A Partitioning Algorithm for Detecting Eventuality Coincidence in Temporal Double recurrence

A logical theory of regular double or multiple recurrence of eventualities, which are regular patterns of occurrences that are repeated, in time, has been developed within the context of temporal reasoning that enabled reasoning about the problem of coincidence. i.e. if two complex eventualities, or eventuality sequences consisting respectively of component eventualities x0, x1,....,xr and y0, y1, ..,ys both recur over an interval k and all eventualities are of fixed durations, is there a subinterval of k over which the occurrence xp and yq for p between 1 and r and q between 1 and s coincide. We present the ideas behind a new algorithm for detecting the coincidence of eventualities xp and yq within a cycle of the double recurrence of x and y. The algorithm is based on the novel concept of gcd partitions that requires the partitioning of each of the incidences of both x and y into eventuality sequences each of which components have a duration that is equal to the greatest common divisor of the durations of x and y. The worst case running time of the partitioning algorithm is linear in the maximum of the duration of x and that of y, while the worst case running time of an algorithm exploring a complete cycle is quadratic in the durations of x and y. Hence the partitioning algorithm works faster than the cyclical exploration in the worst case.