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

This thesis presents accounts of a range of linguistic phenomena in an extended categorial framework, and develops proposals for processing grammars set within this framework. Linguistic phenomena whose treatment we address include word order, grammatical relations and obliqueness, extraction and island constraints, and binding. The work is set within a flexible categorial framework which is a version of the Lambek calculus (Lambek, 1958) extended by the inclusion of additional type-forming operators whose logical behaviour allows for the characterization of some aspect of linguistic phenomena. We begin with the treatment of extraction phenomena and island constraints. An account is developed in which there are many interrelated notions of boundary, and where the sensitivity of any syntactic process to a particular class of boundaries can be addressed within the grammar. We next present a new categorial treatment of word order which factors apart the specification of the order of a h...

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 represents categorial grammar as an implicational type theory in the spirit of Girard's linear logic, and illustrates linguistic applications of a range of typeconstructors over and above implication. The type theoretic perspective is concerned with a correspondence between the logic of types, and computational operations over the objects inhabiting types. In linguistic applications this correspondence is between rules of grammar which are theorems of type inference, and compositional operations in the various algebras in which linguistic objects, i.e. signs, are assumed to have dimensions: syntax, semantics, etc. Rule-to-rule description is familiar from Montague Grammar, but the idea here is to classify signs with structured types satisfying universal type laws determined by the semantics of the type connectives, in contrast to classification by categories satisfying stipulated rules. On this scheme an object language is to be specified by a type assignment to its finite vocabulary: a formal grammar is just a lexicon, plus perhaps some improper type axioms, and a grammar formalism is just a meta-language of types with its uniform logic and interpretation is each linguistic dimension. The aim is to develop a language of types which has sufficient transparency, sensitivity, and generality to implement interesting descriptions of natural language. The paper will illustrate sentence grammar, and also use of the semantic term algebra as a functional programming language for presentation of lexical semantics. 1

This paper describes a method for chart parsing Lam- bek grammars. The method is of particular interest in two regards. FirsSly, it allows efficient processing of grammars which use necessity operators, an extension proposed for handling locality phenomena. Secondly, the method is emily adapted to allow incremental processing of Lambek grammars, a possibility that has hitherto been unavailable.

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this paper, we demonstrate that the multimodal categorial grammars are in fact Turing-complete in their weak generative capacity. The result follows from a straightforward reduction of generalized rewriting systems to a mixed associative and modal categorial calculus. It turns out that we should not be surprised that seemingly arbitrary kinds of operations can be coded in multimodal categorial grammars. In this paper, we show that any computable grammar can be coded as a multimodal categorial grammar. From the standpoint of formal linguistics, this opening of the computational floodgates might even appear to be inevitable. Simply compare the introduction of general transformations in transformational grammars [Peters and Ritchie 1973], metarules in phrase structure grammars [Uszkoreit and Peters 1986], and lexical rules in categorial and phrase structure systems [Carpenter 1991], all of which have been shown to be Turing-complete. Although steps may be taken to restrict the power of these systems to ensure decidability, such moves appear rather ad hoc because of their lack of linguistic motivation. For instance, consider the restrictions against metarule self application [Gazdar et al. 1985], or the finite bound placed on unary phrase structure rules (and by association, empty categories) by [Kaplan and Bresnan 1982]. Natural language syntax is a difficult matter, and no formalism has even come close to providing a universal system in which all and only natural language grammars can be expressed. Perhaps even more discouraging is the fact that no grammars for particular languages have ever been developed that even come close to covering a naturally occurring range of data in a theoretically clean fashion. On the other hand, the grammar fragments that are typically propo...

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

After a brief introduction to categorial grammar in relation to basic sentence structure, subject-verb agreement is implemented within the formalism with predicate-logical types. The apparatus is then applied to an analysis of clitics in terms of lifting. Phenomena such as clitic doubling and subject pro drop are treated in the general formalism of categorial logic. The clause-locality of cliticisation is captured by reference to temporally intensional domains in relation to which cliticclimbing over infinitival verbs is characterised.

This paper contributes to the theory of substructural logics .that are of interest to categorial grammarians. Combining se- mantic ideas of Hepple [1990] and Morrill [1990], proof-theoretic ideas of Venema [1993b; 1993a] and the theory ofequational specifications, a class of resource-preserving logics is defined, for which decidability and completeness theorems are established.