41 pagesInternational audienceWe present an extension of the lambda-calculus with differential constructions. We state and prove some basic results (confluence, 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

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30 pagesInternational audienceWe introduce interaction nets for a fragment of the differential lambda-calculus and exhibit in this framework a new symmetry between the "of course" and the "why not" modalities of linear logic, which is completely similar to the symmetry between the "tensor" and "par" connectives of linear logic. We use algebraic intuitions for introducing these nets and their reduction rules, and then we develop two correctness criteria (weak typeability and acyclicity) and show that they guarantee strong normalization. Finally, we outline the correspondence between this interaction nets formalism and the resource lambda-calculus

International audienceWe define the complete Taylor expansion of an ordinary lambda-term as an infinite linear combination --- with rational coefficients --- of terms of a resource calculus similar to Boudol's resource lambda-calculus. In this calculus, all applications are (multi-)linear in the algebraic sense, i.e. commute with linear combination of the function or the argument. We study the collective behaviour of the beta-reducts of the terms occurring in the Taylor expansion of any ordinary lambda-term, using a uniformity property that they enjoy