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Towards Hilbert's 24th Problem: Combinatorial Proof Invariants
, 2006
"... Proofs Without Syntax [37] introduced polynomialtime checkable combinatorial proofs for classical propositional logic. This sequel approaches Hilbert’s 24 th Problem with combinatorial proofs as abstract invariants for sequent calculus proofs, analogous to homotopy groups as abstract invariants for ..."
Abstract

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Proofs Without Syntax [37] introduced polynomialtime checkable combinatorial proofs for classical propositional logic. This sequel approaches Hilbert’s 24 th 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.
Combinatorial Proof Semantics
"... The paper Proofs Without Syntax [Annals of Mathematics, to appear] introduced the notion of a combinatorial proof for classical propositional logic. The present paper uses combinatorial proofs to define a semantics for classical propositional sequent calculus, an inductive translation from sequent ..."
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The paper Proofs Without Syntax [Annals of Mathematics, to appear] introduced the notion of a combinatorial proof for classical propositional logic. The present paper uses combinatorial proofs to define a semantics for classical propositional sequent calculus, an inductive translation from sequent proofs to combinatorial proofs. The semantics is abstract and efficient: abstract in the sense that it identifies many sequent proofs, and efficient in the sense that combinatorial proofs are polynomialtime checkable and the inductive translation is polynomial.