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

TL;DR

This paper classifies the expressive power of two-sorted first-order logics combining points and intervals over dense and unbounded linear orders, advancing the formal understanding of temporal reasoning frameworks.

Contribution

It provides a complete classification of sub-languages based on their expressive power, clarifying the role of points and intervals in temporal logic over specific linear orders.

Findings

Complete classifications of sub-languages' expressive power.

Clarification of the inclusion of points in interval-based semantics.

Insights into temporal reasoning over dense and unbounded orders.

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. In this Part II, we deal with the cases of all dense and the case of all unbounded linearly ordered sets.