Qualitative spatial reasoning using algebraic topology

Abstract

Many propositions in spatial and qualitative reasoning can be modelled as regions in phase or configuration spaces. Deciding the consistency of k propositions then amounts to deciding the question of whether there exists a point which simultaneously falls into all k corresponding regions. A more difficult problem is to decide whether there is a point which falls only within the given k regions. I call this feasibility of the set of propositions.

In this paper, I present a method for deciding consistency and feasibility for convex regions using only topological inference. It uses Helly's theorem to decide consistency of any set of k propositions based on information about consistency of small subsets. Using methods of algebraic topology, I show a sufficient method to compute a minimal skeleton of feasible places which accurately models the connectivity between feasible environments.

The method has been implemented. I show how to formulate and solve the piano-movers problem, an important problem in spatial reasoning, using the framework.