Weak Composition for Qualitative Spatial and Temporal Reasoning

Abstract

Abstract. It has now been clear for some time that for many qualitative spatial or temporal calculi, for instance the well-known RCC8 calculus, the operation of composition of relations which is used is actually only weak composition, which is defined as the strongest relation in the calculus that contains the real composition. An immediate consequence for qualitative calculi where weak composition is not equivalent to composition is that the well-known concept of pathconsistency is not applicable anymore. In these cases we can only use algebraic closure which corresponds to applying the path-consistency algorithm with weak composition instead of composition. In this paper we analyse the effects of having weak compositions. Starting with atomic CSPs, we show under which conditions algebraic closure can be used to decide consistency in a qualitative calculus, how weak consistency affects different important techniques for analysing qualitative calculi and under which conditions these techniques can be applied. For our analysis we introduce a new concept for qualitative relations, the "closure under constraints". It turns out that the most important property of a qualitative calculus is not whether weak composition is equivalent to composition, but whether the relations are closed under constraints. All our results are general and can be applied to all existing and future qualitative spatial and temporal calculi. We close our paper with a road map of how qualitative calculi should be analysed. As a side effect it turns out that some results in the literature have to be reconsidered.

Ordering Spatio-Temporal Sequences to Meet Transition Constraints: Complexity and Framework

“…A (binary) qualitative temporal or spatial constraint language [15] is based on a finite set B of jointly exhaustive and pairwise disjoint (JEPD) relations defined on a domain D, called the set of base relations. The base relations of set B of a particular qualitative constraint language can be used to represent the definite knowledge between any two entities with respect to the given level of granularity.…”

“…Hence, 2 B represents the total set of relations. 2 B is equipped with the usual set-theoretic operations union and intersection, the converse operation, and the weak composition operation denoted by symbol [15]. We note that the notion of a qualitative constraint language is not consistent in literature, an issue which is nicely explained in the work of Dylla et al in [6].…”

“…A subclass A ⊆ 2 B L is a maximal tractable subclass if there is no other tractable subclass that properly contains A. [15]. The constraint graph of a QCN N = (V, C) is the graph (V, E), denoted by G(N ), for which we have that…”

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“…Therefore, we use the notion of weak composition [7], which is defined for relations R 1 and R 2 as the strongest relation R 1 R 2 in R that contains R 1 • R 2 , i.e. :…”