## Canonical proof nets for classical logic

### BibTeX

@MISC{Mckinley_canonicalproof,

author = {Richard Mckinley},

title = {Canonical proof nets for classical logic},

year = {}

}

### OpenURL

### Abstract

Abstract. Proof nets provide abstract counterparts to sequent proofs modulo rule permutations; the idea being that if two proofs have the same underlying proof-net, they are in essence the same proof. Providing a convincing proof-net counterpart to proofs in the classical sequent calculus is thus an important step in understanding classical sequent calculus proofs. By convincing, we mean that (a) there should be a canonical function from sequent proofs to proof nets, (b) it should be possible to check the correctness of a net in polynomial time, (c) every correct net should be obtainable from a sequent calculus proof, and (d) there should be a cut-elimination procedure which preserves correctness. Previous attempts to give proof-net-like objects for propositional classical logic have failed at least one of the above conditions. In [23], the author presented a calculus of proof nets (expansion nets) satisfying (a) and (b); the paper defined a sequent calculus corresponding to expansion nets but gave no explicit demonstration of (c). That sequent calculus, called LK ∗ in this paper, is a novel one-sided sequent calculus with both additively and multiplicatively formulated disjunction rules. In this paper (a selfcontained extended version of [23]) , we give a full proof of (c) for expansion nets with respect to LK ∗, and in addition give a cut-elimination procedure internal to expansion nets – this makes expansion nets the first notion of proof-net for classical logic satisfying all four criteria. 1