SéMantiques et Syntaxes vectorielles de la logique LinéAIRE. (Vectorial Semantics and Syntax of Linear Logic)

VECTORIAL SEMANTICS AND SYNTAX OF LINEAR LOGICWith finiteness spaces, Ehrhard has shown a semantics of linear logic with a differentiationoperation. In this framework, the interpretation of formulas can be represented as Taylor series.This has led to the introduction of a differential syntax. This thesis in denotational semanticspursues this work through an exploration of vectorial semantics of linear logic and contributesto the semantic and syntactic study of the Taylor formula.In the first part, we tackle semantics. We present the interpretation of linear logic construc-tions in linearised topological vector spaces. We build an intrinsic notion of finiteness spaces,named finitary Lefschetz spaces. We characterise the linearised topological vector spaces whichare reflexive and complete thanks to linear bornologies. Last, we show that the Taylor seriesproperty of finiteness spaces is still true in the linearised topological vector spaces framework.The second part is about differential syntax. The syntactic Taylor formula translate termsinto a sum of differential terms representing different executions. As shown by Ehrhard andRegnier, the terms that are the target of this translation are coherent. We introduce a totalsemantics which catch this relation. We build a linear extension of lambda-calculus, namedbarycentric-calculus, interpreted by the total semantics. Finally, in the differential nets frame-work, we present a non-deterministic algorithm which decides if a finite set of differential netscomes from a linear logic net through the syntactic Taylor formula.