Dynamic topological logic
type: Item
instance of: scholarly article
title: Dynamic topological logic
zbMATH Open document ID: 1067.03028
author: Philip Kremer
author: G. E. Mints
published in: Annals of Pure and Applied Logic
publication date: 2005-01-31T00:00:00Z
review text: Three important research areas meet together in the abstract framework of dynamic topological logic (DTL): the topological semantics for S4, topological dynamics, and temporal logic. It is known that S4 can be understood as the logic of topological spaces, and \(\square\) can be understood as a topological modality (with the meaning of the topological interior). Thus, the topological semantics for S4 is based on topological spaces rather than Kripke frames. On the other hand, topological dynamics studies the asymptotic properties of continuous maps on topological spaces. A dynamic topological system is a topological space \(X\) together with a continuous function \(f\) which can be thought of in temporal terms as moving the points of the topological space. Dynamic topological logics are the logics of dynamic topological systems, just as S4 is the logic of topological spaces, and are defined for a trimodal language with an S4-ish topological modality, and two temporal modalities (`next' and `henceforth') both interpreted using the continuous function \(f\). In particular, `next' expresses \(f\)'s action on \(X\) from one moment to the next, and `henceforth' expresses the asymptotic behaviour of \(f\). The authors introduce the dynamic topological analogues of Kripke models, the dynamic Alexandrov models, in order to set a precise definition of dynamic topological model; then, a semantic definition of the dynamic topological logic generated by a class \(\mathcal{T}\) of topological spaces and/or a class \(\mathcal{F}\) of continuous functions. Later, several specific DTLs are considered, presenting their properties and axiomatizing some of their next-interior fragments. Finally, a sound and complete axiomatization of a DTL is given in a particular trimodal fragment of the language in which the temporal modalities cannot occur in the scope of a topological modality.
reviewed by: Manuel Ojeda-Aciego
Mathematics Subject Classification ID: 03B45
Mathematics Subject Classification ID: 03B44
Mathematics Subject Classification ID: 54H20
zbMATH DE Number: 2131594
zbMATH Keywords: modal logic
zbMATH Keywords: temporal logic
zbMATH Keywords: topological semantics
zbMATH Keywords: topological dynamics
MaRDI profile type: Publication
full work available at URL: j.apal.2004.06.004
OpenAlex ID: W1998261607
cites work: Reasoning About Space: The Modal Way
cites work: Q5767704
cites work: Q5522002
cites work: Q4100724
cites work: Q3909268
cites work: On dynamic topological and metric logics
cites work: An elementary proof of the completeness of PDL
cites work: Semantical Analysis of Modal Logic I Normal Modal Propositional Calculi
cites work: Q3677141
cites work: The algebra of topology
cites work: A proof of topological completeness for S4 in \((0,1)\)
cites work: Q5556395
cites work: Q5734410
cites work: Discrete linear future time without axioms
cites work: Q3734375
cites work: Algebraic postulates and a geometric interpretation for the Lewis calculus of strict implication
cites work: Q3931654
DOI: 10.1016/J.APAL.2004.06.004
description: scientific article; zbMATH DE number 2131594