AN IMPLEMENTATION OF FORMAL SEMANTICS IN THE FORMALISM OF RELATIONAL DATABASES. Claire VANDERHOEFT D~partement de Linguistique G~n(rale, CP175 UniversitY, Libre de Bruxelles 50, avenue Roosevelt 1050 Bruxelles Belgique E-mail: [email protected] I describe the database. Then, I explain how the principles used to design the database meet ABSTRACT the requirements of formal semantics. The fourth part is concerned with entailment while This paper presents an implementation the last part mainly shows how one proceeds to of formal semantics as described in Keenan interpret sentences. and Fa!tz's Boolean Semantics for Natural Langt~age [4]. The main characteristic of this implementation is that it avoids the intermediate step of translating NL into a formal language, 1. BACKGROUND. such as an extended version of predicate calculus. My choice of not using any The topics which this work is intermediate language, which Montague already concerned with have mainly been studied from suggested in Universal Grammar [5], makes three points of view. my implementation free of the problems related A first class of studies covers the to the syntax of such a language like binding problems encountered in trying to translate NL the variables and resolving scope ambiguities. into a formal language. On the one hand, there On the other hand, not translating NL into an is theoretical research aiming at such a intermediate language requires every translation, like PATR [7]. On the other hand, denotation (i.e. semantic value) to be various kinds of inaccuracies of NL explicitly and accurately represented in a translations into logical form in view of database. accessing databases have been discussed, see [6] for example. A second field of research that need be 0. INTRODUCTION. mentioned is concerned with NL interfaces. Famous systems are described in [9] and [1]. In extensional semantics, each There are important differences between these denotation corresponds to an object of the systems and my work since I am not aiming at world° The world is the set of all the accessing a knowledge base at all. The denotations. In the implementation that I shall database that I use encodes NL meanings and it present in this paper, the world will be does so according to linguistic constraints. represented by means of a database, more Traditionally, the database rather encodes a precisely a relational database. certain knowledge independent of the language The structure of the database is used to talk about it. Problems specific to NL designed in such a way that it makes explicit interfaces can be found in [3] and [81. the semantic type of each denotation. From another point of view, there are Although I will not always stick to the standard works which are concerned with the question version of formal semantics when assigning of the organisation of the knowledge base semantic types to syntactic categories, I aim at constituted by NL meanings, see [2]. The accounting for the same range of phenomena difference between my approach and ones like that formal semantics deals with. [2], is that I am sticking to the theory of formal The paper will be divided into five semantics. Consequently, I do not (yet) parts. First, I shall trace back research results address questions about the structure of the to which my contribution can be related. Next, i 377 lexicon nor do I treat pragmatic phenomena like corresponding type. For example, the relation common sense inferences. which corresponds to the type of noun phrases has an attribute whose values are noun phrase denotations. Each denotation is an atomic value of the attribute of the relation. 2. THE DATABASE. Furthermore, each such value actually belongs to an n-tuple belonging to the extension of the The structure of the database is relation. (This is due to the fact that the set- dependent on the semantic properties of the theoretical model of the world, i.e. the denotations. More specifically, the structure of database, contains all the denotations built on the database is dependent on the fact that an ontology constituted by a set of entities, IJ, denotations are classified into different types and the truth values.) and specifically recognized as the denotations The structure of the database captures of such and such syntactic categories. the (degree of) complexity of a denotation by Each denotation of each constituent is a connecting the relation which represents the value in the database. Some of the denotations semantic type assigned to the corresponding result from the composition of other syntactic category with the relations which denotations. Which denotations can be represent the semantic types assigned to the composed with which other ones are properties constituents of a complex expression of the of their type. These properties are not encoded same syntactic category. For example, a as such. The overall structure of the database proper name has a complex denotation because shows how the semantic types combine with it belongs to the syntactic category of noun each other. Consequently, complex phrases. Therefore, its denotation belongs to denotations (denotations of complex the extension of the relation representing the expressions) are represented by atomic type of noun phrases. Since there are noun values, but the fact that they are complex is phrases constituted by a determiner and a deduced from the structure of the database. common noun, the relation representing the Consider the case of noun phrase denotations. type of noun phrases actually connects to the The denotation of a determiner combines with relations representing respectively the types of the denotation of a common noun. This determiners and of common nouns. Hence, the combination yields the denotation of a noun structure of the relation associated to the type of phrase, i.e., an atomic value in the database. noun phrases and, in particular, of proper The representation of this denotation is names, shows that they are complex connected (in the sense of relational expressions. databases) to the representations of the denotations of the noun and of the determiner. To the extent that we need to define the Therefore, it can be recognized as a complex connections which show the respective denotation. complexity of each type of denotation, a relation is actually defined for each semantic The design of the database is dependent type. For example, we shall define relations on the fact that we need an explicit means to like Tn, Tdet, Tnp, Tvp standing, respectively, recognize the type of each denotation for the type of denotations of common nouns, represented in it. Within the formalism of determiners, noun phrases, verb phrases. relational databases, defining types of Still, there is a problem in defining denotations amounts to defining a relation for connections. The problem is that, in formal each such type. semantics, expressions of different syntactic A relation is formally defined as an n- categories can have the same semantic type. tuple of formal attributes. By formal For example, common nouns and intransitive attribute is meant a way to identify the verbs share the same type. Now, the attribute (a position in the relation or a name) complexity of the denotation of a noun phrase and the definition of the set of its possible is encoded in the fact that it is connected to the values. The extension of a relation is the set denotation of a common noun. On the of all well-formed n-tuples of attribute values contrary, the denotation of a verb phrase must for the corresponding formal attributes. (An be connected to the denotation of a simple verb ill-formed n-tuple has at least one non-possible and to the denotations of complements. In value for a formal attribute.) general, we not only need to define relations as counterparts of semantic types, but, where Relations each represent a type. Each expressions of different syntactid categories of them has (at least) one attribute whose collapse into the same type, their types must domain is the set of denotations of the nevertheless correspond to different relations. 3 7~ 2 With respect to the example, the relation Tv - ii) relations that correspond to types of non° (the type; of simple verbs) cannot be the same lexical categories have one attribute whose as the relation Tn, because Tvp connects to Tv domain is the set consisting of all the while Tnp connects to Tn. denotations of all these expressions. Notice that it is true of all the syntactic Moreover, they have other attributes, one for categories that they have one and only one each of their constituents. These attributes relation as semantic counterpart. Sometimes have as domain the extension of the relations however, the relation could be defined as the corresponding to the types of these sum of several relations. For example the constituents. complex relation Tvp has several mutually These principles ensure that denotations exclusive sets of connections. It connects to are submitted to the principle of the relation Tv and the relation Tnp, or to the compositionality which states that the relation Tv and the relation Tpp or the relation denotation of a complex expression is a Tvp and the relation Tpp, etc. compound of the denotations of' its constituents. (It is important to understand that Let me illustrate these principles by we want to be able to check that denotations are showing the definition of two relations. The submitted to compositionality.) How Trip relation is defined as a triple of formal compositionality constrains the definitions of attributes: the relations will now be illustrated on Tn and Tnp= <[np],Tn,Tdet> Tnp. where it is understood that the possible values In the extension of a relation like Tn, all of the first attribute are noun phrase the pairs of values have the property that the denotations, the possible values of the second first value is the denotation of the second one.