Begin, After, and Later: a Maximal Decidable Interval Temporal Logic

TL;DR

This paper classifies the decidability of various fragments of the interval temporal logic HS, identifying maximal decidable fragments over discrete linear orders and providing optimal decision procedures.

Contribution

It extends the understanding of decidability boundaries for HS fragments by analyzing combinations of specific interval relations and their inverses.

Findings

Decidability of ABBar extended to ABBbarLbar fragment.

Maximal decidable fragments identified over strongly discrete linear orders.

Decision procedures proven optimal in complexity.

Abstract

Interval temporal logics (ITLs) are logics for reasoning about temporal statements expressed over intervals, i.e., periods of time. The most famous ITL studied so far is Halpern and Shoham's HS, which is the logic of the thirteen Allen's interval relations. Unfortunately, HS and most of its fragments have an undecidable satisfiability problem. This discouraged the research in this area until recently, when a number non-trivial decidable ITLs have been discovered. This paper is a contribution towards the complete classification of all different fragments of HS. We consider different combinations of the interval relations Begins, After, Later and their inverses Abar, Bbar, and Lbar. We know from previous works that the combination ABBbarAbar is decidable only when finite domains are considered (and undecidable elsewhere), and that ABBbar is decidable over the natural numbers. We extend these results by showing that decidability of ABBar can be further extended to capture the language ABBbarLbar, which lays in between ABBar and ABBbarAbar, and that turns out to be maximal w.r.t decidability over strongly discrete linear orders (e.g. finite orders, the naturals, the integers). We also prove that the proposed decision procedure is optimal with respect to the complexity class.