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We extend the theory of unified correspondence to a very broad class of logics with algebraic semantics given by varieties of normal lattice expansions (LEs), also known as 'lattices with operators'. Specifically, we introduce a very general syntactic definition of the class of Sahlqvist formulas and inequalities, which applies uniformly to each LE-signature and is given purely in terms of the order-theoretic properties of the algebraic interpretations of the logical connectives. Together with this, we introduce a variant of the algorithm ALBA, specific to the setting of LEs, which effectively computes first-order correspondents of LE-inequalities, and is guaranteed to succeed on a wide class of inequalities (the so-called inductive inequalities) which significantly extend the Sahlqvist class. Further, we show that every inequality on which ALBA succeeds is canonical. The projection of these results yields state-of-the-art correspondence theory for many well known substructural logics, such as the Lambek calculus and its extensions, the Lambek-Grishin calculus, the logic of (not necessarily distributive) de Morgan lattices, and the multiplicative-additive fragment of linear logic. theoretical framework which would provide a mathematically grounded way to compare (hierarchies of) Sahlqvisttype classes belonging to different logical settings. 1 Along with this lack of uniformity, each Sahlqvist-type results is tied to a particular choice of relational semantics for the relevant logic. Such a choice could be motivated by the fact that a logic has a uniquely established set-based semantics, but for many logics, like substructural logics, this is not the case. Hence, it is desirable to have a modular Sahlqvist theory that would distinguish core characteristics from incidental details relating to a particular choice of relational semantics.A theory which subsumes the previous results and which satisfies the desiderata of uniformity and modularity is currently emerging, and has been dubbed unified correspondence [7]. It is built on duality-theoretic insights [11] and uniformly exports the state-of-the-art in Sahlqvist theory from normal modal logic to a wide range of logics which include, among others, intuitionistic and distributive lattice-based (normal modal) logics [10], non-normal (regular) modal logics of arbitrary modal signature [39], hybrid logics [14], and mu-calculus [5,6].The breadth of this work has also stimulated many and varied applications. Some are closely related to the core concerns of the theory itself, such as the understanding of the relationship between different methodologies for obtaining canonicity results [38,9], or of the phenomenon of pseudo-correspondence [12]. Other, possibly surprising applications include the dual characterizations of classes of finite lattices [20], and the identification of the syntactic shape of axioms which can be translated into analytic structural rules of a proper display calculus [28]. Finally, the insights of unified correspondence theory have made it possible ...

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Abstract. Modal formulae express monadic second-order properties on Kripke frames, but in many important cases these have first-order equivalents. Computing such equivalents is important for both logical and computational reasons. On the other hand, canonicity of modal formulae is important, too, because it implies frame-completeness of logics axiomatized with canonical formulae.Computing a first-order equivalent of a modal formula amounts to elimination of secondorder quantifiers. Two algorithms have been developed for second-order quantifier elimination: SCAN, based on constraint resolution, and DLS, based on a logical equivalence established by Ackermann.In this paper we introduce a new algorithm, SQEMA, for computing first-order equivalents (using a modal version of Ackermann's lemma) and, moreover, for proving canonicity of modal formulae. Unlike SCAN and DLS, it works directly on modal formulae, thus avoiding Skolemization and the subsequent problem of unskolemization. We present the core algorithm and illustrate it with some examples. We then prove its correctness and the canonicity of all formulae on which the algorithm succeeds. We show that it succeeds not only on all Sahlqvist formulae, but also on the larger class of inductive formulae, introduced in our earlier papers. Thus, we develop a purely algorithmic approach to proving canonical completeness in modal logic and, in particular, establish one of the most general completeness results in modal logic so far.

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 ✩