David Fernández-Duque ; Yoàv Montacute - Dynamic Cantor Derivative Logic

lmcs:10042 - Logical Methods in Computer Science, December 18, 2023, Volume 19, Issue 4 - https://doi.org/10.46298/lmcs-19(4:26)2023
Dynamic Cantor Derivative LogicArticle

Authors: David Fernández-Duque ORCID; Yoàv Montacute ORCID

    Topological semantics for modal logic based on the Cantor derivative operator gives rise to derivative logics, also referred to as d-logics. Unlike logics based on the topological closure operator, d-logics have not previously been studied in the framework of dynamical systems, which are pairs (X,f) consisting of a topological space X equipped with a continuous function f:XX. We introduce the logics wK4C, K4C and GLC and show that they all have the finite Kripke model property and are sound and complete with respect to the d-semantics in this dynamical setting. In particular, we prove that wK4C is the d-logic of all dynamic topological systems, K4C is the d-logic of all TD dynamic topological systems, and GLC is the d-logic of all dynamic topological systems based on a scattered space. We also prove a general result for the case where f is a homeomorphism, which in particular yields soundness and completeness for the corresponding systems wK4H, K4H and GLH. The main contribution of this work is the foundation of a general proof method for finite model property and completeness of dynamic topological d-logics. Furthermore, our result for GLC constitutes the first step towards a proof of completeness for the trimodal topo-temporal language with respect to a finite axiomatisation -- something known to be impossible over the class of all spaces.


    Volume: Volume 19, Issue 4
    Published on: December 18, 2023
    Accepted on: October 2, 2023
    Submitted on: September 14, 2022
    Keywords: Mathematics - Logic,Computer Science - Logic in Computer Science
    Funding:
      Source : OpenAIRE Graph
    • Incentive - LA 14 - 2013; Code: Incentivo/EEI/LA0014/2013

    Classifications

    Mathematics Subject Classification 20201

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