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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., Comment: Adding proof of Theorem 2 to appendix

The Region Connection Calculus (RCC) is a well-known calculus for representing part-whole and topological relations. It plays an important role in qualitative spatial reasoning, geographical information science, and ontology. The computational complexity of reasoning with RCC5 and RCC8 (two fragments of RCC) as well as other qualitative spatial/temporal calculi has been investigated in depth in the literature. Most of these works focus on the consistency of qualitative constraint networks. In this paper, we consider the important problem of redundant qualitative constraints. For a set $\Gamma$ of qualitative constraints, we say a constraint $(x R y)$ in $\Gamma$ is redundant if it is entailed by the rest of $\Gamma$. A prime subnetwork of $\Gamma$ is a subset of $\Gamma$ which contains no redundant constraints and has the same solution set as $\Gamma$. It is natural to ask how to compute such a prime subnetwork, and when it is unique. In this paper, we show that this problem is in general intractable, but becomes tractable if $\Gamma$ is over a tractable subalgebra $\mathcal{S}$ of a qualitative calculus. Furthermore, if $\mathcal{S}$ is a subalgebra of RCC5 or RCC8 in which weak composition distributes over nonempty intersections, then $\Gamma$ has a unique prime subnetwork, which can be obtained in cubic time by removing all redundant constraints simultaneously from $\Gamma$. As a byproduct, we show that any path-consistent network over such a distributive subalgebra is weakly globally consistent and minimal. A thorough empirical analysis of the prime subnetwork upon real geographical data sets demonstrates the approach is able to identify significantly more redundant constraints than previously proposed algorithms, especially in constraint networks with larger proportions of partial overlap relations., Comment: An extended abstract appears in Proceedings of the 14th International Conference on the Principles of Knowledge Representation and Reasoning (KR-14), Vienna, Austria, July 20-24, 2014