Spatial Reasoning with Rectangular Cardinal Direction Relations

It is widely accepted that spatial reasoning plays a central role in various artificial intelligence applications. In this paper we study a recent, quite expressive model presented by Skiadopoulos and Koubarakis in [28, 29] for qualitative spatial reasoning with cardinal direction relations. We consider some interesting open problems of this formalism, mainly concerning finding tractable, expressive enough, subclasses of the full set (i.e., including disjunction) of relations. So far, no such subclass have been found except that of basic relations only. We focus on a small subset of cardinal relations, named rectangular cardinal relations. We investigate the connection between rectangular cardinal relations and Balbiani, Condotta and del Cerro’s Rectangle Algebra [2, 3]. By exploiting such a connection, we show that the set of basic rectangular cardinal relations is tractable but the case of disjunctive rectangular relations is not. Then, we introduce a tractable subset of the set of disjunctive rectangular cardinal relations, called saturated-convex rectangular, for which consistency can be decided in quadratic time, and the minimal network can be found in cubic time. Finally, we prove that the saturated-convex rectangular fragment is indeed a subclass of the general model for cardinal relations.