Enforcing Structure on Temporal Sequences: The Allen Constraint

Recent applications of constraint programming to entertainment, e.g., music or video, call for global constraints describing the structure of temporal sequences. A typical constraint approach is to model each temporal event in the sequence with one variable, and to state constraints on these indexed variables. However, this approach hampers the statement of constraints involving events based on temporal position, since the position depends on preceding events rather than on the index. We introduce Allen, a global constraint relating event indexes with temporal positions. Allen maintains two set-variables: the set of events occurring at a position defined by an Allen relation, and the set of their indexes. These variables enable defining structural and temporal synchronization properties that cannot be stated on indexed variables. We show that a model based on a local scheduling approach does not solve the problem, even for very small instances, highlighting the need for complex filtering. We present a model that uses Multi-valued Decision Diagrams (MDDs) to compile the Allen constraint. We show that this model can be used to state and solve two complex musical tasks: audio track synchronization and musical score generation.