The problem of coincidence in a theory of temporal multiple recurrence

Abstract

Logical theories have been developed which have allowed temporal reasoning about eventualities (a la Galton) such as states, processes, actions, events, processes 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 ,....,x r and y 0 , y 1 , ..,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 t  r and 0  u  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.

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

“…On the strength of a result from the literature [1] that each cycle of recurrence is an exact copy of others, (in other words, the temporal relationship between the i th x p and the j th y q within any cycle of the double recurrence of x and y is an invariant), it follows that a coincidence occurs in one cycle if and only if it exists in all the others. This is a direct consequence of the fact that the duration of any eventuality is xed from occurrence to occurrence.…”

“…This is a direct consequence of the fact that the duration of any eventuality is xed from occurrence to occurrence. Therefore an algorithm alluded to in [1] for the coincidence problem explores a cycle of double recurrence for coincidence. If that fails, then no coincidence can ever occur between x p and y q during the double recurrence of x and y.…”

“…The problem of coincidence of eventualities 1 within the context of temporal double recurrence has been introduced in [1]. Let x and y be sequences of eventualities [6] such that x is the eventuality sequence x 0 , x 1 , …, x n-1 and y is the eventuality sequence y 0 , y 1 , …, y m-1 such that each x i and y j are eventualities and each eventuality has a xed duration in all its occurrences.…”

“…If eventuality sequences x and y both recur over the same interval k, a cycle of such a double recurrence is, roughly speaking, a minimal time subinterval j of k such that each of x 1 0 and y 1 0 start j and j is nished by both the intervals x i n − 1 and y j m − 1 , where n and m are lengths of eventuality sequences x and y respectively. It has been argued in [1] that the duration of a cycle of a double recurrence is the lowest common multiple of the durations of the eventuality sequences x and y.…”

“…A simple example of the coincidence problem within the context of a temporal double recurrence (from [1]) is presented as follows. A factory is in a continuous recurrence between a working state and a resting state.…”