Deciding the consistency of branching time interval networks

Allen’s Interval Algebra (IA) is one of the most prominent formalisms in the area of qualitative temporal reasoning; however, its applications are naturally restricted to linear flows of time. When dealing with nonlinear time, Allen’s algebra can be extended in several ways, and, as suggested by Ragni and Wölfl, a possible solution consists in defining the Branching Algebra (BA) as a set of 19 basic relations (13 basic linear relations plus 6 new basic nonlinear ones) in such a way that each basic relation between two intervals is completely defined by the relative position of the endpoints on a tree-like partial order. While the problem of deciding the consistency of a network of IA-constraints is well-studied, and every subset of the IA has been classified with respect to the tractability of its consistency problem, the fragments of the BA have received less attention. In this paper, we first define the notion of convex BA-relation, and, then, we prove that the consistency of a network of convex BA-relations can be decided via path consistency, and is therefore a polynomial problem. This is the first non-trivial tractable fragment of the BA; given the clear parallel with the linear case, our contribution poses the bases for a deeper study of fragments of BA towards their complete classification.