Spatial logic of tangled closure operators and modal mu-calculus

Abstract

There has been renewed interest in recent years in McKinsey and Tarski's interpretation of modal logic in topological spaces and their proof that S4 is the logic of any separable dense-in-itself metric space. Here we extend this work to the modal mu-calculus and to a logic of tangled closure operators that was developed by Fernández-Duque after these two languages had been shown by Dawar and Otto to have the same expressive power over finite transitive Kripke models. We prove that this equivalence remains true over topological spaces.We extend the McKinsey–Tarski topological ‘dissection lemma’. We also take advantage of the fact (proved by us elsewhere) that various tangled closure logics with and without the universal modality ∀ have the finite model property in Kripke semantics. These results are used to construct a representation map (also called a d-p-morphism) from any dense-in-itself metric space X onto any finite connected locally connected serial transitive Kripke frame.This yields completeness theorems over X for a number of languages: (i) the modal mu-calculus with the closure operator ◇; (ii) ◇ and the tangled closure operators 〈t〉 (in fact 〈t〉 can express ◇); (iii) ◇,∀; (iv) ◇,∀,〈t〉; (v) the derivative operator 〈d〉; (vi) 〈d〉 and the associated tangled closure operators 〈dt〉; (vii) 〈d〉,∀; (viii) 〈d〉,∀,〈dt〉. Soundness also holds, if: (a) for languages with ∀, X is connected; (b) for languages with 〈d〉, X validates the well-known axiom G1. For countable languages without ∀, we prove strong completeness. We also show that in the presence of ∀, strong completeness fails if X is compact and locally connected.