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Fixed point logics and definable topological properties

Published online by Cambridge University Press:  13 December 2023

David Fernández-Duque  and
Quentin Gougeon Open the ORCID record for Quentin Gougeon [Opens in a new window]
David Fernández-Duque
Affiliation:
Department of Philosophy, University of Barcelona, Barcelona, Spain Czech Academy of Sciences, Institute of Computer Science, Prague, Czech Republic
Quentin Gougeon*
Affiliation:
CNRS-INPT-UT3, Toulouse University, Toulouse, France
*
Corresponding author: Quentin Gougeon; Email: quentin.gougeon@irit.fr
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Abstract

Modal logic enjoys topological semantics that may be traced back to McKinsey and Tarski, and the classification of topological spaces via modal axioms is a lively area of research. In the past two decades, there has been interest in extending topological modal logic to the language of the mu-calculus, but previously no class of topological spaces was known to be mu-calculus definable that was not already modally definable. In this paper, we show that the full mu-calculus is indeed more expressive than standard modal logic, in the sense that there are classes of topological spaces (and weakly transitive Kripke frames), which are mu-definable but not modally definable. The classes we exhibit satisfy a modally definable property outside of their perfect core, and thus we dub them imperfect spaces. We show that the mu-calculus is sound and complete for these classes. Our examples are minimal in the sense that they use a single instance of a greatest fixed point, and we show that least fixed points alone do not suffice to define any class of spaces that is not already modally definable.

Keywords

Mu-calculusexpressivitytopological semantics
Type
Paper
Information
Mathematical Structures in Computer Science , Volume 34 , Issue 2 , February 2024 , pp. 81 - 97
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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