We consider the satisfiability and finite satisfiability problems for extensions of the two-variable fragment of firstorder logic in which an equivalence closure operator can be applied to a fixed number of binary predicates. We show that the satisfiability problem for two-variable, first-order logic with equivalence closure applied to two binary predicates is in 2NEXPTIME, and we obtain a matching lower bound by showing that the satisfiability problem for two-variable first-order logic in the presence of two equivalence relations is 2NEXPTIME-hard. The logics in question lack the finite model property; however, we show that the same complexity bounds hold for the corresponding finite satisfiability problems. We further show that the satisfiability (=finite satisfiability) problem for the twovariable fragment of first-order logic with equivalence closure applied to a single binary predicate is NEXPTIME-complete.
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...
We study multimodal logics over universally first-order definable classes of frames. We show that even for bimodal logics, there are universal Horn formulas that define set of frames such that the satisfiability problem is undecidable, even if one or two of the binary relations are transitive.
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