Definite Clause Grammars (DCGs) have proved valuable to computational linguists since they can be used to specify phrase structured grammars. It is well known how to encode DCGs in Horn clauses. Some linguistic phenomena, such as filler-gap dependencies, are difficult to account for in a completely satisfactory way using simple phrase structured grammar. In the literature of logic grammars there have been several attempts to tackle this problem by making use of special arguments added to the DCG predicates corresponding to the grammatical symbols. In this paper we take a different line, in that we account for filler-gap dependencies by encoding DCGs within hereditary Harrop formulas, an extension of Horn clauses (proposed elsewhere as a foundation for logic programming) where implicational goals and universally quantified goals are permitted. Under this approach, filler-gap dependencies can be accounted for in terms of the operational semantics underlying hereditary Harrop formulas, in a way reminiscent of the treatment of such phenomena in Generalized Phrase Structure Grammar (GPSG). The main features involved in this new formulation of DCGs are mechanisms for providing scope to constants and program clauses along with a mild use of λ-terms and λ-conversion. 1

The Lambek calculus, an intuitionistic fragment of Linear Logic, has recently been rediscovered by linguists. Due to its built-in hypothetical reasoning mechanism, it allows for describing a certain range of those phenomena in natural language syntax which involve incomplete subphrases or moved constituents. Previously, it seemed unclear how to extent traditional parsing techniques in order to incorporate reasoning about incomplete phrases, without causing the undesired effect of derivational equivalences. It turned out that the Lambek calculus offers a framework to formulate equivalent but more implementation-oriented calculi where this problem does not occur. In this paper, such a theorem prover for the Lambek calculus, i.e. a parser for Lambek categorial grammars, is defined. Permutations of proof steps which would cause derivational equivalence in a purely sequential formulation do not play a role in a (pseudo-)parallel approach which is based on a lemma table or a "chart". At the...

Instead of incorporating a gap-percolation mechanism for handling certain "movement" phenomena, the extended categorial grammars contain special inference rules for treating these problems. The Lambek categorial grammar is one representative of the grammar family under consideration. It allows for a restricted use of hypothetical reasoning. We define a modification of the Cocke-Younger-Kasami (CKY) parsing algorithm which covers this additional deductive power and analyze its time complexity.

Intuitionistic logic programming is an extension of Horn-clause logic programming in which implications may appear "embedded" on the right-hand side of a rule. Thus, rules of the form A(x) / [B(x) / C(x)] are allowed. These rules are called embedded implications . In this paper, we develop a language in which negation-as-failure is combined with embedded implications in a principled way. Although this combination has been studied by other researchers, Gabbay has argued in [10] that the entire idea is logically incoherent since modus ponens would not be valid in such a system. We show how to solve this problem by drawing a distinction between rules and goals. To specify the semantics of rules and goals, we then develop an analogue of Przymusinski's perfect model semantics for stratified Horn-clause logic [20]. Several modifications are necessary to adapt this idea from classical logic to intuitionistic logic, but we eventually show how to define a preferred model of a stratified intui...

This paper is a study of circumscription, not in classical logic, as usual, but in intuitionistic logic. We first review the intuitionistic circumscription of Horn clause logic programs, which was discussed in previous work, and we then consider the larger class of embedded implications . The ordinary circumscription axiom turns out to be inappropriate for this class of rules, and we analyze two alternatives: (1) prioritized circumscription, which works for stratified embedded implications; and (2) partial circumscription, which is independent of the stratification. We then show that these two approaches coincide by identifying a single structure that serves as the final Kripke model for both circumscription axioms. This means that prioritized circumscription and partial circumscription entail exactly the same set of implicational queries. Several applications of these ideas are described, including: (1) an interpretation of negation-asfailure; (2) a formalization of indefinite reasoni...

We present a new chart parsing method for Lambek grammars, inspired by a method for DTree grammar parsing. The formulae of a Lambek sequent are firstly converted into rules of an indexed grammar formalism, which are used in an Earley-style predictive chart algorithm. The method is non-polynomial, but performs well for practical purposes --- much better than previous chart methods for Lambek grammars. 1 Introduction We present a new chart parsing method for Lambek grammars. The starting point for this work is the observation, in (Hepple, 1998), of certain similarities between categorial grammars and the D-Tree grammar (DTG) formalism of Rambow et al. (1995a). On this basis, we have explored adapting the DTG parsing approach of Rambow et al. (1995b) for use with the Lambek calculus. The resulting method is one in which the formulae of a Lambek sequent that is to be proven are first converted to produce rules of a formalism which combines ideas from the multiset-valued linear indexed gr...

First-order treatments of long-distance phenomena such as relativization typically suffer from overgeneration. Higher order inspired extensions of Prolog have been proposed with varying degrees of success, but still suffer from overgeneration in the case of imbricated structures. We first propose an Assumption Grammar based treatment which deals successfully with this case both for analysis and for generation, and which maintains semantic compositionality as well. We then propose a cleaner, true higher order logic approach which solves the same problems, we argue that this approach is superior to other kinds of grammars dealing with long distance dependencies, and we advocate the development of a mixed platform (-Prolog plus continuation based assumptions) where the best of both worlds can be exploited. 1 Introduction Overgeneration (the acceptance by a language processor of incorrect as well as correct sentences) has long been a problem in logic-based as well as other computational ...