2023/07/31 by Massaioli, Fabio
#F.3.1 #F.3.2 #F.4.1 #FOS: Computer and information sciences #Logic in Computer Science (cs.LO)
paper · doi:10.48550/arxiv.2307.16594
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.