The Undecidability of the Logic of Subintervals

Abstract

The Halpern-Shoham logic is a modal logic of time intervals. Some effort has been put in last ten years to classify fragments of this beautiful logic with respect to decidability of its satisfiability problem. We complete this classification by showing -what we believe is quite an unexpected result -that the logic of subintervals, the fragment of the Halpern-Shoham logic where only the operator "during", or D, is allowed, is undecidable over discrete structures. This is surprising as this, apparently very simple, logic is decidable over dense orders and its reflexive variant is known to be decidable over discrete structures. Our result subsumes a lot of previous undecidability results of fragments that include D. 218 J. Marcinkowski and J. Michaliszyn / The Undecidability of the Logic of Subintervals on it, this logic is probably the most influential time interval logic. But historically it was not the first one. Actually, the earliest papers about intervals in context of modal logic were written by philosophers, e.g., [12]. In computer science, the earliest attempts to formalize time intervals were process logic [21,22] and interval temporal logic [20]. Relations between intervals in linear orders from an algebraic point of view were first studied systematically by Allen [1].The Halpern-Shoham logic is a modal temporal logic, where the elements of a model are no longer -like in classical temporal logics -points in time, but rather pairs of points in time. Any such paircall it [p, q], where q is equal to or later than p -can be viewed as a (closed) time interval, that is, the set of all time points between p and q. HS logic does not assume anything about order -it can be discrete or continuous, linear or branching, complete or not.Halpern and Shoham introduce six modal operators acting on intervals. Their operators are "begins" B, "during" D, "ends" E, "meets" A, "later" L, "overlaps" O and the six inverses of those operators: B,D,Ē,Ā,L,Ō. It is easy to see that the set of operators is redundant. For example, A, B and E can define D (B and E suffice for that -a prefix of my suffix is my infix) and L (here A is enough -"later" means "meets an interval that meets"). The operator O can be expressed using E andB. The expressive power of HS operators has been studied in [16].In their paper, Halpern and Shoham show that (satisfiability of formulae of) their logic is undecidable. Their proof requires logic with three operators (B, E and A are explicitly used in the formulae and, as we mentioned above, once B, E and A are allowed, D and L come for free) so they state a question about decidable fragments of their logic.Considerable effort has been put since then to settle this question. First, it was shown [14] that the BE fragment is undecidable in the dense case. Recently, it was shown that the satisfiability problem for the HS fragments O * , A * D * , B * E * is undecidable in any class of linear orders that contains, for each n > 0, at least one linear order with length greater than n [6], where each X * may be repla...

A decidable weakening of Compass Logic based on cone-shaped cardinal directions

“…An up-to-date account of undecidability results for HS fragments can be found in [5]. Among the known results, we recall the undecidability of the logics D (quantifying over sub-intervals) and O (quantifying over overlapping intervals) -as well as of their transposes -interpreted over infinite discrete temporal domains [5,16], and the undecidability of the logic BE (quantifying over beginning and ending intervals) interpreted over both dense and infinite discrete temporal domains [12,15].…”

Complexity of ITL Model Checking: Some Well-Behaved Fragments of the Interval Logic HS

“…In [8], it has been shown that the satisfiability problem for HS interpreted over all relevant (classes of) linear orders is highly undecidable. Since then, a lot of work has been done on satisfiability for HS fragments, which showed that undecidability rules over them [3], [10], [13]. However, meaningful exceptions exist, e.g., the interval logic of temporal neighbourhood AA and the logic of sub-intervals D [4]- [6], [17].…”

Model checking for fragments of Halpern and Shoham's interval temporal logic based on track representatives

“…They take intervals, instead of points, as their primitive temporal entities. Such a choice gives them the ability to cope with advanced temporal properties, such as actions with duration, accomplishments, and temporal aggregations, which can not be properly dealt with by standard, point-based temporal logics.Expressiveness of ITLs makes them well suited for many applications in a variety of computer science fields, including artificial intelligence (reasoning about action and change, qualitative reasoning, planning, configuration and multi-agent systems, and computational linguistics), theoretical computer science (formal verification, synthesis), and databases (temporal and spatio-temporal databases) [2,10,18,30,8,27,26,19,11]. However, this great expressiveness is a double-edged sword: in most cases the satisfiability problem for ITLs turns out to be undecidable, and, in the few cases of decidable ITLs, the standard proof machinery, like Rabin's theorem, is usually not applicable.The most prominent ITL is Halpern and Shoham's modal logic of time intervals (HS, for short) [13].…”