Algorithmic correspondence and completeness in modal logic. I. The core algorithm SQEMA

TL;DR

The paper introduces SQEMA, a new algorithm for computing first-order equivalents of modal formulas and proving their canonicity, extending the class of formulas for which canonical completeness can be algorithmically established.

Contribution

SQEMA is a novel algorithm that directly operates on modal formulas to compute first-order equivalents and prove canonicity, surpassing previous methods like SCAN and DLS.

Findings

Succeeds on all Sahlqvist formulas

Extends to the larger class of inductive formulas

Establishes a broad class of canonically complete modal formulas

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 second-order 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.