The main concern of this paper is the design of a noetherian and confluent normalization for LK 2 (that is, classical second order predicate logic presented as a sequent calculus). The method we present is powerful: since it allows us to recover as fragments formalisms as seemingly different as Girard's LC and Parigot's , FD ([9, 11, 27, 31]), delineates other viable systems as well, and gives means to extend the Krivine/Leivant paradigm of `programming-with-proofs' ([22, 23]) to classical logic; it is painless: since we reduce strong normalization and confluence to the same properties for linear logic (for non-additive proof nets, to be precise) using appropriate embeddings (so-called decorations); it is unifying: it organizes known solutions in a simple pattern that makes apparent the how and why of their making. A comparison of our method to that of embedding LK into LJ (intuitionistic sequent calculus) brings to the fore the latter's defects for these `deconstructi...

In this paper, we develop a system of typed lambda-calculus for the Zermelo-Fraenkel set theory, in the framework of classical logic. The first, and the simplest system of typed lambda-calculus is the system of simple types, which uses the intuitionistic propositional calculus, with the only connective #. It is very important, because the well known Curry-Howard correspondence between proofs and programs was originally discovered with it, and because it enjoys the normalization property : every typed term is strongly normalizable. It was extended to second order intuitionistic logic, in 1970, by J.-Y. Girard[4], under the name of system F, still with the normalization property. More recently, in 1990, the Curry-Howard correspondence was extended to classical logic, following Felleisen and Griffin [6] who discovered that the law of Peirce corresponds to control instructions in functional programming

When using the Curry-Howard correspondence in order to obtain executable programs from mathematical proofs, we are faced with a difficult problem: to interpret each axiom of our axiom system for mathematics (which may be, for example, second order classical logic, or classical set theory) as an instruction of our programming language. This problem

We show that the decomposition of Intuitionistic Logic into Linear Logic along the equation A ! B = !A ( B may be adapted into a decomposition of classical logic into LLP, the polarized version of Linear Logic. Firstly we build a categorical model of classical logic (a Control Category) from a categorical model of Linear Logic by a construction similar to the co-Kleisli category. Secondly we analyse two standard Continuation-Passing Style (CPS) translations, the Plotkin and the Krivine's translations, which are shown to correspond to two embeddings of LLP into LL.

In 1990 J-L. Krivine introduced the notion of storage operators. They are λ-terms which simulate call-by-value in the call-by-name strategy and they can be used in order to modelize assignment instructions. J-L. Krivine has shown that there is a very simple second order type in AF 2 type system for storage operators using Gődel translation of classical to intuitionistic logic. In order to modelize the control operators, J-L. Krivine has extended the system AF 2 to the classical logic. In his system the property of the unicity of integers representation is lost, but he has shown that storage operators typable in the system AF 2 can be used to find the values of classical integers. In this paper, we present a new classical type system based on a logical system called mixed logic. We prove that in this system we can characterize, by types, the storage operators and the control operators. We present also a similar result in the M. Parigot’s λµ-calculus. 1

Introduction In this lecture, we shall consider a very well known typed #-calculus system, which is the second order #-calculus (also called "system F") of Girard [2], rediscovered by Reynolds [16] in a computer science frame. We shall extend it in two ways: . Types will be formulas of second order predicate calculus, and not only, as in system F, second order propositional calculus [5, 6]. In a certain sense, this is a harmless extension, since the #-terms which are typable are the same. This kind of extension has already been considered by D. Leivant [11]. . A much more serious extension is the following: the underlying logic will be classical logic, and not only, as in system F, intuitionistic logic. Extraction of programs from classical proofs has been considered, since two or three years by several people (C. Murthy [12], J.Y. Girardapproach has the following features: 1. We

In 1990 Krivine (1990b) introduced the notion of storage operators. They are λ-terms which simulate call-by-value in the call-by-name strategy. Krivine (1990b) has shown that there is a very simple type in the AF 2 type system for storage operators using Gődel translation from classical to intuitionistic logic. Parigot (1993a) and Krivine (1994) have shown that storage operators play an important tool in classical logic. In this paper, we present a synthesis of various results on this subject. 1

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We present a new method to extract from a classical proof of ∀x(I[x] → ∃y(O[y] ∧ S[x, y])) a program computing y from x. This method applies when O is a data type and S is a decidable predicate. Algorithms extracted this way are often far better than a stupid enumeration of all the possible outputs and this is verified on a non trivial example: a proof of Dickson’s lemma. 1 Introduction. Since Griffin and Felleisen [7, 8], we know a relation between classical proofs and programs. However, it is not true that from a classical proof of the existence of an object you can compute this object. This would clearly be a contradiction with the existence of provably total but non computable functions (such as a function saying

Le contenu computationnel des preuves: No-counterexample interpretation et spécification des théorèmes de l’arithmétique mémoire sous la direction de Jean-Louis Krivine