Taylor expansion in linear logic is invertible

Each Multiplicative Exponential Linear Logic (MELL) proof-net can be expanded into a differential net, which is its Taylor expansion. We prove that two different MELL proof-nets have two different Taylor expansions. As a corollary, we prove a completeness result for MELL: We show that the relational model is injective for MELL proof-nets, i.e. the equality between MELL proof-nets in the relational model is exactly axiomatized by cut-elimination. In the seminal paper by Harvey Friedman [18], it has been shown that equality between simply-typed lambda-terms in the full typed structure MX over an infinite set X is completely axiomatized by β and η: for any simply-typed lambda-terms v and u, we have (MX v = u ⇔ v ≃βη u). Some variations, refinements and generalizations of this result have been provided by Gordon Plotkin [28] and Alex Simpson [31]. A natural problem is to know whether a similar result could be obtained for Linear Logic. Such a result can be seen as a “separation” theorem. To obtain such separation theorems, it is a prerequesite to have a “canonical” syntax. When Jean-Yves Girard introduced Linear Logic (LL) [19], he not only introduced a sequent calculus system but also “proof-nets”. Indeed, as for LJ and LK (sequent calculus systems for intuitionnistic and classical logic, respectively), different proofs in LL sequent calculus can represent “morally” the same proof: proof-nets were introduced to find a unique representative for these proofs. The technology of proof-nets was completely satisfactory for the multiplicative fragment without units. For proof-nets having additives, contractions or weakenings, it was easy to exhibit different proof-nets that should be identified. Despite some flaws, the discovery of proof-nets was striking. In particular, Vincent Danos proved by syntactical means in [6] the confluence of these proof-nets for the Multiplicative Exponential Linear Logic fragment (MELL). For additives, the problem to have a satisfactory notion of proof-net has been addressed in [22]. For MELL, a “new syntax” was introduced in [7]. In the original syntax, the following properties of the weakening and of the contraction did not hold: • the associativity of the contraction; For the multiplicative fragment with units, it has been recently shown [21] that, in some sense, no satisfactory notion of proof-net can exist. Our proof-nets have no jump, so they identify too many sequent calculus proofs, but not more than the relational semantics.