Algorithmic correspondence for intuitionistic modal mu-calculus, Part 1

Modal mu-calculus [5] is a logical framework combining simple modalities with fixed point operators, enriching the expressivity of modal logic so as to deal with infinite processes like recursion. It has a simple syntax, an easily given semantics, and is decidable. Many expressive modal and temporal logics such as PDL, CTL and CTL∗ can be seen as fragments of the modal mu-calculus. Sahlqvist-style frame-correspondence theory for modal mu-calculus has recently been developed in [6]. The correspondence results in [6] are developed purely model-theoretically. However, they can be naturally encompassed within the existing algebraic approach to correspondence theory [2, 3, 4], and generalized to mu-calculi on a weaker-than-classical (and, particularly, intuitionistic) base. We focus in particular on the language of bi-intuitionistic modal mu-calculus, and we enhance the algorithm, or calculus for correspondence, ALBA [3] for the elimination of monadic second order variables, so as to guarantee its success over a class including the Sahlqvist mu-formulas defined in [6]. The algorithm ALBA: an example. Consider the transitivity axiom p→ p. As is discussed at length in [3, 2], every piece of argument used to prove this correspondence on frames can be translated by duality to their complex algebras, which are perfect distributive lattices with operators. The validity condition on frames translates into its complex algebra as A |= ∀p[ p ≤ p], followed by the chain of equivalences