Cut-Restriction: From Cuts to Analytic Cuts

Cut-elimination is the bedrock of proof theory with a multitude of applications from computational interpretations to proof analysis. It is also the starting point for important meta-theoretical investigations into decidability, complexity, disjunction property, interpolation, and more. Unfortunately cut-elimination does not hold for the sequent calculi of most non-classical logics. It is well-known that the key to applications is the subformula property (a typical consequence of cut-elimination) rather than cut-elimination itself. With this in mind, we introduce cut-restriction, a procedure to restrict arbitrary cuts to analytic cuts (when elimination is not possible). The algorithm applies to all sequent calculi satisfying language-independent and simple-to-check conditions, and it is obtained by adapting age-old cut-elimination. Our work encompasses existing results in a uniform way, subsumes Gentzen’s cut-elimination, and establishes new analytic cut properties.