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

The computational properties of qualitative spatial reasoning have been investigated to some degree. However, the question for the boundary between polynomial and NP-hard reasoning problems has not been addressed yet. In this paper we explore this boundary in the "Region Connection Calculus" RCC-8. We extend Bennett's encoding of RCC-8 in modal logic. Based on this encoding, we prove that reasoning is NPcomplete in general and identify a maximal tractable subset of the relations in RCC-8 that contains all base relations. Further, we show that for this subset path-consistency is sufficient for deciding consistency. 1 Introduction When describing a spatial configuration or when reasoning about such a configuration, often it is not possible or desirable to obtain precise, quantitative data. In these cases, qualitative reasoning about spatial configurations may be used. One particular approach in this context has been developed by Randell, Cui, and Cohn [20], the so-called Region Connecti...

this paper have been presented at the 4th European Summer School in Logic, Language and Information, University of Essex, 1992; at the Tempus Summer School for Algebraic and Categorical Methods in Computer Science, Masaryk University, Brno, 1993; and the Summer School in Logic Methods in Concurrency, Aarhus University, 1993. I would like to thank the organisers and the participants of these summer schools, and of the Banff higher order workshop. I would also like to thank Julian Bradfield for use of his Tex tree constructor for building derivation trees and Carron Kirkwood, Faron Moller, Perdita Stevens and David Walker for comments on earlier drafts.

An algorithm is presented which eliminates second-order quantiers over predicate variables in formulae of type 9P 1 ; . . . ; Pn where is an arbitrary formula of first-order predicate logic. The resulting formula is equivalent to the original formula -- if the algorithm terminates. The algorithm can for example be applied to do interpolation, to eliminate the second-order quantiers in circumscription, to compute the correlations between structures and power structures, to compute semantic properties corresponding to Hilbert axioms in non classical logics and to compute model theoretic semantics for new logics. An earlier version of the paper has been published in [GO92b].

In this paper we introduce a modal lan- guage L T for imposing constraints on trees, and an extension LT(L r) for imposing con- straints on trees decorated with feature structures. The motivation for introducing these languages is to provide tools for formalising grammatical frameworks perspicuously, and the paper illustrates this by showing how the leading ideas of arsa can be captured in LT(Lr).

In recent years, a great deal of attention has been devoted to logics of "commonsense" reasoning. Among the candidates proposed, circumscription has been perceived as an elegant mathematical technique for modeling nonmonotonic reasoning, but di cult to apply in practice. The major reason for this is the 2nd-order nature of circumscription axioms and the di culty in nding proper substitutions of predicate expressions for predicate variables. One solution to this problem is to compile, where possible, 2nd-order formulas into equivalent 1st-order formulas. Although some progress has been made using this approach, the results are not as strong as one might desire and they are isolated in nature. In this article, we provide a general method which can be used in an algorithmic manner to reduce circumscription axioms to 1st-order formulas. The algorithm takes as input an arbitrary 2nd-order formula and either returns as output an equivalent 1st-order formula, or terminates with failure. The class of 2nd-order formulas, and analogously the class of circumscriptive theories which can be reduced, provably subsumes those covered by existing results. We demonstrate the generality of the algorithm using circumscriptive theories with mixed quanti ers (some involving Skolemization), variable constants, non-separated formulas, and formulas with n-ary predicate variables. In addition, we analyze the strength of the algorithm and compare it with existing approaches providing formal subsumption results.

We show how labelled deductive systems can be combined with a logical framework to provide a natural deduction implementation of a large and well-known class of propositional modal logics (including K, D, T , B, S4, S4:2, KD45, S5). Our approach is modular and based on a separation between a base logic and a labelling algebra, which interact through a fixed interface. While the base logic stays fixed, different modal logics are generated by plugging in appropriate algebras. This leads to a hierarchical structuring of modal logics with inheritance of theorems. Moreover, it allows modular correctness proofs, both with respect to soundness and completeness for semantics, and faithfulness and adequacy of the implementation. We also investigate the tradeoffs in possible labelled presentations: We show that a narrow interface between the base logic and the labelling algebra supports modularity and provides an attractive proof-theory (in comparision to, e.g., semantic embedding) but limits th...

Natural language generation (NLG) is first and foremost a reasoning task. In this reasoning, a system plans a communicative act that will signal key facts about the domain to the hearer. In generating action descriptions, this reasoning draws on characterizations both of the causal properties of the domain and the states of knowledge of the participants in the conversation. This dissertation shows how such characterizations can be specified declaratively and accessed efficiently in NLG. The heart of this dissertation is a study of logical statements about knowledge and action in modal logic. By investigating the proof-theory of modal logic from a logic programming point of view, I show how many kinds of modal statements can be seen as straightforward instructions for computationally manageable search, just as Prolog clauses can. These modal statements provide sufficient expressive resources for an NLG system to represent the effects of actions in the world or to model an addressee whose knowledge in some respects exceeds and in other respects falls short of its own. To illustrate the use of such statements, I describe how the SPUD sentence planner exploits a modal knowledge base to

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...

In this paper we propose an ecient algorithmic solution to the problem of determining a Bisimulation Relation on a nite structure working both on the explicit and on the implicit (symbolic) representation. As far as the explicit case is concerned, starting from a set-theoretic point of view we propose an algorithm that optimizes the solution to the Relational Coarsest Partition Problem given by Paige and Tarjan in 1987; its use in model-checking packages is discussed and tested. For well structured graphs our algorithm reaches a linear worst-case behaviour. The same ideas can be elaborated for the development of the algorithm for the symbolic case.