Toward a comprehensive treatment of temporal constraints about periodic events

In this article, we propose an Allen-like approach to deal with different types of temporal constraints about periodic events. We consider the different components of such constraints (thus, unlike Allen, we also take into account quantitative constraints) including frame times, user-defined periods, qualitative temporal constraints, and numeric quantifiers and the interactions between such components. We propose a specialized high-level formalism to represent temporal constraints about periodic events; temporal reasoning on the formalism is performed by a path-consistency algorithm repeatedly applying our operations of inversion, intersection, and composition and by a specialized reasoner about periods and numeric quantification. The high-level formalism has been designed in such a way that different types of temporal constraints about periodic events can be represented in a compact and (hopefully) user-friendly way and path-consistency-based temporal reasoning on the formalism can be performed in polynomial time. We also prove that our definitions of inversion, intersection, and composition and, thus, of our path-consistency algorithm, are correct. This article also sketches the general architecture of the temporal manager for periodic events (TeMP+), that has been designed on the basis of our approach. As a working example, we show an application of our approach to scheduling in a school.