A Spatial Odyssey of the Interval Algebra: 1. Directed Intervals

Allen’s well-known Interval Algebra has been developed for temporal representation and reasoning, but there are also interesting spatial applications where intervals can be used. A prototypical example are traffic scenarios where cars and their regions of influence can be represented as intervals on a road as the underlying line. There are several differences of temporal and spatial intervals which have to be considered when developing a spatial interval algebra. In this paper we analyze the first important difference: as opposed to temporal intervals, spatial intervals can have an intrinsic direction with respect to the underlying line. We develop an algebra for qualitative spatial representation and reasoning about directed intervals, identify tractable subsets, and show that path-consistency is sufficient for deciding consistency for a particular subset which contains all base relations.