On Distributive Subalgebras of Qualitative Spatial and Temporal Calculi

Qualitative calculi play a central role in representing and reasoning about qualitative spatial and temporal knowledge. This paper studies distributive subalgebras of qualitative calculi, which are subalgebras in which (weak) composition distributives over nonempty intersections. The well-known subclass of convex interval relations is an example of distributive subalgebras. It has been proven for RCC5 and RCC8 that path consistent constraint network over a distributive subalgebra is always minimal and strongly n-consistent in a qualitative sense (weakly globally consistent). We show that the result also holds for the four popular qualitative calculi, i.e. Point Algebra, Interval Algebra, Cardinal Relation Algebra, and Rectangle Algebra. Moreover, this paper gives a characterisation of distributive subalgebras, which states that the intersection of a set of \(m\ge 3\) relations in the subalgebra is nonempty if and only if the intersection of every two of these relations is nonempty. We further compute and generate all maximal distributive subalgebras for those four qualitative calculi mentioned above. Lastly, we establish two nice properties which will play an important role in efficient reasoning with constraint networks involving a large number of variables.