A Note on Algebraic Closure and Closure under Constraints

A-closure is the equivalent of path consistency for qualitative spatiotemporal calculi with weak composition. We revisit existing attempts to characterize the question whether a-closure is a complete method for deciding consistency of CSPs over such calculi. Renz and Ligozat’s characterization via closure under constraints has been refuted by Westphal, Hué and Wölfl. However, for many commonly used calculi, completeness of a-closure and closure under constraints coincide. We show that it is unlikely to obtain an effective procedure to decide closure under constraints by an enumeration process – and thus completeness of a-closure if the characterization were true. We further provide a sufficient condition for closure under constraints via properties of the automorphism group of the set of relations in a calculus.