We show that—unlike products of ‘transitive’ modal logics which are usually undecidable—their ‘expanding domain’ relativisations can be decidable, though not in primitive recursive time. In particular, we prove the decidability and the finite expanding product model property of bimodal logics interpreted in two-dimensional structures where one component—call it the ‘flow of time’—is • a finite linear order or a finite transitive tree and the other is composed of structures like • transitive trees/partial orders/quasi-orders/linear orders or only finite such structures expanding (...) over time. The decidability proof is based on Kruskal’s tree theorem, and the proof of non-primitive recursiveness is by reduction of the reachability problem for lossy channel systems. The result is used to show that the dynamic topological logic interpreted in topological spaces with continuous functions is decidable if the number of function iterations is assumed to be finite. (shrink)

In this paper we shall introduce a simple temporal logic suitable for reasoning about the temporal aspects of parallel universes, parallel processes, distributed systems, or multiple agents. We will use a variant of the mosaic method to prove decidability of this logic. We also show that the logic does not have the finite model property. This shows that the mosaic method is sometimes a stronger way of establishing decidability.

Dynamic topological logic provides a context for studying the confluence of the topological semantics for S4, topological dynamics, and temporal logic. The topological semantics for S4 is based on topological spaces rather than Kripke frames. In this semantics, □ is interpreted as topological interior. Thus S4 can be understood as the logic of topological spaces, and □ can be understood as a topological modality. Topological dynamics studies the asymptotic properties of continuous maps on topological spaces. Let a dynamic topological system (...) be a topological space X together with a continuous function f. f can be thought of in temporal terms, moving the points of the topological space from one moment to the next. Dynamic topological logics are the logics of dynamic topological systems, just as S4 is the logic of topological spaces. Dynamic topological logics are defined for a trimodal language with an S4-ish topological modality □, and two temporal modalities, ○ and *, both interpreted using the continuous function f. In particular, ○ expresses f’s action on X from one moment to the next, and * expresses the asymptotic behaviour of f. (shrink)

We consider the problem of the product finite model property for binary products of modal logics. First we give a new proof for the product finite model property of the logic of products of Kripke frames, a result due to Shehtman. Then we modify the proof to obtain the same result for logics of products of Kripke frames satisfying any combination of seriality, reflexivity and symmetry. We do not consider the transitivity condition in isolation because it leads to infinity axioms (...) when taking products. (shrink)

In this paper we improve the results of [2] by proving the product f.m.p. for the product of minimal n-modal and minimal n-temporal logic. For this case we modify the finite depth method introduced in [1]. The main result is applied to identify new fragments of classical first-order logic and of the equational theory of relation algebras, that are decidable and have the finite model property.

The paper studies many-dimensional modal logics corresponding to products of Kripke frames. It proves results on axiomatisability, the finite model property and decidability for product logics, by applying a rather elaborated modal logic technique: p-morphisms, the finite depth method, normal forms, filtrations. Applications to first order predicate logics are considered too. The introduction and the conclusion contain a discussion of many related results and open problems in the area.

We use the apparatus of the canonical formulas introduced by Zakharyaschev [10] to prove that all finitely axiomatizable normal modal logics containing K4.3 are decidable, though possibly not characterized by classes of finite frames. Our method is purely frame-theoretic. Roughly, given a normal logic L above K4.3, we enumerate effectively a class of frames with respect to which L is complete, show how to check effectively whether a frame in the class validates a given formula, and then apply a Harropstyle (...) argument to establish the decidability of L, provided of course that it has finitely many axioms. (shrink)