Dynamic topological logic

Abstract

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 (interior), and two temporal modalities, (next) and * (henceforth), 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 .

Dynamic Topological Logic

“…The current paper is part of a research programme whose first results were announced in three conference abstracts, [11,12], and [13]. (These results are reproduced and proved below.)…”

Computable CTL * for Discrete-Time and Continuous-Space Dynamic Systems

“…Here, atomic propositions are open sets of states and each formulae must result in an open set. This, unlike in [11], requires an altered interpretation of negations, i. e. they results in the interior for the set complement.…”

“…One of the topological logics is a branching-time logic called Dynamical Topological Logic (DT L) [11]. A distinguishing feature of this logic is that it contains one topological modality, namely the interior operator.…”

Dynamic topological logic of metric spaces

“…The continuity of f gives rise to a non-trivial interaction between the modalities, giving rise to the 'next-interior' logic S4C introduced in [2] and, later, to Dynamic Topological Logic (DT L), introduced in [7], adding a 'henceforth' modality to the language.…”

“…Variations in the semantics generate a complex jungle of logics; even DT L interpreted over the real line is not yet fully understood (but see [7,10,11]). …”