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35
A New Deconstructive Logic: Linear Logic
, 1995
"... 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 a ..."
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Cited by 102 (11 self)
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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 `programmingwithproofs' ([22, 23]) to classical logic; it is painless: since we reduce strong normalization and confluence to the same properties for linear logic (for nonadditive proof nets, to be precise) using appropriate embeddings (socalled 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...
A CurryHoward foundation for functional computation with control
 In Proceedings of ACM SIGPLANSIGACT Symposium on Principle of Programming Languages
, 1997
"... We introduce the type theory ¯ v , a callbyvalue variant of Parigot's ¯calculus, as a CurryHoward representation theory of classical propositional proofs. The associated rewrite system is ChurchRosser and strongly normalizing, and definitional equality of the type theory is consistent, compatib ..."
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Cited by 77 (3 self)
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We introduce the type theory ¯ v , a callbyvalue variant of Parigot's ¯calculus, as a CurryHoward representation theory of classical propositional proofs. The associated rewrite system is ChurchRosser and strongly normalizing, and definitional equality of the type theory is consistent, compatible with cut, congruent and decidable. The attendant callbyvalue 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 firstclass continuations. Prooftheoretically 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...
A formulaeastypes interpretation of subtractive logic
 Journal of Logic and Computation
, 2004
"... We present a formulaeastypes interpretation of Subtractive Logic (i.e. biintuitionistic logic). This presentation is twofold: we first define a very natural restriction of the λµcalculus which is closed under reduction and whose type system is a constructive restriction of the Classical Natural ..."
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Cited by 23 (1 self)
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We present a formulaeastypes interpretation of Subtractive Logic (i.e. biintuitionistic logic). This presentation is twofold: we first define a very natural restriction of the λµcalculus 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 firstclass coroutines (a restricted form of firstclass continuations). Keywords: CurryHoward isomorphism, Subtractive Logic, control operators, coroutines. 1
Subtractive Logic
, 1999
"... This paper is the first part of a work whose purpose is to investigate duality in some related frameworks (cartesian closed categories, lambdacalculi, intuitionistic and classical logics) from syntactic, semantical and computational viewpoints. We start with category theory and we show that any ..."
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Cited by 20 (1 self)
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This paper is the first part of a work whose purpose is to investigate duality in some related frameworks (cartesian closed categories, lambdacalculi, 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 ...
Arithmetical proofs of strong normalization results for symmetric λcalculi
"... symmetric λµcalculus ..."
A CPSTranslation of the λµCalculus
, 1994
"... We present a translation of Parigot's λµcalculus [10] into the usual λcalculus. This translation, which is based on the socalled 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 log ..."
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Cited by 15 (1 self)
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We present a translation of Parigot's λµcalculus [10] into the usual λcalculus. This translation, which is based on the socalled 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 byproduct, we get the normalization of second order typed λµcalculus.
La valeur d’un entier classique en λµcalcul
 Archive for Mathematical Logic
, 1997
"... de mathématiques, équipe de logique, ..."
Non deterministic classical logic: the λµ ++calculus
"... In this paper, we present an extension of λµcalculus 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 parallelor. 1 ..."
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Cited by 8 (8 self)
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In this paper, we present an extension of λµcalculus 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 parallelor. 1
Une Preuve Formelle Et Intuitionniste Du Theoreme De Completude De La Logique Classique
, 1996
"... Introduction Il est bien connu que la correspondance de CurryHoward 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 ..."
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Cited by 6 (0 self)
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Introduction Il est bien connu que la correspondance de CurryHoward 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.
Completeness and Partial Soundness Results for Intersection & Union Typing for λµ ˜µ
 Annals of Pure and Applied Logic
"... This paper studies intersection and union type assignment for the calculus λµ ˜µ [17], a proofterm syntax for Gentzen’s classical sequent calculus, with the aim of defining a typebased semantics, via setting up a system that is closed under conversion. We will start by investigating what the minima ..."
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Cited by 6 (6 self)
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This paper studies intersection and union type assignment for the calculus λµ ˜µ [17], a proofterm syntax for Gentzen’s classical sequent calculus, with the aim of defining a typebased semantics, via setting up a system that is closed under conversion. We will start by investigating what the minimal requirements are for a system for λµ ˜µ to be closed under subject expansion; this coincides with System M ∩ ∪ , the notion defined in [19]; however, we show that this system is not closed under subject reduction, so our goal cannot be achieved. We will then show that System M ∩ ∪ is also not closed under subjectexpansion, but can recover from this by presenting System M C as an extension of M ∩ ∪ (by adding typing rules) and showing that it satisfies subject expansion; it still lacks subject reduction. We show how to restrict M ∩ ∪ so that it satisfies subjectreduction as well by limiting the applicability to type assignment rules, but only when limiting reduction to (confluent) callbyname or callbyvalue reduction M ∩ ∪ ; in restricting the system, we sacrifice subject expansion. These results combined show that a sound and complete intersection and union type assignment system cannot be defined for λµ ˜µ with respect to full reduction.