Contents 1 Introduction: grammatical reasoning 1 2 Linguistic inference: the Lambek systems 5 2.1 Modelinggrammaticalcomposition ............................ 5 2.2 Gentzen calculus, cut elimination and decidability . . . . . . . . . . . . . . . . . . . . 9 2.3 Discussion: options for resource management . . . . . . . . . . . . . . . . . . . . . . 13 3 The syntax-semantics interface: proofs and readings 16 3.1 Term assignment for categorial deductions . . . . . . . . . . . . . . . . . . . . . . . . 17 3.2 Natural language interpretation: the deductive view . . . . . . . . . . . . . . . . . . . 21 4 Grammatical composition: multimodal systems 26 4.1 Mixedinference:themodesofcomposition........................ 26 4.2 Grammaticalcomposition:unaryoperations ....................... 30 4.2.1 Unary connectives: logic and structure . . . . . . . . . . . . . . . . . . . . . . . 31 4.2.2 Applications: imposing constraints, structural relaxation

Recent work in categorial grammar has seen proposals for a wide range of systems, differing in their `resource sensitivity'...

We introduce a method of memoisation of categorial proof nets. Exploiting the planarity of non-commutative proof nets, and unifiability as a correctness criterion, parallelism is simulated through construction of a proof net matrix of most general unifiers for modules, in a manner analogous to the Cocke-Younger-Kasami algorithm for context free grammar. 1 Memoisation of categorial proof nets: parallelism in categorial processing 1 Introduction If the evolutionary tendency of grammatical formalisms could be summed up in one word, that word could well be lexicalism. The lexicon was once considered the locus of all and only idiosyncratic information; it may be hard now to find any proponents at all of such a view. Rather, one hears of the balance or tradeoff between lexicon and syntax: the tenet that the lexicon should comprise only what is idiosyncratic is, simply, no longer held. The notion that there is a compromise to be struck between lexicon and syntax is in turn rejected in the...

1 A sign-based categorial framework This paper investigates discontinuous type constructors within the framework of a signbased generalization of categorial type calculi. The paper takes its inspiration from Oehrle’s (1988) work on generalized compositionality for multidimensional linguistic objects, and, we hope, may establish a bridge between work in Unification Categorial Grammar or HPSG, and the research that views categorial grammar from the perspective of substructural type logics. Categorial sequents are represented as composed of multidimensional signs, modelled as tuples of the form 〈Type, Semantics, Syntax〉 They simultaneously characterize the semantic and structural properties of linguistic objects in terms of a type-assignment labelled with semantic information (a lambda term) and structural, syntactic information. As argued elsewhere (Moortgat 1988), the structural information refers to phonological structuring of linguistic material, rather than to syntactic structure in the conventional sense. For the purposes of this paper, structural information is simplified to a string description.

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We show how categorial deduction can be implemented in higher-order (linear) logic programming, thereby realising parsing as deduction for the associative and non-associative Lambek calculi. This provides a method of solution to the parsing problem of Lambek categorial grammar applicable to a variety of its extensions. The present work deals with the parsing problem for Lambek calculus and its extensions as developed

We consider the task of theorem proving in Lambek calculi and their generalisation to "multimodal residuation calculi". These form an integral part of categorial logic, a logic of signs stemming from categorial grammar, on the basis of which language processing is essentially theorem proving. The demand of this application is not just for efficient processing of some or other specific calculus, but for methods that will be generally applicable to categorial logics. It is proposed that multimodal cases be treated by dealing with the highest common factor of all the connectives as linear (propositional) validity. The prosodic (sublinear) aspects are encoded in labels, in effect the term-structure of quantified linear logic. The correctness condition on proof nets ("long trip condition") can be implemented by SLD resolution in linear logic with unification on labels/terms limited to one way matching. A suitable unification strategy is obtained for calculi of discontinuity by normalisation...

This paper solves some puzzles in the formalisation of logic fo.r discontinuity in categorial grammax. A 'tuple' operation introduced in [Solias, 1992] is defined as a mode of prosodic combination which has associated projection functions, and consequently can support a property of unique prosodic decomposability. Discontinuity operators are defined model-theoretically by a residuation scheme which is paxticulaxly ammenable proof-theoretically. This enables a formulation which both improves on the logic for wrapping and infixing of [Moortgat, 1988] which is only partial, and resolves some problems of determinacy of insertion point in the application of these proposals to in-situ binding phenomena. A discontinuous product is also defined by the residuation scheme, enabling formulation of rules of both use and proof for a 'substring' product that would have been similarly doomed to partial logic. We show

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The multiplicative fragment of linear logic has found a number of applications in computational linguistics: in the "glue language" approach to LFG semantics, and in the formulation and parsing of various categorial grammars. These applications call for efficient deduction methods. Although a number of deduction methods for multiplicative linear logic are known, none of them are tabular meth- ods, which bring a substantial efficiency gain by avoiding redundant computation (c.f. chart methods in CFG parsing): this paper presents such a method, and discusses its use in relation to the above applications.