Proof Nets with Explicit Negation for Multiplicative Linear Logic

Multiplicative linear logic (MLL) was introduced in Gi87] as a one-sided sequent calculus: linear negation is a notion that is deened, via De Morgan identities. One obtains proof nets for MLL by identifying derivations in the one-sided calculus that are equal up to a permutation of inference rules. In this paper we consider a similar quotient for the formulation of MLL as a two-sided sequent calculus: to the usual set of links we add links also for the left rules. As a consequence, negation need no longer be deened, but can be treated as a basic connective. We develop the fundamental theory (substructures, empires and sequential-ization) for this variation on the notion of proof net, and show how to obtain Girard's sequentialization theorem for the standard proof nets in one-sided se-quent calculus as a corollary.