The finest of its class: The natural point-based ternary calculus ℒℜ, for qualitative spatial reasoning

Abstract

In this paper, a ternary qualitative calculus ℒℜ. for spatial reasoning is presented that distinguishes between left and right. A theory is outlined for ternary point-based calculi in which all the relations are invariant when all points are mapped by rotations, scalings, or translations (RST relations). For this purpose, we develop methods to determine arbitrary transformations and compositions of RST relations. We pose two criteria which we call practical and natural. 'Practical' means that the relation system should be closed under transformations, compositions and intersections and have a finite base that is jointly exhaustive and pairwise disjoint. This implies that the well-known path consistency algorithm [10] can be used to conclude implicit knowledge. 'Natural' calculi are close to our natural way of thinking because the base relations and their complements are connected. The main result of the paper is the identification of a maximally refined calculus amongst the practical natural RST calculi, which turns out to be very similar to Ligozat's flip-flop calculus. From that it follows, e.g., that there is no finite refinement of the TPCC calculus by Moratz et al that is closed under transformations, composition, and intersection. © Springer-Verlag Berlin Heidelberg 2005.