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<Paper uid="J90-1002">
  <Title>AN INTERPRETATION OF NEGATION IN FEATURE STRUCTURE DESCRIPTIONS</Title>
  <Section position="2" start_page="0" end_page="0" type="intro">
    <SectionTitle>
1 INTRODUCTION
</SectionTitle>
    <Paragraph position="0"> A number of linguistic theories and computational approaches to parsing natural language have employed the notion of associating informational elements, consisting of features and their values, with phrases. Such elements, called feature structures, have been used in linguistic theories, such as Generalized Phrase Structure Grammar (GPSG; Gadzar et al. 1985) and Lexical Functional Grammar (Kaplan and Bresnan 1983), and in computational formalisms, such as Functional Unification Grammar (Kay 1979) and PATR-II (Shieber 1984).</Paragraph>
    <Paragraph position="1"> Rounds and Kasper introduced a logical formalism to describe feature structures with disjunctive specification (Kasper 1987; Kasper and Rounds 1986; Rounds and Kasper 1986). The language is a form of modal propositional logic. To define the semantics of this language, feature structures are formally defined as acyclic finite automata. The detailed definition is given in Section 2. A fundamental property of the semantics is that it is monotonic in the sense that the set of automata satisfying a given formula is upward-closed under the operation of subsumption. This is important, because we consider a formula to be only a partial description of a feature structure. This property is precisely formulated in Section 2.</Paragraph>
    <Paragraph position="2"> Several researchers have expressed a need for extending this logic to include the operators of negation and implication. These two are related in that, in most logical systems, it is possible to use one to define the other (in the presence of a disjunction operator). In this paper, we shall concentrate on the problem of extending the logic to include negation, while also showing that it yields a satisfactory interpretation of implication.</Paragraph>
    <Paragraph position="3"> Karttunen (1984), for instance, provides examples of feature structures in which a negation operator might be useful. For instance, the most natural way to represent the number and person attributes of a verb such as sleep would be to say that it is not third person singular, rather than expressing it as a disjunction of the other possibilities. We express this agreement constraint by the following formula: agreement : -1 (person : third A number : singular) (1) Pereira (1987) provides the following example formula that expresses the semantic constraint that the subject and object of a clause cannot be coreferential unless the object is a reflexive pronoun: Computational Linguistics Volume 16, Number 1, March 1990 11 Anuj Dawar and K. Vijay-Shanker An Interpretation of Negation in Feature Structure Descriptions obj : type : reflexive k/-7 (subj : ref ~ obj : ref) (2) This constraint can, in fact, be represented just as naturally as the following implication: (subj : ref ~ obj : ref) ~ obj : type : reflexive (3) Similarly, the feature co-occurrence constraints in GPSG (Gadzar et el. 1985) include implications of the form 3 l ~ q~ (where q~ is some description). While a formula of the form 3 l is not part of the Rounds-Kasper logic, we intend it here as asserting the existence of a feature l in a structure. This would normally be expressed in the Rounds-Kasper formalism by the formula I:NIL. As we see later, formulae of the kind we have in this example, i.e. in which an existential appears negated, require special treatment and will motivate an extension to the feature structure formalism.</Paragraph>
    <Paragraph position="4"> Various interpretations have been suggested that define a semantics for these operators (see Section 3), but none has gained universal acceptance. Pereira (1987) set forth certain properties that any such interpretation should satisfy. We suggested that three-valued logic provides us with a framework appropriate for defining the semantics of a feature description logic (which we will call FDL) that includes a negation operator (Dawar and Vijay-Shanker 1989). We also showed that the three-valued framework (based on Kleene's three-valued logic; Kleene 1952) is powerful enough to express most of the existing definitions of negation and implication. It is therefore possible to compare these different approaches. We also presented one particular three-valued interpretation for FDL, motivated by the approach to negation given by Karttunen (1984), that meets the conditions stated by Pereira.</Paragraph>
    <Paragraph position="5"> In the present work, we give an exposition of these results, and we also examine another three-valued interpretation for FDL, obtained by using a modified notion of the feature structures that serve as models. This new interpretation, while preserving the desirable properties of the previous one, also provides a satisfactory semantics for the problematic case, mentioned above, of formulae with a negated existential.</Paragraph>
    <Paragraph position="6"> In Section 2 we present an exposition of the Rounds-Kasper logic. In Section 3 we examine some existing approaches to defining the semantics of negation, and we also present the framework of three-valued logic within which we define our own interpretation. In Section 4 we exhibit the modified notion of feature structures as models for FDL, and we give the semantics of FDL in terms of these modified feature structures. Finally, in Section 5, we present a proof system for the language and esta61ish some computational results.</Paragraph>
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