Fixed point logics and definable topological properties

0580737 - ÚI 2025 RIV GB eng J - Journal Article
Fernández-Duque, David - Gougeon, Q.
Fixed point logics and definable topological properties.
Mathematical Structures in Computer Science. Roč. 34, č. 2 (2024), s. 81-97. ISSN 0960-1295. E-ISSN 1469-8072
Institutional support: RVO:67985807
Keywords : expressivity * Mu-calculus * topological semantics
OECD category: Pure mathematics
Impact factor: 0.5, year: 2022
Method of publishing: Limited access
https://doi.org/10.1017/S0960129523000385

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.
Permanent Link: https://hdl.handle.net/11104/0349498