Towards Hilbert's 24th Problem: Combinatorial Proof Invariants: (Preliminary version)

Proofs Without Syntax [Hughes, D.J.D. Proofs Without Syntax. Annals of Mathematics 2006 (to appear), http://arxiv.org/abs/math/0408282 (v3). August 2004 submitted version also available: [35]] introduced polynomial-time checkable combinatorial proofs for classical propositional logic. This sequel approaches Hilbert's 24thProblem with combinatorial proofs as abstract invariants for sequent calculus proofs, analogous to homotopy groups as abstract invariants for topological spaces.The paper lifts a simple, strongly normalising cut elimination from combinatorial proofs to sequent calculus, factorising away the mechanical commutations of structural rules which litter traditional syntactic cut elimination.Sequent calculus fails to be surjective onto combinatorial proofs: the paper extracts a semantically motivated closure of sequent calculus from which there is a surjection, pointing towards an abstract combinatorial refinement of Herbrand's theorem.