In the present paper, the algorithmic correspondence theory developed in Conradie and Palmigiano [9] is extended to mu-calculi with a non-classical base. We focus in particular on the language of bi-intuitionistic modal mu-calculus. We enhance the algorithm ALBA introduced in Conradie and Palmigiano [9] so as to guarantee its success on the class of recursive mu-inequalities, which we introduce in this paper. Key to the soundness of this enhancement are the order-theoretic properties of the algebraic interpretation of the fixed point operators. We show that, when restricted to the Boolean setting, the recursive mu-inequalities coincide with the "Sahlqvist mu-formulas" defined in van Benthem, Bezhanishvili and Hodkinson [22]. IntroductionModal mu-calculus [17] 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. Modal mu-calculus has become a fundamental logical tool in theoretical computer science and has been extensively studied [3], and applied for instance in the context of temporal properties of systems, and of infinite properties of concurrent systems. Many expressive modal and temporal logics such as PDL, CTL, CTL * can be seen as fragments of the modal mu-calculus [3,15]. It provides a unifying framework connecting modal and temporal logics, automata theory and the theory of games, where fixed point constructions can be used to talk about the long term strategies of players, as discussed in [23].Correspondence theory studies the relationships between classical first-and second-order logic, and modal logic, interpreted on Kripke frames. A modal and a first-order formula correspond if they define the same class of structures. Specifically, Sahlqvist theory is concerned with the identification of syntactically specified classes of modal formulas which correspond to first-order formulas. Sahlqvist-style frame-correspondence theory for modal mu-calculus has recently been developed in [22]. Such analysis strengthens the general mathematical theory of the mu-calculus, facilitates the transfer of results ✩

We extend unified correspondence theory to Kripke frames with impossible worlds and their associated regular modal logics. These are logics the modal connectives of which are not required to be normal: only the weaker properties of additivity ✸x ∨ ✸y = ✸(x ∨ y) and multiplicativity ✷x ∧ ✷y = ✷(x ∧ y) are required. Conceptually, it has been argued that their lacking necessitation makes regular modal logics better suited than normal modal logics at the formalization of epistemic and deontic settings. From a technical viewpoint, regularity proves to be very natural and adequate for the treatment of algebraic canonicity Jónsson-style. Indeed, additivity and multiplicativity turn out to be key to extend Jónsson's original proof of canonicity to the full Sahlqvist class of certain regular distributive modal logics naturally generalizing Distributive Modal Logic [19,9]. Most interestingly, additivity and multiplicativity are key to Jónsson-style canonicity also in the original (i.e. normal) DML. Our contributions include: the definition of Sahlqvist inequalities for regular modal logics on a distributive lattice propositional base; the proof of their canonicity following Jónsson's strategy; the adaptation of the algorithm ALBA to the setting of regular modal logics on two non-classical (distributive lattice and intuitionistic) bases; the proof that the adapted ALBA is guaranteed to succeed on a syntactically defined class which properly includes the Sahlqvist one; finally, the application of the previous results so as to obtain proofs, alternative to Kripke's, of the strong completeness of Lemmon's epistemic logics E2-E5 with respect to elementary classes of Kripke frames with impossible worlds.

The present paper proposes a new introductory treatment of the very well known Sahlqvist correspondence theory for classical modal logic. The first motivation for the present treatment is pedagogical: classical Sahlqvist correspondence is presented in a uniform and modular way, and, unlike the existing textbook accounts, extends itself to a class of formulas laying outside the Sahlqvist class proper. The second motivation is methodological: the present treatment aims at highlighting the algebraic and order-theoretic nature of the correspondence mechanism. The exposition remains elementary and does not presuppose any previous knowledge or familiarity with the algebraic approach to logic. However, it provides the underlying motivation and basic intuitions for the recent developments in the Sahlqvist theory of nonclassical logics, which compose the so-called unified correspondence theory.

By de Vries duality, the category of compact Hausdorff spaces is dually equivalent to the category of de Vries algebras (complete Boolean algebras endowed with a proximity-like relation). We provide an alternative "modal-like" duality by introducing the concept of a Gleason space, which is a pair (X, R), where X is an extremally disconnected compact Hausdorff space and R is an irreducible equivalence relation on X. Our main result states that the category of Gleason spaces is equivalent to the category of compact Hausdorff spaces, and is dually equivalent to the category of de Vries algebras.

The theory of canonical extensions typically considers extensions of maps A ! B to maps A ! B. In the present paper, the theory of canonical extensions of maps A ! B to maps A ! B is developed, and is applied to obtain a new canonicity proof for those inequalities in the language of Distributive Modal Logic (DML) on which the algorithm ALBA [9] is successful.