The paper is a overview of the major qualitative spatial representation and reasoning techniques. We survey the main aspects of the representation of qualitative knowledge including ontological aspects, topology, distance, orientation and shape. We also consider qualitative spatial reasoning including reasoning about spatial change. Finally there is a discussion of theoretical results and a glimpse of future work. The paper is a revised and condensed version of [33, 34].

The need for spatial representations and spatial reasoning is ubiquitous in AI – from robot planning and navigation, to interpreting visual inputs, to understanding natural language – in all these cases the need to represent and reason about spatial aspects of the world is of key importance. Related fields of research, such as geographic information science

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

In recent years there has been increasing interest in constructing cognitive vision systems capable of interpreting the high level semantics of dynamic scenes. Purely quantitative approaches to the task of constructing such systems have met with some success. However, qualitative analysis of dynamic scenes has the advantage of allowing easier generalisation of classes of different behaviours and guarding against the propagation of errors caused by uncertainty and noise in the quantitative data. Our aim is to integrate quantitative and qualitative modes of representation and reasoning for the analysis of dynamic scenes. In particular, in this paper we outline an approach for constructing cognitive vision systems using qualitative spatial-temporal representations including prototypical spatial relations and spatio-temporal event descriptors automatically inferred from input data. The overall architecture relies on abduction: the system searches for explanations, phrased in terms of the learned spatio-temporal event descriptors, to account for the video data.

We consider an extension of linear-time temporal logic (LTL) with constraints interpreted over a concrete domain. We use a new automata-theoretic technique to show pspace decidability of the logic for the constraint systems (Z, <, =) and (N, <, =). Along the way, we give an automata-theoretic proof of a result of [BC02] when the constraint system D satisfies the completion property. Our decision procedures extend easily to handle extensions of the logic with past operators and constants, as well as an extension of the temporal language itself to monadic second order logic. Finally, we show that the logic...

We study two-dimensional Cartesian products of modal logics determined by infinite or arbitrarily long finite linear orders and prove a general theorem showing that in many cases these products are undecidable, in particular, such are the squares of standard linear logics like K4:3, S4:3, GL:3, Grz:3, or the logic determined by the Cartesian square of any infinite linear order. This theorem solves a number of open problems of Gabbay and Shehtman [7]. We also prove a sufficient condition for such products to be not recursively enumerable and give a simple axiomatisation for the square K4:3 K4:3 of the minimal liner logic using non-structural Gabbay-type inference rules.

. Region Based Geometry (RBG) is an axiomatic theory of qualitative congurations of spatial regions. It is based on Tarski's Geometry of Solids, in which the parthood relation and the concept of sphere are taken as primitive. Whereas in Tarski's theory the combination of mereological and geometrical axioms involves set theory, in RBG the interface is achieved by purely 1st-order axioms. This means that the elementary sublanguage of RBG is extremely expressive, supporting inferences involving both mereological and geometrical concepts. Categoricity of the RBG axioms is proved: all models are isomorphic to a standard interpretation in terms of Cartesian spaces over R. 1. Introduction Many researchers in the eld of Qualitative Spatial Reasoning (QSR) have argued that it is useful to have representations in which spatial regions are the basic entities [10, 8]. This ontology contrasts with the approach of classical geometry, where lines, surfaces and regions are typically thought of as ...

There exist a number of qualitative constraint calculi that are used to represent and reason about temporal or spatial configurations. However, there are only very few approaches aiming to create a spatio-temporal constraint calculus. Similar to Bennett et al., we start with the spatial calculus RCC-8 and Allen's interval calculus in order to construct a qualitative spatio-temporal calculus. As we will show, the basic calculus is NP-complete, even if we only permit base relations. When adding the restriction that the size of the spatial regions persists over time, or that changes are continuous, the calculus becomes more useful, but the satisfiability problem appears to be much harder. Nevertheless, we are able to show that satisfiability is still in NP.

Recently, a hierarchy of spatio-temporal languages based on the propositional temporal logic PTL and the spatial languages RCC-8, BRCC-8 and S4u has been introduced. Although a number of results on their computational properties were obtained, the most important questions were left open.

Information about the size of spatial regions is often easily accessible and, when combined with other types of spatial information, it can be practically very useful. In this paper we introduce four classes of qualitative and metric size constraints, and we study their integration with the Region Connection Calculus RCC-8, a widely studied approach for qualitative spatial reasoning with topological relations. Reasoning about RCC-8 relations is NP-hard, but three large maximal tractable subclasses of RCC-8, called b H8 , C8 and Q8 respectively, have been identied. We propose an O(n 3 ) time path-consistency algorithm based on a novel technique for combining RCC-8 relations and qualitative size relations forming a Point Algebra, where n is the number of spatial regions. This algorithm is correct and complete for deciding consistency when the topological relations are either in b H8 , C8 or Q8 , and has the same complexity as the best known method for deciding consistency...