The Region Connection Calculus (RCC) [41] 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 Γ of qualitative constraints, we say a constraint (xRy) in Γ is redundant if it is entailed by the rest of Γ. A prime subnetwork of Γ is a subset of Γ which contains no redundant constraints and has the same solution set as Γ. 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 Γ is over a tractable subalgebra S of a qualitative calculus. Furthermore, if S is a subalgebra of RCC5 or RCC8 in which weak composition distributes over nonempty intersections, then Γ has a unique prime subnetwork, which can be obtained in cubic time by removing all redundant constraints simultaneously from Γ. 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 con- * Corresponding Author
Abstract. 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 ≥ 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.
In this paper we show that the Voronoi-based 9-intersection model (V9I) proposed by Chen et al. (2001) is more expressive than what has been believed before. Given any two spatial entities A, B, the V9I relation between A and B is represented as a 3 × 3 Boolean matrix. For each pair of types of spatial entities points, lines, and regions, we first show that most Boolean matrices do not represent a V9I relation by using topological constraints and the definition of Voronoi regions, and then find illustrations for all the remaining matrices. This guarantees that our method is sound and complete. In particular, we show that there are 18 V9I relations between two areas with connected interior, while there are only nine 4intersection relations. Our investigations also show that, unlike many other spatial relation models, V9I relations are context-or shape-sensitive. That is, the existence of other entities or the shape of the entities may affect the validity of certain relations. Keywords: Voronoi region; 9-intersection; topological relation; qualitative spatial reasoning * This is a draft version from the authors. The published version is online at the website of IJGIS. See
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