SÉMANTIQUES ET SYNTAXES VECTORIELLES DE LA LOGIQUE LINÉAIRE

Christine Tasson 1, *
Abstract : With finiteness spaces, Ehrhard has shown a semantics of linear logic with a differentiation operation. 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 semantics pursues this work through an exploration of vectorial semantics of linear logic and contributes to the semantic and syntactic study of the Taylor formula. In the first part, we tackle semantics. We present the interpretation of linear logic constructions in linearised topological vector spaces. We build an intrinsic notion of finiteness spaces, named finitary Lefschetz spaces.We characterise the linearised topological vector spaces which are reflexive and complete thanks to linear bornologies. Last, we show that the Taylor series property 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 terms into a sum of differential terms representing different executions. As shown by Ehrhard and Regnier, the terms that are the target of this translation are coherent. We introduce a total semantics which catch this relation. We build a linear extension of lambda-calculus, named barycentric-calculus, interpreted by the total semantics. Finally, in the differential nets framework, we present a non-deterministic algorithm which decides if a finite set of differential nets comes from a linear logic net through the syntactic Taylor formula.
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https://tel.archives-ouvertes.fr/tel-00440752
Contributor : Christine Tasson <>
Submitted on : Monday, January 4, 2010 - 10:38:01 PM
Last modification on : Friday, January 4, 2019 - 5:32:59 PM
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Christine Tasson. SÉMANTIQUES ET SYNTAXES VECTORIELLES DE LA LOGIQUE LINÉAIRE. Informatique [cs]. Université Paris-Diderot - Paris VII, 2009. Français. ⟨tel-00440752v3⟩

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