A General Reasoning Scheme for Underspeci ed Representations

In this paper we present an underspeci ed logic, i.e. a pair consisting of a proper underspeci ed semantic representation formalism and a deductive component that directly operates on these structures. We show how the main features of this formalism can be imported into other existing underspeci ed semantic representation formalisms. We also show that deduction rules may be imported there along the same lines. The set of importable rules will of course depend on the completeness properties of the particular formalisms. The current paper is a short version of [10]. 1 The landscape of Underspeci ed Semantic Representations Underspeci ed semantic representations have attracted increasing interest within computational linguistics. Several formalisms have been developed that allow to represent sentence or text meanings with that degree of speci city that is determined by the context of interpretation. As the context changes they must allow for (partial) disambiguation steps performed by a process of re nement that goes hand in hand with the construction algorithm. And as the interpretation of phrases often 1 relies on deductive principles and thus any construction algorithm must be able to integrate the results of deductive processes, any semantic formalism should be equipped with a deductive component that operates directly on its semantic forms. We call a meaning of a representation formalism L underspeci ed, if it represents an ambiguous natural language sentence or text in a more compact manner than by a disjunction of all its readings. L is called semantic if its representations are model-theoretically interpretable or if it comes with a disambiguation device that turns underspeci ed representations into sets of model-theoretically interpretable representations. 2 If L's disambiguation steps produce representations of L only, then L is called closed. And if L's disambiguation device produces all possible re nements of any , then L is called complete. Completeness is of course dependent on the particular natural language fragment L is supposed to cover. In this paper we restrict ourselves to the fragment of simple sentences containing singular inde nite as well as quanti ed NPs, 3 relative clauses and negation. To give an example what completeness involves let us consider a sentence with three quanti ed NPs with underspecifed scoping relations. Then L must be able to represent all 2 3! = 64 re nements, i.e. partial and complete disambiguations of this sentence. For many formalisms the question whether they are complete wrt. to a particular fragment, or not, is not decided yet. We, therefore, take a very liberal view and interpret 'complete' more in the sense of 'intended to be comlete' than in the sense of a precise characterization of expressive power. A formalism L is called proper if it is closed and complete. It is c-deductive (or 'classically deductive') if there is an inference mechanism for the disjunction of fully speci ed formulas the underspeci ed formula is supposed to represent. The formalism is called u-deductive if it is equipped with a deductive component that operates directly on the underspeci ed forms. 1 E.g. in order to apply nominal and temporal resolution, consistency checks, integration of world knowledge or other non-compositional interpretation principles. 2 Note that the second disjunct requires that either the underspeci ed representations themselves or the disambiguation algorithm is subject to certain demands on wellformedness, as, e.g., the so-called 'free-variable constraint' ([11], [9]). Although we think that this is a very important distinction (in particular under computational aspects) we do not distinguish here between those formalisms which are provided with a check of such meta-level constraints directly for underspeci ed representations and those formalisms whose well-formedness test requires all the total disambiguations. 3 With the additional assumption that the interpretation of inde nite NPs is clause-bounded. Table 1: Comparison of various underspeci ed formalisms with respect to some desirable logical properties. LFG MG MRS QLF UDRS USDL UL semantic yes yes no yes yes yes yes closed yes yes yes yes yes yes yes complete no no no almost yes yes yes proper no no no almost yes yes yes c-deductive yes yes no yes yes yes yes u-deductive no no no no yes no yes cu-deductive no no no no no no yes If the deduction on the underspeci ed formulas can be merged with disambiguation steps, it is named cu-deductive. Table 1 gives a classi cation of some underspeci ed formalisms according to these properties. LFG stands for the linear logic approach to LFG semantics [6]. MG means Montague Grammar [7]. MRS is the Minimal Recursion Semantics of [4]. Quasi Logical Forms QLF and underspeci cation has been explored in [2]. For Underspeci ed Discourse Representation Structures UDRS see [13]. USDL is one of the formalisms which have been described in the section on underspeci cation in [3]. UL is the U(nderspeci ed) L(ogic), we present in this paper. As can be judged from the available literature, almost all formalisms are semantic. The completeness property will be discussed subsequently for each formalism. Obviously, all the 'semantic' formalisms are classically deductive, but only UDRS's and UL are u-deductive. And only UL is cu-deductive. The underspeci ed logic UL is a pair consisting of a proper underspeci ed semantic representation formalism L, and a deductive component that directly operates on these structures. For the purpose of this paper and also for the sake of comparison we have split up the representations of a formalism L into three components, B, C, and D. M speci es the building blocks of the representation language, and C tells us how these building blocks are to be put together. D is the disambiguation device which implements the construction of the individual meaning representations from a meaning description hB;Ci. In the remainder of this section we present the Linear Logic approach to LFG semantics 4 from the point of view of B, C, and D. Section 2 will then explain the deductive principles of our underspeci ed logic UL and will show how these principles can be imported into LFG. 1.1 Linear Logic approach to LFG Semantics In the case of [6]'s linear logic approach to LFG semantics M consists of linear logic formulas built up from semantic projections (i.e. formulas of the form h ; Y with h referring to an f-structure and Y being a variable or a formula of higher order predicate logic). C re ects the hierarchical ordering of the underlying f-structure. The structure in (2) is the f-structure of the ambiguous sentence (1). Every boy saw a movie. (1)