An Integrated First-Order Theory of Points and Intervals over Linear Orders (Part I)

TL;DR

This paper develops a unified first-order logical framework for points and intervals over linear orders, classifying the expressive power of its sub-languages and addressing the classical debate on points within interval semantics.

Contribution

It introduces a comprehensive two-sorted first-order language for points and intervals, providing a complete classification of its sub-languages' expressive capabilities.

Findings

Complete classification of sub-languages by expressive power

Clarification of the role of points in interval-based semantics

Framework allows shifting perspectives between points and intervals

Abstract

There are two natural and well-studied approaches to temporal ontology and reasoning: point-based and interval-based. Usually, interval-based temporal reasoning deals with points as a particular case of duration-less intervals. A recent result by Balbiani, Goranko, and Sciavicco presented an explicit two-sorted point-interval temporal framework in which time instants (points) and time periods (intervals) are considered on a par, allowing the perspective to shift between these within the formal discourse. We consider here two-sorted first-order languages based on the same principle, and therefore including relations, as first studied by Reich, among others, between points, between intervals, and inter-sort. We give complete classifications of its sub-languages in terms of relative expressive power, thus determining how many, and which, are the intrinsically different extensions of two-sorted first-order logic with one or more such relations. This approach roots out the classical problem of whether or not points should be included in a interval-based semantics.