Minimality and convexity properties in spatial CSPs

The research in qualitative reasoning and in spatial CSP is always investigated in the backdrop of its temporal counterpart - qualitative temporal reasoning and TCSP. Unlike the case of interval algebra (IA), the composition table of RCC, IA´s so-called spatial counterpart, is in general neither complete nor extensional, the compositional consistency can be still a valid reasoning mechanism. Even in such a restricted situation, many of the known properties of IA have not been investigated for validity in the context of RCC. We address, in this paper two such properties-convexity and minimality. The importance of minimality cannot be underestimated as in a minimal network every label is feasible and hence determining all the consistent scenarios can be accomplished very efficiently. It is known that path consistency does not yield a minimal network for tractable classes of RCC-8. We represent RCC-8 relations as a partially ordered set and exploit the properties of partial ordering to derive very interesting theoretical results. We show here that there exists a convex class of relations of RCC-8 for which path consistency yields a minimal network. Our results are very important as it gives a sufficient condition for minimality and useful to generate all consistent scenarios whenever compositional consistency is a valid reasoning mechanism