Abstract. In many applications and in many fields, algorithms can considerably be speed up if the underlying arithmetical computations are considered carefully. In this article, we present a theoretical analysis of discrete affine transformations in higher dimension. More precisely, we investigate the arithmetical paving structure induced by the transformation to design fast algorithms.

International audienceIn many applications and in many fields, algorithms can considerably be speed up if the underlying arithmetical computations are considered carefully. In this article, we present a theoretical analysis of affine transformations in dimension 3. More precisely, we investigate the arithmetical paving induced by the transformation to design fast algorithms

Abstract. We show how Modified Bar-Recursion, a variant of Spector's Bar-Recursion due to Berger and Oliva can be used to realize the Axiom of Choice in Parigot's Lambda-Mu-calculus, a direct-style language for the representation and evaluation of classical proofs. We rely on Hyland-Ong innocent games. They provide a model to perform the usual infinitary reasoning on Bar-Recursion needed for the realization of classical choice, and where, moreover, the standard datatype of natural numbers is in the image of a CPS-translation. IntroductionPeano's Arithmetic in all finite types (PA ω ) is a multisorted version of firstorder Peano's Arithmetic, with one sort for each simple type, together with the constants of Gödel's System T and their defining equations. When augmenting PA ω with the Axiom of Choice (CAC), we obtain a system known to contain large parts of classical analysis (see e.g. [9,16]). A similar system can be obtained by extending Peano's Arithmetic to Second-Order Logic (see e.g. [16]).We are interested here in the realizability interpretation of PA ω + CAC. Realizability is a mathematical tool, part of the Curry-Howard correspondence, used to extract computational content from formal proofs.The usual route to get a computational interpretation of (some extension of) PA ω is to apply a negative translation, yielding proofs in (some extension of) Heyting's Arithmetic in all finite types (HA ω , the intuitionist variant of PA ω , see e.g. [19]), followed by a computational interpretation of the translated proofs. Realizability for HA ω can be obtained in simply-typed settings, typically using Gödel's System T. In this way, CAC is translated to a formula which can be realized by combining a realizer of the Intuitionistic Axiom of Choice (IAC) with a realizer of the Double Negation Shift (DNS, see Sect. 3). Intuitionistic choice is easily realizable, and realizers of DNS can be obtained by adapting Spector's Bar-Recursion to realizability [3,4].We are interested here in a computational interpretation of PA ω + CAC based on a realizability interpretation directly for classical proofs. It has been noted by Griffin [6] that the control operator call/cc of the functional language Scheme ⋆ UMR 5668 CNRS ENS Lyon UCBL INRIA 2 can be typed using Peirce's Law, which gives full Classical Logic when added to Intuitionistic Logic. Since then, there have been much work on calculi for Classical Logic, starting from Parigot's λµ-calculus [14]. Moreover, Krivine has developed a notion of Classical Realizability for Second-Order Peano's Arithmetic which relies on Girard's System F [10] (see also [13,12]).In this paper, we investigate a version of Spector's Bar-Recursion in a classical realizability setting for PA ω , obtained by adapting Krivine's Realizability to a simply-typed extension of Parigot's λµ-calculus. Our main point concerns Bar-Recursion. Handling Bar-Recursion in realizability (typically to show that it realizes DNS) involves some reasoning on infinite non-constructive objects. This infinitary reasoning can ...