Two Algorithms for Deciding Coincidence In Double Temporal Recurrence of Eventuality Sequences

Let two sequences of eventualities x (signifying the sequence, x0,x1, x2,...,xn-1) and y (signifying the sequence, y0, y1, y2,..,yn-1) both recur over the same time interval and it is required to determine whether or not a subinterval exists within the said interval which is a common subinterval of the intervals of occurrence of xp and yq. This paper presents two algorithms for solving the problem. the first explores an arbitrary cycle of the double recurrence for the existence of such an interval. its worst case running time is quadratic. The other algorithm is based on the novel notion of gcd-partitions and has a linear worst case running time. If the eventuality sequence pair (W,z) is a gcd-partition for the double recurrence (x, y),then, from a certain property of gcd-partitions, within any cycle of the double recurrence, there exists r and s such that intervals of occurrence of xp and yq are non-disjoint with the interval of co-occurrence of wr and zs. As such, a coincidence between xp and yq occurs within a cycle of the double recurrence if and only if such r and s exist so that the interval of co-occurrence of wr and zs shares a common interval with the common interval of occurrences of xp and yq. The algorithm systematically reduces the number of wr and zs pairs to be explored in the process of finding the existence of the coincidence.