Sahlqvist Correspondence for Modal mu-calculus

Abstract

We define analogues of modal Sahlqvist formulas for the modal mucalculus, and prove a correspondence theorem for them.

Preservation of Sahlqvist fixed point equations in completions of relativized fixed point Boolean algebras with operators

“…Fixed point operators are only allowed in the negative terms substituted in the skeleton. Currently we are unable to extend Lemma 4.5 and the claim of Theorem 4.7 to skeletons that involve fixed point operators (e.g., as in the Sahlqvist fixed point formulas of [3]). We leave it as an (interesting) open problem whether the notions of skeletons or boxed atoms could be generalized so that they involve fixed point operators and so that the analogues of Lemma 4.5 and Theorem 4.7 for these terms still hold.…”

Structures for Epistemic Logic

“…For common knowledge, the corresponding property is not first order definable, but van Benthem explains in [11] how it corresponds with a property in First-Order Logic with Least Fixed Points, see also [17].…”

“…As µx2x has no free variables, validity and satisfiability for this formula are equivalent. The fact that scatteredness of a space can be captured by fixed point formulas is not very surprising as scatteredness is a topological analogue of dual wellfoundedness [43] and it was shown in [11] (see also [17]) that µx2x together with the transitivity axiom expresses dual well-foundedness of a Kripke structure. A similar observation in algebraic terms (for the so-called diagonalisable algebras) has been made already in [57].…”

Johan van Benthem on Logic and Information Dynamics

“…Analogues of this theory have been recently developed for other classes of logics. We mention here in this sense van Benthem's work on fixed-point logics and their correspondent fragments of second-order logic [67,72,104]. More generally, the study of systematic translations [33] and of transfer results between logics is central to the contemporary model-theoretic and algebraic 7 approaches to Logic.…”