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Algebraic Totality, towards Completeness

  • Christine Tasson
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5608)

Abstract

Finiteness spaces constitute a categorical model of Linear Logic (LL) whose objects can be seen as linearly topologised spaces, (a class of topological vector spaces introduced by Lefschetz in 1942) and morphisms as continuous linear maps. First, we recall definitions of finiteness spaces and describe their basic properties deduced from the general theory of linearly topologised spaces. Then we give an interpretation of LL based on linear algebra. Second, thanks to separation properties, we can introduce an algebraic notion of totality candidate in the framework of linearly topologised spaces: a totality candidate is a closed affine subspace which does not contain 0. We show that finiteness spaces with totality candidates constitute a model of classical LL. Finally, we give a barycentric simply typed lambda-calculus, with booleans \({\mathcal{B}}\) and a conditional operator, which can be interpreted in this model. We prove completeness at type \({\mathcal{B}}^n\to{\mathcal{B}}\) for every n by an algebraic method.

Keywords

Topological Vector Space Linear Logic Product Topology Continuous Linear Function Compact Open Topology 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Christine Tasson
    • 1
  1. 1.Preuves, Programmes, SystèmesFrance

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