Modes of Connection

A connection based approach to common-sense topological description and reasoning. The support of the EPSRC under grants GR/K65041 and GR/M56807 is gratefully acknowledged. We are also grateful for comments made on an earlier version of this paper References 1. Asher N and Vieu L. Toward a geometry of common sense: A semantics and a complete axiomatization of mereotopology. Carving up space: Steps towards the construction of an absolutely complete theory of spatial regions. 14 each other. In most previous work (an exception is 13]), partial overlap has always been taken to be a single relation (usually denoted PO(x,y)), just as connection itself is usually taken to be a single relation. Whilst recognising that there are potentially innnitely many varieties of partial overlap relation, Galton parameterised these using a matrix notation: x a b o where x; a; b and o are the numbers of connected components of x \ y, xny, ynx, compl(x y). He investigates all matrices with numbers no greater than two; of the 54 theoretical possibilities, just 23 are physically realisable. However although two of these cases are in fact relations of external connection and one is a relation of disconnection (rather than partial overlap), the calculus does not in fact allow the distinctions to be made that we address in this paper. It is also worth pointing to the work of Egenhofer and Franzosa 16], who present a calculus which allows, at the cost of arbitrary complexity of course, the possibility of distinguishing any topological distinct situations. (Compare 9] for a related proposal.) 9 Further Work The analysis of connection presented here and in 14] certainly does not exhaust the possibilities. For example, we have not explicitly investigated irregular regions or regions of higher order topological genus (i.e., regions with holes) and this may bear explicit investigation. Equally, the analysis could be extended to connection relations between spatial entities of diiering dimensions (cf. 23, 20, 10, 17]). Another possibility is to extend the analysis to cover for instance the notion of \weak connection" deened in 1], the Brentanian notion of connection 6], or the notion of connection through \\at" boundaries 30]. Yet another notion of connection is given by a pair of linked (interlocked) tori. Finally, we intend to complete work on the framework outlined in 14] integrating all the dimensions, exploring the various elements of the framework, and placing other existing work in the literature …