Decidable and Undecidable Fragments of Halpern and Shoham’s Interval Temporal Logic: Towards a Complete Classification

Abstract

Abstract. Interval temporal logics are based on temporal structures where time intervals, rather than time instants, are the primitive ontological entities. They employ modal operators corresponding to various relations between intervals, known as Allen's relations. Technically, validity in interval temporal logics translates to dyadic second-order logic, thus explaining their complex computational behavior. The full modal logic of Allen's relations, called HS, has been proved to be undecidable by Halpern and Shoham under very weak assumptions on the class of interval structures, and this result was discouraging attempts for practical applications and further research in the field. A renewed interest has been recently stimulated by the discovery of interesting decidable fragments of HS. This paper contributes to the characterization of the boundary between decidability and undecidability of HS fragments. It summarizes known positive and negative results, it describes the main techniques applied so far in both directions, and it establishes a number of new undecidability results for relatively small fragments of HS.

Decidability of the Logics of the Reflexive Sub-interval and Super-interval Relations over Finite Linear Orders

“…Decidability of temporal neighbourhood logics over various classes of linear orders can be proved by reducing their satisfiability problems to that of the two-variable fragment of first-order logic over the same classes of linear orders [6]. In fact, neighbourhood temporal logic turns out to be a maximal decidable fragment of HS, when interpreted over any class of linear orders that contains at least one linear order with an infinitely ascending/descending sequence [4], [7], [8]. In this paper, we focus on interval logics of subinterval and super-interval structures.…”

“…It can easily be shown that the proposed techniques cannot be directly lifted to the strict and proper cases. Moreover, the undecidability of various extensions of the logics of sub-interval and superinterval structures has been recently proved [4], [7], [12].…”

Reasoning with Time Intervals: A Logical and Computational Perspective

“…Hence, F must contain at least one modality in the set { E , E }. This prevents modalities B and B to be included in F , as they would immediately yield undecidability [42] in the finite case. Then, it follows that F can contain only modalities from the set { A , A , E , E , L , L }, and thus it must belong to the diagram, which is a contradiction.…”

Maximal Decidable Fragments of Halpern and Shoham’s Modal Logic of Intervals

“…An optimal (NEXPTIME) tableau-based decision procedure for AĀ over the integers has been given in [5] and later extended to the classes of all (resp., dense, discrete) linear orders [6], while a decidable metric extension of the future fragment of AĀ over the natural numbers has been proposed in [7] and later extended to the full logic [4]. Finally, a number of undecidable extensions of AĀ have been given in [2,3].…”