Pure Sequent Calculi

Abstract

Analyticity, also known as the subformula property, typically guarantees decidability of derivability in propositional sequent calculi. To utilize this fact, two substantial gaps have to be addressed: (i) What makes a sequent calculus analytic? and (ii) How do we obtain an efficient decision procedure for derivability in an analytic calculus? In the first part of this article, we answer these questions for pure calculi —a general family of fully structural propositional sequent calculi whose rules allow arbitrary context formulas. We provide a sufficient syntactic criterion for analyticity in these calculi, as well as a productive method to construct new analytic calculi from given ones. We further introduce a scalable decision procedure for derivability in analytic pure calculi by showing that it can be (uniformly) reduced to classical satisfiability. In the second part of the article, we study the extension of pure sequent calculi with modal operators. We show that such extensions preserve the analyticity of the calculus and identify certain restricted operators (which we call “Next” operators) that are also amenable for a general reduction of derivability to classical satisfiability. Our proofs are all semantic, utilizing several strong general soundness and completeness theorems with respect to non-deterministic semantic frameworks: bivaluations (for pure calculi) and Kripke models (for their extension with modal operators).