The interval representation problem

Abstract

We define the interval representation problem as the problem of representing the relationships which exist between the truth of an assertion over an interval and the truth of an assertion (either the same or a different one) over its internal points and/or subintervals. For example, "ran a mile without stopping" is true over an interval if and only if "running" is true at each point in the interval and the distance travelled due to "running" equals a mile. There is a relationship between "ran a mile without stopping" over an interval and "running" at the interval's interior points. The interval representation problem is a fundamental problem that must be addressed by first-order temporal logics. In this article, we look at previous attempts to solve the problem and conclude that it remains an open problem. We present a first-order temporal logic called GCH and discuss its approach to the interval representation problem. GCH uses a different approach for representing information associated with an interval. We do not directly associate these assertions with intervals. The representation of these assertions is based on the assumption that what is true at every point in an interval completely determines what is true over the interval. We use the Riemann integral to represent interval-based information.

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