License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.CSL.2022.19
URN: urn:nbn:de:0030-drops-157397
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/15739/
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Fernández-Duque, David ; Montacute, Yoàv

Dynamic Cantor Derivative Logic

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LIPIcs-CSL-2022-19.pdf (0.7 MB)

Abstract

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: X → X.
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 T_D 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.

BibTeX - Entry

@InProceedings{fernandezduque_et_al:LIPIcs.CSL.2022.19,
  author =	{Fern\'{a}ndez-Duque, David and Montacute, Yo\`{a}v},
  title =	{{Dynamic Cantor Derivative Logic}},
  booktitle =	{30th EACSL Annual Conference on Computer Science Logic (CSL 2022)},
  pages =	{19:1--19:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-218-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{216},
  editor =	{Manea, Florin and Simpson, Alex},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2022/15739},
  URN =		{urn:nbn:de:0030-drops-157397},
  doi =		{10.4230/LIPIcs.CSL.2022.19},
  annote =	{Keywords: dynamic topological logic, Cantor derivative, temporal logic, modal logic}
}

Keywords: dynamic topological logic, Cantor derivative, temporal logic, modal logic
Collection: 30th EACSL Annual Conference on Computer Science Logic (CSL 2022)
Issue Date: 2022
Date of publication: 27.01.2022

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