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...

We introduce the type theory ¯ v , a call-by-value variant of Parigot's ¯-calculus, as a Curry-Howard representation theory of classical propositional proofs. The associated rewrite system is Church-Rosser and strongly normalizing, and definitional equality of the type theory is consistent, compatible with cut, congruent and decidable. The attendant call-by-value programming language ¯pcf v is obtained from ¯ v by augmenting it by basic arithmetic, conditionals and fixpoints. We study the behavioural properties of ¯pcf v and show that, though simple, it is a very general language for functional computation with control: it can express all the main control constructs such as exceptions and first-class continuations. Proof-theoretically the dual ¯ v -constructs of naming and ¯-abstraction witness the introduction and elimination rules of absurdity respectively. Computationally they give succinct expression to a kind of generic (forward) "jump" operator, which may be regarded as a unif...

We present a formulae-as-types interpretation of Subtractive Logic (i.e. bi-intuitionistic logic). This presentation is two-fold: we first define a very natural restriction of the λµ-calculus which is closed under reduction and whose type system is a constructive restriction of the Classical Natural Deduction. Then we extend this deduction system conservatively to Subtractive Logic. From a computational standpoint, the resulting calculus provides a type system for first-class coroutines (a restricted form of first-class continuations). Keywords: Curry-Howard isomorphism, Subtractive Logic, control operators, coroutines. 1

This paper is the first part of a work whose purpose is to investigate duality in some related frameworks (cartesian closed categories, lambda-calculi, intuitionistic and classical logics) from syntactic, semantical and computational viewpoints. We start with category theory and we show that any bicartesian closed category with coexponents is degenerated (i.e. there is at most one arrow between two objects). The remainder of the paper is devoted to logical issues. We examine the propositional calculus underlying the type system of bicartesian closed categories with coexponents and we show that this calculus corresponds to subtractive logic: a conservative extension of intuitionistic logic with a new connector (subtraction) dual to implication. Eventually, we consider first order subtractive logic and we present an embedding of classical logic into subtractive logic. Introduction This paper is the first part of a work whose purpose is to investigate duality in some related ...

We present a translation of Parigot's λµ-calculus [10] into the usual λ-calculus. This translation, which is based on the so-called continuation passing style, is correct with respect to equality and with respect to evaluation. At the type level, it induces a logical interpretation of classical logic into intuitionistic one, akin to Kolmogorov's negative translation. As a by-product, we get the normalization of second order typed λµ-calculus.

symmetric λµ-calculus

de mathématiques, équipe de logique,

Introduction Il est bien connu que la correspondance de Curry-Howard permet d'associer un programme, sous la forme d'un #-terme, a toute preuve intuitionniste, formalisee dans le calcul des predicats du second ordre (voir, par exemple [3]). Cette correspondance a ete etendue, assez recemment, a la logique classique moyennant une extension convenable du #-calcul (voir [1,4,5,6]). Chaque theoreme formalise en logique du second ordre correspond donc a une specification de programme. Il se pose alors le probleme, en general tout a fait non trivial, de trouver la specification associ ee a un theoreme donne ; autrement dit, de determiner le comportement operationnel commun aux #-termes associes aux diverses demonstrations formelles du theoreme consid ere. Cette question est resolue ici pour le theoreme de completude de la logique classique. La premiere etape consiste a formaliser convenablement ce theoreme en logique du second ordre.

In this paper, we present an extension of λµ-calculus called λµ ++-calculus which has the following properties: subject reduction, strong normalization, unicity of the representation of data and thus confluence only on data types. This calculus allows also to program the parallel-or. 1

. We show that the one can consider proof of the Gentzen's LK as the continuation passing style(CPS) programs; and the cut-elimination procedure for LK as computation. To be more precise, we observe that Strongly Normalizable(SN) and Church-Rosser(CR) cut-elimination procedure for (intuitionistic decoration of) LKT and LKQ, as presented in Danos et al.(1993), precisely corresponds to call-by-name(CBN) and call-by-value(CBV) CPS calculi, respectively. This can also be seen as an extension to classical logic of Zucker-Pottinger-Mints investigation of the relations between cut-elimination and normalization. 1 Introduction Continuation Passing Style(CPS): Since Griffin's influential work [12] on the Curry-Howard correspondence between classical proofs and CPS programs, there has been a lot of interest on programming in classical proofs. It is because these classical calculi relate to important programming concepts such as non-local exit or exception handling. In Griffin's result, Plotkin...