# The Problem of Coincidence in A Theory of Temporal Multiple Recurrence

## TL;DR

This paper addresses the problem of coincidence in temporal multiple recurrence within a logical framework, providing a formal solution based on temporal projection over recurrence cycles.

## Contribution

It formalizes the coincidence problem for two recurring sequences of fixed-duration eventualities and offers a solution using temporal projection techniques.

## Key findings

- Formal definition of the coincidence problem for multiple recurrence sequences.
- A solution based on temporal projection over recurrence cycles.
- Formalization within a first-order logical theory.

## 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 formalising 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 x0, x1,....,xr and y0, y1, ..,ys both recur over an interval k and all eventualities are of fixed durations, is there a sub-interval of k over which the incidence xt and yu for t between 0..r and s between 0..s coincide. The solution presented here formalises the intuition that a solution can be found by temporal projection over a cycle of the multiple recurrence of both sequences.

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Source: https://tomesphere.com/paper/1705.00969