On Distributive Subalgebras of Qualitative Spatial and Temporal Calculi
This work addresses foundational issues in qualitative reasoning for AI applications, such as spatial and temporal knowledge representation, but is incremental as it builds on existing theories of distributive subalgebras.
The paper tackles the problem of characterizing and generating distributive subalgebras in qualitative spatial and temporal calculi, proving that path consistent constraint networks over these subalgebras are minimal and globally consistent, with examples provided for calculi like RCC5 and RCC8.
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. It has been proven for RCC5 and RCC8 that path consistent constraint network over a distributive subalgebra is always minimal and globally consistent (in the sense of strong $n$-consistency) in a qualitative sense. The well-known subclass of convex interval relations provides one such an example of distributive subalgebras. This paper first gives a characterisation of distributive subalgebras, which states that the intersection of a set of $n\geq 3$ 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 Point Algebra, Interval Algebra, RCC5 and RCC8, Cardinal Relation Algebra, and Rectangle Algebra. Lastly, we establish two nice properties which will play an important role in efficient reasoning with constraint networks involving a large number of variables.