reduces to the newer one, however, in cases {like coordinate structures} Figure 1: Summary of Grammar System Properties where there are several head constituents. Furthermore, by utilizing an order structure on the do- Though notational differences and some grammatical main of constants C, it may be possible to model that trou- devices are glossed over here, the comparison is useful as blesome coordination phenomenon, number agreement in a first step in unifying the various formalisms under one coordinated noun phrases [8,15]. semantic umbrella. Furthermore, this analysis elicits the need to distinguish carefully between the domain of fea- ture structures F and that of descriptions. This distinction 7. Conclusion is not clear in the published accounts of GPSG and LFG, which imprecision is responsible for a number of uncertain- We have approached the problem of analyzing the ties in the interpretation of operators and conventions in meaning of grammar formalisms by applying the techniques those formalisms. of denotational semantics taken from work on the semantics of computer languages. This has enabled us to In addition to formal insights, linguistic insights have • account rigorously for intrinsically partial descrip- also been gleaned from this work. First of all, we note tions, 'that while the systems make crucial use of unification, gen- eralization is also a well-defined notion therein and might • derive directly notions of unification, instantiation indeed be quite useful. In fact, it was this availability of the and generalization, generalization operation that suggested a simplified account • relate feature systems in linguistics with type systems of coordination facts in English now being used in GPSG in , [15] and in an extension of PATR-II [8]. Though the issues • show that feature systems in GPSG, I, FG and PATR- of coordination and agreement are discussed in greater de- II are special cases of a single construction, tail in these two works, we present here a simplified view of the use of generalization in a GPSG coordination analysis. • give semantics to a variety of mechanisms in grammar formalisms, and Circa 1982 GPSG [6] analyzed coordination by using a • introduce operations for modeling linguistic phenom- special principle, the conjunct realization principle (CRP), ena that have not previously been considered. to achieve partial instantiation of head features {including agreement} on the parent category. This principle, together We plan to develop the approach further to give ac- with the head feature convention (HFC) and control agree- counts of negative and disjunctive constraints [8], besides ment principle {CAP), guaranteed agreement between the the simple equational constraints discussed here. head noun of a subject and the head verb of a predicate in On the basis of these insights alone, it should be clear English sentences. The HFC, in particular, can be stated that the view of grammar formalisms as programming lan- in our notation as (0 head) = (n head) for n the head of 0. guages offers considerable potential for investigation. But, even more importantly, the linguistic discipline enforced A more recent analysis [4,15] replaced the conjunct re- by a rigorous approach to the design and analysis of gram- alization principle with a modified head feature conven- mar formalisms may make possible a hitherto unachievable tion that required a head to be more instantiated than the standard of research in this area. parent, that is: (0 head) E (n head) for all constituents n which are heads of 0. Making coordinates heads of their parent achieved the effect of the CRP. Unfortunately, since the HFC no longer forced identity of agreement, a new principle--the completeness principle (NCP), [1] Ait-Kaci, H. "A New Model of Computation Based which required that NP's be fully instantiated--was re- on a Calculus of Type Subsumption." Dept. of Com- quired to guarantee that the appropriate agreements were puter and Information Science, Univer:ity of Penn- maintained. sylvania (November 1983). Making use of the order structure of the domains we [2] Bresnan, J. and R. Kaplan. "Lexical-Functional have just built, we can achieve straightforwardly the effect Grammar: A for Granmlatical Repre- of the CRP and the old HFC without any notion of the sentation." In J. Bresnan, Ed., The ,%Icntal Represen- NCP. Our final version of the HFC merely requires that tation of Grammatical Relations, MIT Press, Cam- the parent's head features be the generalization of the head bridge, Massachusetts (1982), pp. 173-281.

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