Intuitionistic Differential Nets and Resource Lambda-Calculus

We define pure intuitionistic differential nets, extending Ehrhard and Regnier’s differential interaction netswith the exponential box of Linear Logic.Normalization of the exponential reduction and confluence of the full one is proved. Though interesting and independent on their own, these results are directed and adjusted to give a translation of Boudol’s untyped λ-calculus with multiplicities extended with a linear-non linear reduction à la Ehrhrad andRegnier’s differentialλ-calculus. Such reduction comes in two flavours: baby-step and giant-step β-reduction. The translation, based on Girard’s encoding A → B ∼ !A ⊸ B and as such extending the usual one for λ-calculus and proof nets, is in a sense injective and surjective and enjoys bisimulation for giant-step β-reduction, a result from which we also derive confluence of both the reductions.