On the semantics of proofs in classical sequent calculus

These pre-print (unpublished) notes providing extended draft material for a forthcoming shorter and updated paper. The talk I gave at LC2023 in Milan is also based upon these notes.

Abstract

We discuss the problem of finding non-trivial invariants of non-deterministic, symmetric cut-reduction procedures in the classical sequent calculus. We come to the conclusion that (an enriched version of) the propositional fragment of GS4 — i.e. the one-sided variant of Kleene’s context-sharing style sequent system G4, where independent rule applications permute freely — is an ideal framework in which to attack the problem. We show that the graph induced by axiom rules linking dual atom occurrences is preserved under arbitrary rule permutations in the cut-free fragment of GS4. We then refine the notion of axiom-induced graph so as to extend the result to derivations with cuts, and we exploit the invertibility of logical rules to define a global normalisation procedure that preserves the refined axiom-induced graphs, thus yielding a non-trivial invariant of cut-elimination in GS4. Finally, we build upon the result to devise a new proof system for classical propositional logic, where the rule permutations of GS4 reduce to identities.