Abstract. We give a calculus of proof-nets for classical propositional logic. These nets improve on a proposal due to Robinson by validating the associativity and commutativity of contraction, and provide canonical representants for classical sequent proofs modulo natural equivalences. We present the relationship between sequent proofs and proof-nets as an annotated sequent calculus, deriving formulae decorated with expansion/deletion trees. We then see a subcalculus, expansion nets, which in addition to these good properties has a polynomial-time correctness criterion. 1

The present report is a, somewhat lengthy, addendum to Danos et al.(1997), where the elimination of cuts from derivations in sequent calculus for classical logic was studied `from the point of view of linear logic'. To that purpose a formulation of classical logic was used, that - as in linear logic - distinguishes between multiplicative and additive versions of the binary connectives. The main novelty here is the observation that this type-distinction is not essential: we can allow classical sequent derivations to use any combination of additive and multiplicative introduction rules for each of the connectives, and still have strong normalization and conuence of tq-reductions.

We rst dene the polarized proof-nets, an extension of MELL proof-nets for the polarized fragment of linear logic; the main dierence with usual proof-nets is that we allow structural rules on any negative formula. The essential properties (conuence, strong normalization in the typed case) of polarized proof-nets are proved using a reduction preserving translation into usual proof-nets. We then give a reduction preserving encoding of Parigot's -terms for classical logic as polarized proof-nets. It is based on the intuitionistic translation: A ! B !A ( B, so that it is a straightforward extension of the usual translation of -calculus into proof-nets. We give a reverse encoding which sequentializes any polarized proof-net as a -term. In the last part of the paper, we extend the -equivalence for -calculus to -calculus. Interestingly, this new -equivalence relation identies normal -terms. We eventually show that two terms are equivalent i they are translated as the same ...

We give the precise correspondence between polarized linear logic and polarized classical logic. The properties of focalization and reversion of linear proofs [AP91, Gir91a, DJS97] are at the heart of our analysis: we show that the tq-protocol of normalization (dened in [DJS97]) for the classical systems LK pol and LK ; pol perfectly ts normalization of polarized proof-nets. In section 6, some more semantical considerations allow to recover LC as a renement of multiplicative LK pol . MSC: 03F05; 03F07; 03F52 Keywords: Classical Logic, Linear Logic, Cut-Elimination, Proof-Nets, Denotational Semantics, Polarization, Focalization, Reversion.

Abstract not found