I present a method for reasoning about spatial relationships on the basis of entailments in propositional logic. Formalisms for representing topological and other spatial information (e.g. [2] [10] [11]) have generally employed the 1st-order predicate calculus. Whilst this language is much more expressive than 0-order (propositional) calculi it is correspondingly harder to reason with. Hence, by encoding spatial relationships in a propositional representation automated reasoning becomes more effective. I specify representations in both classical and intuitionistic propositional logic, which --- together with well-defined meta-level reasoning algorithms --- provide for efficient reasoning about a large class of spatial relations. 1 INTRODUCTION This work has developed out of research done by Randell, Cui and Cohn (henceforth RCC) on formalising spatial and temporal concepts used in describing physical situations [11]. A set of classical 1st-order logic axioms has been formulated in whi...

While the computational properties of qualitative temporal reasoning have been analyzed quite thoroughly, the computational properties of qualitative spatial reasoning are not very well investigated. In fact, almost no completeness results are known for qualitative spatial calculi and no computational complexity analysis has been carried out yet. In this paper, we will focus on the so-called RCC approach and use Bennett's encoding of spatial reasoning in intuitionistic logic in order to show that consistency checking for the topological base relations can be done efficiently. Further, we show that path-consistency is sufficient for deciding global consistency. As a side-effect we prove a particular fragment of propositional intuitionistic logic to be tractable.

. We consider the modal logics KT and S4, the tense logic K t , and the fragment IPC (^;!) of intuitionistic logic. For these logics backward proof search in the standard sequent or tableau calculi does not terminate in general. In terms of the respective Kripke semantics, there are several kinds of non-termination: loops inside a world (KT), innite resp. looping branches (S4, IPC (^;!) ), and innite branching degree (K t ). We give uniform sequent-based calculi that contain specically tailored loop-checks such that the eciency of proof search is not deteriorated. Moreover all these loop-checks are easy to implement and can be combined with optimizations. Note that our calculus for S4 is not a pure contraction-free sequent calculus, but this (theoretical) defect does not hinder its application in practice. 1 Introduction For many non-classical propositional logics, backward proof search in the usual sequent calculi does not terminate in general. For all the logics we consider in th...

We characterize provability in intuitionistic predicate logic in terms of derivation skeletons and constraints and study the problem of instantiations of a skeleton to valid derivations. We prove that for two different notions of a skeleton the problem is respectively polynomial and NP-complete. As an application of our technique, we demonstrate PSPACE-completeness of the prenex fragment of intuitionistic logic. We outline some applications of the proposed technique in automated reasoning. y y Copyright c fl 1995, 1996 Andrei Voronkov. This technical report and other technical reports in this series can be obtained at http://www.csd.uu.se/~thomas/reports.html or at ftp.csd.uu.se in the directory pub/papers/reports. Some reports can be updated, check one of these addresses for the latest version. Section 1 Introduction The characterization of provability for classical logic in terms of matrices was given by Kanger [9, 10] and Prawitz [19, 20] and is in a fact a reformulation of the...

. We present a new proof-theoretic approach to bounding the complexity of the decision problem for propositional modal logics. We formalize logics in a uniform way as sequent systems and then restrict the structural rules for particular systems. This, combined with an analysis of the accessibility relation of the corresponding Kripke structures, yields decision procedures with bounded space requirements. As examples we give O(n log n) space procedures for the modal logics K and T. 1 Introduction We present a new proof-theoretic approach to bounding the complexity of the decision problem for propositional modal logics. We formalize logics in a uniform way as cut-free labelled sequent systems and then restrict the structural rules for particular systems. This, combined with an analysis of the accessibility relation of the corresponding Kripke structures, yields decision procedures with space requirements that are easily bounded. As examples we give O(n log n) space decision procedures f...

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this paper every formula is equivalent to a formula in negation normal form

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The purpose of the paper is to present the Logics Workbench, an interactive system aiming to facilitate the access to logical formalisms for nonspecialists as well as specialists. It is an integrated system which provides a library of the most important propositional calculi and many algorithms in this area. Special emphasis is put on a clear design of the human interface and a powerful information system, which covers online help and documentation.

. This paper examines the problem of testing consistency of sets of topological relations which are instances of the RCC-8 relation set (Randell, Cui and Cohn 1992). Representations of these relations as constraints within a number of logical frameworks are considered. It is shown that, if the arguments of the relations are interpreted as non-empty open sets within an arbitrary topological space, a complete consistency checking procedure can be provided by means of a composition table. This result is contrasted with the case where regions are required to be planar and bounded by Jordan curves, for which the consistency problem is known to be NP-hard. In order to investigate the completeness of compositional reasoning, the notion of k-compactness of a set of relations w.r.t. a theory is introduced. This enables certain consistency properties of relational networks to be examined independently of any specific interpretation of the domain of entities constrained by the relations. 1. Int...