Logique et Interaction : une Étude Sémantique de la Totalité

Abstract : This thesis deals with the problem of using total strategies for the interpretation of proofs. The first part is about finitary logic. The analysis begins within a syntactical framework : we define a unary strongly normalizing lambda calculus, for which we recall the pointer abstract machine (PAM). We reduce the problem of preservation of totality by composition to a finiteness theorem on objects we call pointer structures. We give three different proofs of this result. The first uses a reduction to the normalization of the unary lambda calculus via the PAM, the second extracts from the problem a simple reduction on trees of integers, whereas the third uses a combinatorial argument of Coquand. The second part is a study of a sequent calculus mu-LJ with inductive and coinductive definitions, in which we give a simulation of Gödel's system T. We define the notion of mu-closed categories, which are models of mu-LJ. In the framework of games we define open arenas (arenas with free type variables). To each of these arenas we associate an open functor on the category of arenas and innocent strategies. We then describe on open arenas a loop construction, which we relate to McCusker's model of recursive types. Loops are enriched with winning conditions imported from parity games, which provide initial algebras and terminal coalgebras to open functors and builds a mu-closed category of games and total innocent strategies. Finally, we give an extension of mu-LJ to an infinite syntax, for which our model is fully complete.
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Pierre Clairambault. Logique et Interaction : une Étude Sémantique de la Totalité. Autre [cs.OH]. Université Paris-Diderot - Paris VII, 2010. Français. ⟨tel-00459307⟩

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