Polarized Proof-Nets: Proof-Nets for LC

We deene a notion of polarization in linear logic (LL) coming from the polarities of Jean-Yves Girard's classical sequent calculus LC 4]. This allows us to deene a translation between the two systems. Then we study the application of this polarization constraint to proof-nets for full linear logic described in 7]. This yields an important simpli-cation of the correctness criterion for polarized proof-nets. In this way we obtain a system of proof-nets for LC. The study of cut-elimination takes an important place in proof-theory. Much work is spent to deal with commutation of rules for cut-elimination in sequent calculi. The introduction of proof-nets (see 7] for instance) solves commutation problems and allows us to deene a clear notion of reduction and complexity. In 4], Jean-Yves Girard deenes the sequent calculus LC using polarities. LC is a reenement of LK with a deterministic cut-elimination. J.-Y. Girard leaves open the following problem about the syntax: \Find a better syntax (which would be to LC what typed-calculus is to LJ) for normalization. .. ]. A kind of proof-nets could be the solution, and the fact that proof-nets are not available for full linear logic could be compensated by the fact that only certain linear conngurations are used." In this paper we address this problem but the situation is now slightly different since proof-nets for full linear logic are given in 7]. In these proof-nets, the boxes for additives are replaced by weights on the nodes giving less sequen-tialization information. To use these proof-nets, we will rst deene a translation from LC to the fragment LLP of LL deened by restricting to polarized formulas. The \particular linear conngurations" of LC correspond to the polarization of LLP. We then turn to the study of proof-structures for LLP and show that the restriction to polarized formulas induces a natural orientation, the orientation of polarization, which is respected by the paths of LL's correctness condition (Orientation Lemma). This yields a striking simpliication of the correctness condition which allows us to get rid of the notion of switches. In particular it turns out to be cubic in the size of polarized proof-nets whereas the LL condition is immediately seen to be exponential.