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

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 describe a computational framework for a grammar architecture in which different linguistic domains such as morphology, syntax, and semantics are treated not as separate components but compositional domains. Word and phrase formation are modeled as uniform processes contributing to the derivation of the semantic form. The morpheme, as well as the lexeme, has lexical representation in the form of semantic content, tactical constraints, and phonological realization. The model is based on Combinatory Categorial Grammars. 1

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 paper shows that movement or equivalent computational structure-changing operations of any kind at the level of logical form can be dispensed with entirely in capturing quantifier scope ambiguity. It offers a new semantics whereby the effects of quantifier scope alternation can be obtained by an entirely monotonic derivation, without typechanging rules. The paper follows Fodor (1982), Fodor and Sag (1982), and Park (1995, 1996) in viewing many apparent scope ambiguities as arising from referential categories rather than true generalized quantifiers.

types in this paper are built up from ground types s, np and n with the help of implication, and thus have forms such as np s, n((np s)s), etc. A restriction on signs is that a sign of abstract type A should have a term of type A in its i-th dimension. The values of the function : for ground types can be chosen on a per grammar basis and in this paper are as in Table 2. For complex types, the rule is that (AB) = A B . This means, for example, that np(np s) = np(np s) = (t)((t)t) and that np(np s) = e(e(st)). As a consequence, (2c) should be of type np(np s). Similarly, (2a) and (2b) can be taken to be of type np, (3a) and (3b) are of types np s and s respectively, etc. In general, if M has abstract type AB and N abstract type A, then the pointwise application M(N) is de ned and has type B.

Abstract. We show how to encode context-free string grammars, linear contextfree tree grammars, and linear context-free rewriting systems as Abstract Categorial Grammars. These three encodings share the same constructs, the only difference being the interpretation of the composition of the production rules. It is interpreted as a first-order operation in the case of context-free string grammars, as a secondorder operation in the case of linear context-free tree grammars, and as a thirdorder operation in the case of linear context-free rewriting systems. This suggest the possibility of defining an Abstract Categorial Hierarchy. 1.

Categorial Grammars'. Section 4 then continues with a closer look at possible ways to set up a particular Lambda Grammar, lling in some design choices. In particular we will opt for a three dimensional grammar there; one component will deal with dominance and precedence, one with semantics, and one with syntactic features. These choices bring us in close contact with the traditional architecture of Lexical-Functional Grammar (LFG, (Kaplan and Bresnan 1982), for further connections with LFG see (Oehrle 1999) and (Muskens 2001a), which is based upon the present system) and indeed the LFG architecture inspires our answer to question 4 above. Section 4 also works out the logics of the three grammatical components in some detail and thus illustrates one possible set of answers to question 3. For the semantic component we choose a standard type logic with possible worlds; for the feature component a type logic over the rst-order theory of features ((Johnson 1991)); and the multimodal approach to grammar that is found in most modern versions of the Lambek Calculus (see (Moortgat 1997) and references therein) will serve as a basis of the component dealing with dominance and precedence. The multimodal approach is thus moved from the general level of combing signs to one of the special dimensions of the grammar, another illustration of the modularity of the set-up. The chapter ends with a short conclusion.

Over the past thirty years, the phenomenon of long-distance dependence has become one of the most well studied phenomena. Requiring as it does correlation between some position in a string and the c-commanding operator which determines its interpretation, it is uncontroversially assumed across different theoretical frameworks to involve an operator-variable binding phenomenon as in standard predicate logics (cf. Chomsky 1981, Morrill 1994, Pollard & Sag 1991, Lappin & Johnson 1996). However, it is known to display a number of properties which distinguish it from the logical operation of quantifier variable binding, and these discrepancies are taken to be indicative of the syntactic idiosyncracy of natural language formalisms. Investigation of these properties has led to the postulation of increasing numbers of discrete phenomena. There has been little attempt until recently to ask the question as to why the overall cluster of wh processes exist ( for recent partial attempts, cf Cheng 1991, Müller & Sternefeld 1994, 1 1996, and Cole 1996). The primary purpose of this paper is to propose an answer to this question. Having set out an array of largely familiar data in Section 1, in Section 2 of the paper we develop the LDS-NL framework within which the analysis is set. This is a formal deductive framework being established as a model of the process of utterance interpretation. Then in Section 3 we present a unified account of the crossover phenomenon, and in Sections 4-5 we briefly indicate analyses of reconstruction, wh-in situ and multiple wh questions, and partial wh movement phenomena, showing how a typology of wh variation emerges. In all cases, the solution will make explicit reference to the discrete stages whereby interpretation is incrementally built up in moving on a leftright basis from the initial empty state to the completed specification of a logical form corresponding to the interpretation of the string in context. The account is thus essentially procedural. In closing we reflect on the direction which this conclusion suggests- that the boundaries between syntax, semantics and pragmatics need to be redrawn, with syntax redefined as the dynamic projection of structure within an abstract parsing schema.

ions Michael Moortgat Dick Oehrle OTS, Utrecht and University of Arizona, Tucson 1 Introduction Our goal in this paper is to develop a general type logical perspective on grammatical phenomena involving discontinuous dependencies. We start from the point of view that grammatical structures may be regarded as `structured resource-premises' in a logic of grammatical deduction, a logic which is characteristically multi-modal. That is, structures are built up by a family of n-ary operations, called modes of composition; each mode is associated with a corresponding product and its residuals; the base logic of products and residuals is characterized in a uniform way for every mode; the properties of the logic as a whole depend, then, on the structural sensitivity of each individual mode and the interaction postulates which govern communication among different modes. Discontinuous dependencies involve a form of grammatical composition that requires communication between resources that do...