Begin, after, and later: A maximal decidable interval temporal logic

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 (B), after (A), later (L) and their inverses Ā, B̄ and L̄. We know from previous works that the combination ABB̄Ā is decidable only when finite domains are considered (and undecidable elsewhere), and that ABB̄ is decidable over the natural numbers. We extend these results by showing that decidability of ABB̄ can be further extended to capture the language ABB̄L̄, which lies in between ABB̄ and ABB̄Ā, 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 EXPSPACE complexity class.