A Logical Semantics for Feature Structures

Unification-based grammar formalisms use structures containing sets of features to describe linguistic objects. Although computational algorithms for unification of feature structures have been worked out in experimental research, these algcwithms become quite complicated, and a more precise description of feature structures is desirable. We have developed a model in which descriptions of feature structures can be regarded as logical formulas, and interpreted by sets of directed graphs which satisfy them. These graphs are, in fact, transition graphs for a special type of deterministic finite automaton. This semantics for feature structures extends the ideas of Pereira and Shieber [11], by providing an interpretation for values which are specified by disjunctions and path values embedded within disjunctions. Our interpretati6n differs from that of Pereira and Shieber by using a logical model in place of a denotational semantics. This logical model yields a calculus of equivalences, which can be used to simplify formulas. Unification is attractive, because of its generality, but it is often computations/]), inefficient. Our mode] allows a careful examination of the computational complexity of unification. We have shown that the consistency problem for formulas with disjunctive values is NP-complete. To deal with this complexity, we describe how disjunctive values can be specified in a way which delays expansion to disjunctive normal form. 1 B a c k g r o u n d : U n i f i c a t i o n in G r a m m a r Several different approaches to natural language grammar have developed the notion of feature structures to describe linguistic objects. These approaches include linguistic theories, such as Generalized Phrase Structure Grammar (GPSG) [2], Lexical Functional Grammar (LFG) [4], and Systemic Grammar [3]. They also include grammar formalisms which have been developed as computational tools, such as Functional Unification Grammar (FUG) [7], and PATR-II [14]. In these computational formalisms, unificat/on is the primary operation for matching and combining feature structures. Feature structures are called by several different names, including f-structures in LFG, and functional descriptiona in FUG. Although they differ in details, each approach uses structures containing sets of attributes. Each attribute is composed of a label/value pair. A value may be an atomic symbol, hut it may also be a nested feature structure. The intuitive interpretation of feature structures may be clear to linguists who use them, even in the absence of a precise definition. Often, a precise definition of a useful notation becomes possible only after it has been applied to the description of a variety of phenomena. Then, greater precision may become necessary for clarification when the notation is used by many different investigators. Our model has been developed in the context of providing a precise interpretation for the feature structures which are used in FUG and PATR-II. Some elements of this logical interpretation have been partially described in Kay's work [8]. Our contribution is to give a more complete algebraic account of the logical properties of feature structures, which can be used explicitly for computational manipulation and mathematical analysis. Proofs of the mathematical soundness and completeness of this logical treatment, along with its relation to similar logics, can be found in [12]. 2 D i s j u n c t i o n a n d N o n L o c a l V a l u e s Karttunen [5] has shown that disjunction and negation are desirable extensions to PATR-II which are motivated by a wide range of linguistic