We present an extension of the lambda-calculus with differential constructions motivated by a model of linear logic discovered by the first author and presented in [Ehr01]. We state and prove some basic results (confluence, weak normalization in the typed case), and also a theorem relating the usual Taylor series of analysis to the linear head reduction of lambda-calculus.

In this tutorial, we introduce the Curry-Howard (proof-program) correspondence which is usually restricted to intuitionistic logic. We explain how to extend this correspondence to the whole of mathematics and we build a simple suitable machine for this.

Abstract. This paper presents a logical approach to the translation of functional calculi into concurrent process calculi. The starting point is a type system for the π-calculus closely related to linear logic. Decompositions of intuitionistic and classical logics into this system provide typepreserving translations of the λ- and λµ-calculus, both for call-by-name and call-by-value evaluation strategies. Previously known encodings of the λ-calculus are shown to correspond to particular cases of this logical embedding. The realisability interpretation of types in the π-calculus provides systematic soundness arguments for these translations and allows for the definition of type-safe extensions of functional calculi. 1

We present an extension of the lambda-calculus with di erential constructions. We state and prove some basic results (con uence, strong normalization in the typed case), and also a theorem relating the usual Taylor series of analysis to the linear head reduction of lambda-calculus. Keywords. Lambda-calculus, linear logic, denotational semantics, linear head reduction. Prerequisites. This paper assumes from the reader some basic knowledge in lambda-calculus and an elementary (but not technical) knowledge of di erential calculus. Notations. Following [Kri93], we denote by (s)t the lambda-calculus application of s to t. The expression (s)t1... tn denotes the term ( · · · (s)t1 · · ·)tn when n ≥ 1, and s when n = 0. Accordingly, if A1,..., An and A are types, both expressions A1,..., An → A and A1 → · · · → An → A denote the type A1 → ( · · · (An → A) · · ·). If a1,..., an are elements of some given set S, we denote by [a1,..., an] the corresponding multi-set over S. If x and y are variables, δx,y is equal to 1 if x = y and to 0 otherwise. We denote by N + the set of positive integers {1, 2,...}.

second order string ACG with deterministic tree walking transducers