When logic programming is based on the proof theory of intuitionistic logic, it is natural to allow implications in goals and in the bodies of clauses. Attempting to prove a goal of the form D ⊃ G from the context (set of formulas) Γ leads to an attempt to prove the goal G in the extended context Γ ∪ {D}. Thus during the bottom-up search for a cut-free proof contexts, represented as the left-hand side of intuitionistic sequents, grow as stacks. While such an intuitionistic notion of context provides for elegant specifications of many computations, contexts can be made more expressive and flexible if they are based on linear logic. After presenting two equivalent formulations of a fragment of linear logic, we show that the fragment has a goal-directed interpretation, thereby partially justifying calling it a logic programming language. Logic programs based on the intuitionistic theory of hereditary Harrop formulas can be modularly embedded into this linear logic setting. Programming examples taken from theorem proving, natural language parsing, and data base programming are presented: each example requires a linear, rather than intuitionistic, notion of context to be modeled adequately. An interpreter for this logic programming language must address the problem of splitting contexts; that is, when attempting to prove a multiplicative conjunction (tensor), say G1 ⊗ G2, from the context ∆, the latter must be split into disjoint contexts ∆1 and ∆2 for which G1 follows from ∆1 and G2 follows from ∆2. Since there is an exponential number of such splits, it is important to delay the choice of a split as much as possible. A mechanism for the lazy splitting of contexts is presented based on viewing proof search as a process that takes a context, consumes part of it, and returns the rest (to be consumed elsewhere). In addition, we use collections of Kripke interpretations indexed by a commutative monoid to provide models for this logic programming language and show that logic programs admit a canonical model. 1

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 present a system for generating parsers based directly on the metaphor of parsing as deduction. Parsing algorithms can be represented directly as deduction systems, and a single deduction engine can interpret such deduction systems so as to implement the corresponding parser. The method generalizes easily to parsers for augmented phrase structure formalisms, such as definiteclause grammars and other logic grammar formalisms, and has been used for rapid prototyping of parsing algorithms for a variety of formalisms including variants of tree-adjoining grammars, categorial grammars, and lexicalized context-free grammars.

In this paper we present a general parsing strategy that arose from the development of an Earicy-type parsing algorithm for TAGs (Schabes and Joshi 1988) and from recent linguistic work in TAGs (Abeille 1988). In our approach

The paper proposes a theory relating syntax, semantics, and intonational prosody, and covering a wide range of English intonational tunes and their semantic interpretation in terms of focus and information structure. The theory is based on a version of combinatory categorial grammar which directly pairs phonological and logical forms without intermediary representational levels.

Linear logic, introduced by Girard, is a refinement of classical logic with a natural, intrinsic accounting of resources. We show that unlike most other propositional (quantifier-free) logics, full propositional linear logic is undecidable. Further, we prove that without the modal storage operator, which indicates unboundedness of resources, the decision problem becomes pspace-complete. We also establish membership in np for the multiplicative fragment, np-completeness for the multiplicative fragment extended with unrestricted weakening, and undecidability for certain fragments of noncommutative propositional linear logic. 1 Introduction Linear logic, introduced by Girard [14, 18, 17], is a refinement of classical logic which may be derived from a Gentzen-style sequent calculus axiomatization of classical logic in three steps. The resulting sequent system Lincoln@CS.Stanford.EDU Department of Computer Science, Stanford University, Stanford, CA 94305, and the Computer Science Labo...

tem of rewrite rules or "productions" like (2), which have their origin in early work in recursion theory by Post, among others. (1) Dexter likes Warren. (2) S ! NP VP VP ! TV NP TV ! flikes;sees; : : :g Categorial Grammar (CG), together with its close cousin Dependency Grammar (which also originated in the 1950s, in work by Tesniere) stems from an alternative approach to context-free grammar pioneered by Bar-Hillel 1953 and Lambek 1958, with earlier antecedents in Ajdukiewicz 1935 and still earlier work by Husserl and Russell in category theory and the theory of types. Categorial Grammars capture the same information by associating a functional type or category with all grammatical entities. For example, all transitive verbs are associated via the lexicon with a category that can be written as follows: (3) likes := (SnNP)=NP The no

This paper introduces a logical system, called BV, which extends multiplicative linear logic by a non-commutative self-dual logical operator. This extension is particularly challenging for the sequent calculus, and so far it is not achieved therein. It becomes very natural in a new formalism, called the calculus of structures, which is the main contribution of this work. Structures are formulae subject to certain equational laws typical of sequents. The calculus of structures is obtained by generalising the sequent calculus in such a way that a new top-down symmetry of derivations is observed, and it employs inference rules that rewrite inside structures at any depth. These properties, in addition to allowing the design of BV, yield a modular proof of cut elimination.

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...

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