Injectivity of relational semantics for (connected) MELL proof-structures via Taylor expansion

Given a syntax S endowed with some rewrite rules, and given a denotational semantics D for S (i.e. a semantics which gives to any term t of S an interpretation t D that is invariant under the rewrite rules), we say that D is injective with respect to S if, for any two normal terms t and t of S, t D = t D implies t = t. In categorical terms, injectivity corresponds to faithfulness of the " interpretation-functor " from S to D; it is a natural and well studied question for denotational semantics of λ-calculi and term rewriting systems (see [7,10]). All (positive) results of injectivity of denotational models with respect to some syntax fit in the general perspective of abolishing the traditional distinction between syntax and semantics. Starting from investigations on denotational semantics of System F (second order typed λ-calculus), in 1987 Girard [8] introduced linear logic (LL), a refinement of intuitionistic logic. He defines two new modalities, ! and ?, giving a logical status to structural rules and allowing one to distinguish between linear resources (i.e. usable exactly once during the cut-elimination process) and resources available at will (i.e. erasable and duplicable during the cut-elimination process). One of the main features of LL is the possibility of representing proofs geometrically (so as the λ-calculus terms) by means of particular graphs called proof-structures. Among proof-structures it is possible to characterize " in a geometric way " the ones corresponding to proofs in LL sequent calculus through the Danos-Regnier correctness criterion [2] (see also [11]): roughly speaking, a proof-structure is a proof-net (i.e. it corresponds to a proof in LL sequent calculus) if and only if it fulfils some conditions about acyclicity and connectedness (ACC). 4 Ehrhard [3] introduced finiteness spaces, a denotational model of LL (and λ-calculus) which interprets formulas by topological vector spaces and proofs by analytical functions: in this model the operations of differentiation and the Taylor expansion make sense. Ehrhard and Regnier [4,5,6] internalized these operations in the syntax and thus introduced differential linear logic DiLL 0 (which 4 Strictly speaking, this equivalence holds only in some fragments of LL, for example the multiplicative one (MLL) without ⊥. In larger fragments of LL, as for instance MELL (the multiplicative-exponential fragment of LL, sufficiently expressive to encode the λ-calculus) one only has that all proof-nets are ACC proof-structures, but to obtain the converse ACC …