We describe an interval logic for reasoning about space. The logic simplifies an earlier theory developed by Randell and Cohn, and that of Clarke upon which the former was based. The theory supports a simpler ontology, has fewer defined functions and relations, yet does not suffer in terms of its useful expressiveness. An axiomatisation of the new theory and a comparison with the two original theories is given.

. The field of Qualitative Spatial Reasoning is now an active research area in its own right within AI (and also in Geographical Information Systems) having grown out of earlier work in philosophical logic and more general Qualitative Reasoning in AI. In this paper (which is an updated version of [25]) I will survey the state of the art in Qualitative Spatial Reasoning, covering representation and reasoning issues as well as pointing to some application areas. 1 What is Qualitative Reasoning? The principal goal of Qualitative Reasoning (QR) [129] is to represent not only our everyday commonsense knowledge about the physical world, but also the underlying abstractions used by engineers and scientists when they create quantitative models. Endowed with such knowledge, and appropriate reasoning methods, a computer could make predictions, diagnoses and explain the behaviour of physical systems in a qualitative manner, even when a precise quantitative description is not available 1 ...

. Although Qualitative Reasoning has been a lively subfield of AI for many years now, it is only comparatively recently that substantial work has been done on qualitative spatial reasoning; this paper lays out a guide to the issues involved and surveys what has been achieved. The papers is generally informal and discursive, providing pointers to the literature where full technical details may be found. 1 What is Qualitative Reasoning? The principal goal of Qualitative Reasoning (QR) [86] is to represent not only our everyday commonsense knowledge about the physical world, but also the underlying abstractions used by engineers and scientists when they create quantitative models. Endowed with such knowledge, and appropriate reasoning methods, a computer could make predictions, diagnoses and explain the behaviour of physical systems in a qualitative manner, even when a precise quantitative description is not available 1 or is computationally intractable. The key to a qualitative ...

The Region-Connection Calculus (RCC) is a well established formal system for qualitative spatial reasoning. It provides an axiomatization of space which takes regions as primitive, rather than as constructions from sets of points. The paper introduces boolean connection algebras (BCAs), and proves that these structures are equivalent to models of the RCC axioms. BCAs permit a wealth of results from the theory of lattices and boolean algebras to be applied to RCC. This is demonstrated by two theorems which provide constructions for BCAs from suitable distributive lattices. It is already well known that regular connected topological spaces yield models of RCC, but the theorems in this paper substantially generalize this result. Additionally, the lattice theoretic techniques used provide the first proof of this result which does not depend on the existence of points in regions. Keywords: Region-Connection Calculus, Qualitative Spatial Reasoning, Boolean Connection Algebra, Mer...

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

In recent years, there has been renewed interest in the development of formal languages for describing mereological (part-whole) and topological relationships between objects in space. Typically, the nonlogical primitives of these languages are properties and relations such as `x is connected' or `x is a part of y', and the entities over which their variables range are, accordingly, not points, but regions: spatial entities other than regions are admitted, if at all, only as logical constructs of regions. This paper considers two ørst-order mereotopological languages, and investigates their expressive power. It turns out that these languages, notwithstanding the simplicity of their primitives, are surprisingly expressive. In particular, it is shown that inønitary versions of these languages are adequate to express (in a sense made precise below) all topological relations over the domain of polygons in the closed plane.

In this paper, we present a spatial logic which can be used to reason about topological and spatial relationships among objects in spatial databases. The main advantages of such a formalism are its rigorousness, clear semantics and sound inference mechanism. We also show how the formalism can be extended to include orientation and metrical information. Comparisons with other formalisms are discussed.

We categorize in recursion-theoretic terms the expressivity of a number of first-order languages that allow quantification over regions in Euclidean space. Specifically we show the following: 1. Let U be any class of closed regions in Euclidean space that includes all simple polygons. Let C(x, y) be the relation, “Region x is connected to region y, ” and let Convex(x) be the property, “Region x is convex. ” Then any relation over U that is analytical and invariant under affine transformations is first-order definable in the structure 〈U, C, Convex〉. 2. Let U be as in (1), and let Closer(x, y, z) be the relation “Region x is closer to y than to z. ” Then any relation over U that is analytical and invariant under orthogonal transformations is first-order definable in the structure 〈U, Closer〉. 3. Let U be the class of finite unions of intervals in the real line. Then any relation over U that is analytical and invariant under linear transformations is first-order definable in the structure 〈U, Closer〉. 4. If the class of regions is restricted to be polygons with rational vertices, then results analogous to (1-3) hold, substituting “arithmetical relation ” for “analytical relation”.

We introduce a family of languages intended for representing knowledge and reasoning about metric (and more general distance) spaces. While the simplest language can speak only about distances between individual objects and Boolean relations between sets, the more expressive ones are capable of capturing notions such as `somewhere in (or somewhere out of) the sphere of a certain radius', `everywhere in a certain ring', etc. The computational complexity of the satisfiability problem for formulas in our languages ranges from NP-completeness to undecidability and depends on the class of distance spaces in which they are interpreted. Besides the class of all metric spaces, we consider, for example, the spaces R \Theta R and N \Theta N with their natural metrics. 1 Introduction The concept of `distance between objects' is one of the most fundamental abstractions both in science and in everyday life. Imagine for instance (only imagine) that you are going to buy a house in London. You then i...

The field of Qualitative Spatial Reasoning is now an active research area in its own right within AI (and also in Geographical Information Systems) having grown out of earlier work in philosophical logic and more general Qualitative Reasoning in AI. In this paper (which is a slightly updated version of [ Cohn, 1997 ] ) I will survey the state of the art in Qualitative Spatial Reasoning, covering representation and reasoning issues as well as pointing to some application areas. 1 What is Qualitative Reasoning? The principal goal of Qualitative Reasoning (QR) [ Weld and De Kleer, 1990 ] is to represent not only our everyday commonsense knowledge about the physical world, but also the underlying abstractions used by engineers and scientists when they create quantitative models. Endowed with such knowledge, and appropriate reasoning methods, a computer could make predictions, diagnoses and explain the behaviour of physical systems in a qualitative manner, even when a precise quantitative...