The problem of coincidence in a theory of temporal multiple recurrence

Logical theories have been developed which have allowed temporal reasoning about eventualities (à la Galton) such as states, processes, actions, events and complex eventualities such as sequences and recurrences of other eventualities. This paper presents the problem of coincidence within the framework of a first order logical theory formalizing temporal multiple recurrence of two sequences of fixed duration eventualities and presents a solution to it.

The coincidence problem is described as: if two complex eventualities (or eventuality sequences) consisting respectively of component eventualities $x_0,x_1,\dots ,x_r$ and $y_0,y_1,\dots ,y_s$ both recur over an interval k and all eventualities are of fixed durations, is there a subinterval of k over which the incidence $x_t$ and $y_u$ for $0\le t\le r$ and $0\le u\le s$ coincide? The solution presented here formalizes the intuition that a solution can be found by temporal projection over a cycle of the multiple recurrence of both sequences.