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

In this paper we study Lambek systems as grammar logics: logics for reasoning about structured linguistic resources. The structural parameters of precedence, dominance and dependency generate a cube of resource-sensitive categorial type logics. From the pure logic of residuation NL, one obtains L, NLP and LP in terms of Associativity, Commutativity, and their combination. Each of these systems has a dependency variant, where the product is split up into a left-headed and a right-headed version. We develop a theory of systematic communication between these systems. The communication is two-way: we show how one can fully recover the structural discrimination of a weaker logic from within a system with a more liberal resource management regime, and how one can reintroduce the structural flexibility of a stronger logic within a system with a more articulate notion of structure-sensitivity. In executing this programme we follow the standard logical agenda: the categorial formula language is enriched with extra control operators, so-called structural modalities, and on the basis of these control operators, we prove embedding theorems for the two directions of substructural communication. But our results differ from the Linear Logic style of embedding with S4-like modalities in that we realize the communication in both directions in terms of a

In this paper we compare grammatical inference in the context of simple and of mixed Lambek systems. Simple Lambek systems are obtained by taking the logic of residuation for a family of multiplicative connectives =; ffl; n, together with a package of structural postulates characterizing the resource management properties of the ffl connective. Different choices for Associativity and Commutativity yield the familiar logics NL, L, NLP, LP. Semantically, a simple Lambek system is a unimodal logic: the connectives get a Kripke style interpretation in terms of a single ternary accessibility relation modeling the notion of linguistic composition for each individual system. The simple systems each have their virtues in linguistic analysis. But none of them in isolation provides a basis for a full theory of grammar. In the second part of the paper, we consider two types of mixed Lambek systems. The first type is obtained by combining a number of unimodal systems into one multimodal logic. The...

this paper I present the EMILE 3.0 algorithm

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...

The idea that to perceive an object is to perceive its affordances---that is, the interactions of the perceiver with the world that the object supports or affords---is attractive from the point of view of theories in cognitive science that emphasize the fundamental role of actions in representing an agent's knowledge about the world. However, in this general form, the notion has so far lacked a formal expression. This paper offers a representation for objects in terms of their affordances using Linear Dynamic Event Calculus, a formalism for reasoning about causal relations over events. It argues that a representation of this kind, linking objects to the events which they are characteristically involved in, underlies some universal operations of natural language syntactic and semantic composition that are postulated in Combinatory Categorial Grammar (CCG). These observations imply that the language faculty is more directly related to prelinguistic cognitive apparatus used for planning action than formal theories in either domain have previously seemed to allow.

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

This paper takes as its starting point the work of Moortgat (1991) and aims to provide a linguisticallymotivated extension to the basic Lambek calculus that will allow, among other things, for an elegant treatment of various `discontinuous constituency' phenomena, including `tough'-constructions in English, cross-serial agreement in Swiss German and quantifier scoping. The proposal is contrasted favorably with related proposals by Moortgat, Morrill and Solias (1993) and Hepple (1994). Keywords: categorial grammar, Lambek calculus, discontinuous constituents, labelled deductive systems 1 Preliminaries This paper takes as its starting point the work of Moortgat (1991) and proposes an alternative extension to the basic Lambek Calculus that will allow, among other things, for the treatment of discontinuous constituents (exemplified herein by toughclass adjective phrases in English and cross-serial dependencies in Swiss German) in terms of operations on headed strings (along the lines of...

The paper addresses constraints on long-distance extraction in categorial grammar, involving formulation and application of logical extensions of Lambek calculus. Structural facilitation, i.e. controlled import of structural properties from higher in the hierarchy of substructural logics, is complemented by a proposal for structural inhibition: controlled import of structural properties from lower in the hierarchy. A treatment is developed which includes island constraints, licensing of subject extraction, `assure'-type "extraction only" valencies, `whom'-binding of downstairs but not upstairs nominative positions, and variation in the penetrative power of fillers. 1 Structural Facilitation and Structural Inhibition Coordination of non-constituents, prosodic phrasing, and incremental interpretation have been taken to motivate a dissolution of constituent structure in categorial grammar (Steedman 1987; Dowty 1987). The associative Lambek calculus instantiates the most extreme position...