Duality for modal μ-logics

Abstract

The modal μ-calculi are extensions of propositional modal logics with least and greatest fix-point operators. They have significant expressive power, being capable to encode properly temporal notions alien to standard modal systems. We focus here on Kozen's [21] μ-calculi, both finitary and infinitary. Based on an extension of the classical modal duality to the case of positive modal algebras that we present, we prove a Stone-type duality for positive modal μ-calculi which specializes to a duality for the Boolean modal μ-logics. Thus we extend while also improving on results published in [3]. The main improvements are: (1) extension to the negation-free case, (2) a presentation of the algebraic models of the logics in a syntax-free manner, (3) an explicit duality for the case of the finitary μ-calculus, missing in [3], and (4) a completeness result for the (negation-free or not) finitary modal μ-calculus in Kripke semantics. The special case of completeness for the Boolean μ-calculus is an improvement over that presented in [3] but weaker than the theorem of [35]. The duality presented here seems to be closer to Abramsky's domain theory in logical form [1] as the latter is based on a more general duality for distributive lattices. And it has the potential to extensions for modal μ-calculi on a non-classical (intuitionistic, relevant) prepositional basis, yielding appropriate completeness theorems.