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

This chapter provides a modern overview of the field of hybrid logic. Hybrid logics are extensions of standard modal logics, involving symbols that name individual states in models. The first results that are nowadays considered as part of the field date back to the early work of Arthur

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.

In this paper we study the expressive power and definability for (extended) modal languages interpreted on topological spaces. We provide topological analogues of the van Benthem characterization theorem and the Goldblatt-Thomason definability theorem in terms of the well established first-order topological language Lt.

This chapter introduces modal logic as a tool for talking about graphs, or to use more traditional terminology, as a tool for talking about Kripke models and frames. We want the reader to gain an intuitive appreciation of this perspective, and a firm grasp of the key technical ideas (such as bisimulations) which underly it. We introduce the syntax and semantics of basic modal logic, discuss its expressivity at the level of models, examine its computational properties, and then consider what it can say at the level of frames. We then move beyond the basic modal language, examine the kinds of expressivity offered by a number of richer modal logics, and try to pin down what it is that makes them all ‘modal’. We conclude by discussing an example which brings many of the ideas we discuss into play: games.

Monotonic modal logics form a generalization of normal modal logics...

It was proved in McKinsey and Tarski [7] that every finite wellconnected closure algebra is embedded into the closure algebra of the power set of the real line R. Pucket [10] extended this result to all finite connected closure algebras by showing that there exists an open map from R to any finite connected topological space. We simplify his proof considerably by using the correspondence between finite topological spaces and finite quasi-ordered sets. As a consequence, we obtain that the propositional modal system S4 of Lewis is complete with respect to Boolean combinations of countable unions of convex subsets of R, which is strengthening of McKinsey and Tarski's original result. We also obtain that the propositional modal system Grz of Grzegorczyk is complete with respect to Boolean combinations of open subsets of R. Finally, we show that McKinsey and Tarski's result can not be extended to countable connected closure algebras by proving that no countable Alexandro# space containing an infinite ascending chain is an open image of R.

A framework to deal with spatial patterns at the qualitative level of mereotopology is proposed. The main...

In: F. Esposito (Eds.), Congress of the Italian Association for Artificial Intelligence (AI*IA 2001), LNAI 2175, pp. 99-110, Springer, 2001.

For an Euclidean space R , let L n denote the modal logic of chequered . For every n 1, we characterize L n using the more familiar Kripke semantics, thus implying that each L n is a tabular logic over the well-known modal system Grz of Grzegorczyk.