this paper argues for the rich world of representation that lies between these two extremes." Levesque and Brachman (1985) 1 Introduction Time and space belong to those few fundamental concepts that always puzzled scholars from almost all scientific disciplines, gave endless themes to science fiction writers, and were of vital concern to our everyday life and commonsense reasoning. So whatever approach to AI one takes [ Russell and Norvig, 1995 ] , temporal and spatial representation and reasoning will always be among its most important ingredients (cf. [ Hayes, 1985 ] ). Knowledge representation (KR) has been quite successful in dealing separately with both time and space. The spectrum of formalisms in use ranges from relatively simple temporal and spatial databases, in which data are indexed by temporal and/or spatial parameters (see e.g. [ Srefik, 1995; Worboys, 1995 ] ), to much more sophisticated numerical methods developed in computational geom

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In this paper, we construct and investigate a hierarchy of spatio-temporal formalisms that result from various combinations of propositional spatial and temporal logics such as the propositional temporal logic the spatial logics RCC-8, BRCC-8, S4 u and their fragments. The obtained results give a clear picture of the trade-off between expressiveness and `computational realisability' within the hierarchy. We demonstrate how di#erent combining principles as well as spatial and temporal primitives can produce NP-, PSPACE-, EXPSPACE-, 2EXPSPACE-complete, and even undecidable spatio-temporal logics out of components that are at most NP- or PSPACE-complete.

This paper reviews and compares constructive and algebraic approaches in the study of rough sets. In the constructive approach, one starts from a binary relation and defines a pair of lower and upper approximation operators using the binary relation. Different classes of rough set algebras are obtained from different types of binary relations. In the algebraic approach, one defines a pair of dual approximation operators and states axioms that must be satisfied by the operators. Various classes of rough set algebras are characterized by different sets of axioms. Axioms of approximation operators guarantee the existence of certain types of binary relations producing the same operators. 1

... this paper we study two special cases of this framework: full systems and hypercubes. Both model static situations in which no agent has any information about another agent's state. Full systems and hypercubes are an appropriate model for the initial congurations of many systems of interest. We establish a correspondence between full systems and hypercube systems and certain classes of Kripke frames. We show that these classes of systems correspond to the same logic. Moreover, this logic is also the same as that generated by the larger class of weakly directed frames. We provide a sound and complete axiomatization, S5WDn , of this logic, and study its computational complexity. Finally, we show that under certain natural assumptions, in a model where knowledge evolves over time, S5WDn characterises the properties of knowledge not just at the initial conguration, but also at all later congurations. In particular, this holds for homogeneous broadcast systems, which capture settings in which agents are initially ignorant of each others local states, operate synchronously, have perfect recall, and can communicate only by broadcasting.

We study a propositional bimodal logic consisting of two S4 modalities and [a], together with the interaction axiom scheme #a## # #a##. In the intended semantics, the plain is given the McKinsey-Tarski interpretation as the interior operator of a topology, while the labelled [a] is given the standard Kripke semantics using a reflexive and transitive binary relation Ra . The interaction axiom expresses the property that the Ra relation is lower semi-continuous with respect to the topology. The class of topological Kripke frames axiomatised by the logic includes all frames over Euclidean space where Ra is the positive flow relation of a di#erential equation. We establish the completeness of the axiomatisation with respect to the intended class of topological Kripke frames, and investigate tableau calculi for the logic, although decidability is still an open question. 1 Introduction We study a propositional bimodal logic consisting of two S4 modalities # and [a], to...

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The aim of this note is to attract attention to the most important open problems and new directions of research in this exciting and promising area. 1 Distance spaces Recall that a metric space is a pair (\Delta; d), where \Delta is a nonempty set (of points) and d is a function from \Delta \Theta \Delta into the set R *0 (of non-negative real numbers) satisfying the following

This is a review of those aspects of the theory of varieties of Boolean algebras with operators (BAO's) that emphasise connections with modal logic and structural properties that are related to natural properties of logical systems.

Summary. The completeness of the modal logic S4 for all topological spaces as well as for the real line R, the n-dimensional Euclidean space R n and the segment (0, 1) etc. (with ✷ interpreted as interior) was proved by McKinsey and Tarski in 1944. Several simplified proofs contain gaps. A new proof presented here combines the ideas published later by G. Mints and M. Aiello, J. van Benthem, G. Bezhanishvili with a further simplification. The proof strategy is to embed a finite rooted Kripke structure K for S4 into a subspace of the Cantor space which in turn encodes (0, 1). This provides an open and continuous map from (0, 1) onto the topological space corresponding to K. The completeness follows as S4 is complete with respect to the class of all finite rooted Kripke structures. 1