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.
