Dominic Hughes - Selected Papers

Towards Hilbert's 24th Problem: Combinatorial Proof Invariants (Preliminary version)
[pdf - ps.gz]
Proceedings of WoLLIC 2006, ENTCS 165, 2006.

Proofs Without Syntax [Ann. Math. '06] introduced polynomial-time checkable combinatorial proofs for classical propositional logic. This sequel approaches Hilbert's 24th Problem 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.