Proof Nets with Explicit Negation for Multiplicative Linear Logic

Abstract

Multiplicative linear logic (MLL) was introduced in [Gi87] as a one-sided sequent calculus: linear negation is a notion that is defined, 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 defined, but can be treated as a basic connective. We develop the fundamental theory (substructures, empires and sequentialization) 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 sequent calculus as a corollary. [email protected] URL: http://www.math.uu.nl/people/puite/ Contents 1 Introduction 1 2 Proof structures of MLL 3 3 Proof nets of MLL 11 4 The proof net of a derivati..