Algorithmic correspondence for intuitionistic modal mu-calculus, Part 2

Sahlqvist-style correspondence results remain a perennial theme and an active topic of research within modal logic. Recently, there has been interest in extending the classical results in this area to the modal mu-calculus [7]. For instance, in [8] van Benthem, Bezhanishvili and Hodkinson define a class of Sahlqvist formulas for the modal mu-calculus, all of which have frame correspondents in first-order logic with least fixed points (FO+LFP). The analysis pursued in that paper is model theoretic in the tradition of the well known Sahlqvist-van Benthem algorithm. Parallel to the model theoretic approach to this type of result, there exists an algebraic-algorithmic approach (see e.g., [3, 4, 5]) which derives correspondence (and canonicity) results by means of ‘calculi of correspondence’ consisting of simple derivation rules which depend for their soundness on the order theoretic properties of the operations interpreting the logical connectives in the algebraic semantics. As indicated in Part 1 [2], these rules are divided into approximation and adjunction rules, together with the Ackermann rules used to eliminate propositional variables. As also indicated in [2], the major challenge to extending the algebraic-algorithmic approach to the modal mu-calculus is obtaining sound approximation and adjunction rules for formulas involving fixed point binders: in the binder-free setting each connective can be treated in isolation on its own order-theoretic merits; in the presence of binders the the analysis must take into account the order-theoretic behaviour of whole formulas. As a result the derivation rules become significantly more involved.