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Logique dans le Facteur Hyperfini: Géométrie de l'Interaction et Complexité

Abstract : This work is a study of the geometry of interaction in the hyperfinite factor introduced by Jean-Yves Girard, and of its relations with ancient constructions. We start by showing how to obtain purely geometrical adjunctions as an identity between sets of cycles appearing between graphs. It is then possible, by chosing a function that measures those cycles, to obtain a numerical adjunction. We then show how to construct, on the basis of such a numerical adjunction, a geometry of interaction for multiplicative additive linear logic where proofs are interpreted as graphs. We also explain how to define from this construction a denotational semantics for MALL, and a notion of truth. We extend this setting in order to deal with exponential connectives and show a full soundness result for a variant of elementary linear logic (ELL). Since the constructions on graphs we define are parametrized by a function that measures cycles, we then focus our study to two particular cases. The first case turns out to be a combinatorial ver- sion of GoI5, and we thus obtain a geometrical caracterisation of its orthogonality which is based on Fuglede-Kadison determinant. The second particular case we study will giveus a refined version of ol- der constructions of geometry of interaction, where orthogonality is based on nilpotency. This allows us to show how these two versions of GoI, which seem quite different, are related and understand that the respective adjunctions are both consequences of a unique geometrical property. We then study the notion of subjective truth. We first define a slightly modified version of GoI5 where the notion of subjective truth is dependent on the choice a maximal abelian subalgebra (masa). We can show in this setting that there is a correspondance between Dixmier's classification of masas (regular, semi-regular, singular) and the fragments of linear logic one can interpret. We then explain how this notion of truth related to the notion of truth of GoI5, which depends on a choice of representation for the hyperfinite factor of type II∞. Finally, we study a proposition made by Girard to study complexity classes through the geometry of interaction in the hyperfinite factor. In particular, we detail Girard's caracterisation of the co-NL com- plexity class by showing how to encode by operators a problem which is complete for this complexity class.
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Submitted on : Friday, December 21, 2012 - 1:57:25 PM
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Thomas Seiller. Logique dans le Facteur Hyperfini: Géométrie de l'Interaction et Complexité. Logique [math.LO]. Aix-Marseille Université, 2012. Français. ⟨tel-00768403⟩

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