Abstract. We give a simple term calculus for the multiplicative exponential fragment of Classical Linear Logic, by extending Barber and Plotkin’s system for the intuitionistic case. The calculus has the nonlinear andlinear implications as the basic constructs, andthis design choice allows a technically managable axiomatization without commuting conversions. Despite this simplicity, the calculus is shown to be sound andcomplete for category-theoretic models given by ∗-autonomous categories with linear exponential comonads. 1

Abstract. We present a rewriting system for the linear lambda calculus corresponding to the {!, ⊸}-fragment of intuitionistic linear logic. This rewriting system is shown to be strongly normalizing, and Church-Rosser modulo the trivial commuting conversion. Thus it provides a simple decision method for the equational theory of the linear lambda calculus. As an application we prove the strong normalization of the simply typed computational lambda calculus by giving a reduction-preserving translation into the linear lambda calculus. 1

We show that the f!;(g-fragment of Intuitionistic Linear Logic is full in the f!; (g-fragment, both formulated as linear lambda calculi. The proof is a mild extension of our previous technique used for showing the fullness of Girard's translation from Intuitionistic Logic into Intuitionistic Linear Logic, and makes use of double-parameterized logical predicates.