Algorithmic Correspondence and Canonicity for Possibility Semantics (Abstract)

Unified Correspondence. Correspondence and completeness theory have a long history in modal logic, and they are referred to as the “three pillars of wisdom supporting the edifice of modal logic” [22, page 331] together with duality theory. Dating back to [20,21], the Sahlqvist theorem gives a syntactic definition of a class of modal formulas, the Sahlqvist class, each member of which defines an elementary (i.e. first-order definable) class of Kripke frames and is canonical. Since modal logic on the frame level is essentially second-order, computing the first-order correspondence of a modal formula is a kind of second-order quantifier elimination. Recently, a uniform and modular theory which subsumes the above results and extends them to logics with a non-classical propositional base has emerged, and has been dubbed unified correspondence [5]. It is built on duality-theoretic insights [9] 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 and general (non-distributive) lattice-based (modal) logics [6,8], non-normal (regular) modal logics based on distributive lattices of arbitrary modal signature [19], hybrid logics [12], many valued logics [16] and bi-intuitionistic and lattice-based modal mu-calculus [1,3,2]. Unified correspondence theory has two components: the first one is a very general syntactic definition of Sahlqvist and inductive formulas, which applies uniformly to each logical signature and is given purely in terms of the order-theoretic properties of the algebraic interpretations of the logical connectives; the second one is the Ackermann lemma based algorithm ALBA, which is a generalization of SQEMA based on order-theoretic and algebraic insights, which effectively computes first-order correspondents of input formulas/inequalities, and is guaranteed to succeed on the Sahlqvist and inductive classes of formulas/inequalities. The algorithm aims at eliminating all propositional variables, which are, on the relational semantics side, second-order variables, and rewrite the formula into a quasi-inequality which contains only nominals and co-nominals, which are, on the relational semantics side, essentially first-order. In this sense, unified correspondence theory is essentially second-order quantifier elimination on the algebraic side. The breadth of this work has stimulated many and varied applications. Some are closely related to the core concerns of the theory itself, such as understanding the relationship between different methodologies for obtaining canonicity results [18,7], the phenomenon of pseudocorrespondence [10], and the investigation of 106