Relation algebras and their application in temporal and spatial reasoning

Abstract Qualitative temporal and spatial reasoning is in many cases based on binary relations such as before, after, starts, contains, contact, part of, and others derived from these by relational operators. The calculus of relation algebras is an equational formalism; it tells us which relations must exist, given several basic operations, such as Boolean operations on relations, relational composition and converse. Each equation in the calculus corresponds to a theorem, and, for a situation where there are only nitely many relations, one can construct a composition table which can serve as a look up table for the relations involved. Since the calculus handles relations, no knowledge about the concrete geometrical objects is necessary. In this sense, relational calculus is pointless. Relation algebras were introduced into temporal reasoning by Allen [1] and into spatial reasoning by Egenhofer and Sharm

On the complemented disk algebra

The importance of relational methods in temporal and spatial reasoning has been widely recognised in the last two decades. A quite large part of contemporary spatial reasoning is concerned with the research of relation algebras generated by the "part of" and "connection" relations in various domains. This paper is devoted to the study of one particular relation algebra appeared in the literature, viz. the complemented disk algebra. This algebra was first described by Düntsch [I. Düntsch, A tutorial on relation algebras and their application in spatial reasoning, Given at COSIT, August 1999, Available from: 〈http://www.cosc.brocku.ca/ ~duentsch/papers/relspat.html〉] and then, Li et al. [Y. Li, S. Li, M. Ying, Relational reasoning in the Region Connection Calculus, Preprint, 2003, Available from: http://arxiv.org/abs/cs/0505041] showed that closed disks and their complements provides a representation. This set of regions is rather restrictive and, thus, of limited practical values. This paper will provide a general method for generating representations of this algebra in the framework of Region Connection Calculus. In particular, connected regions bounded by Jordan curves and their complements is also such a representation. © 2005 Elsevier Inc. All rights reserved