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Control Categories and Duality: on the Categorical Semantics of the LambdaMu Calculus
, 1999
"... this paper, we study the relationship between the callbyname and callbyvalue paradigms for Parigot's calculus. The calculus is an extension of the simplytyped lambda calculus with certain sequential control operators. We show that, in the presence of product and disjunction types, the callby ..."
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Cited by 84 (3 self)
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this paper, we study the relationship between the callbyname and callbyvalue paradigms for Parigot's calculus. The calculus is an extension of the simplytyped lambda calculus with certain sequential control operators. We show that, in the presence of product and disjunction types, the callbyname and callbyvalue calculi are isomorphic to each other, in the sense that there exist syntactic translations between them that preserve the operational semantics and that are mutually inverse up to isomorphism of types. These translations take the form of a
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 semantic view of classical proofs  typetheoretic, categorical, and denotational characterizations (Extended Abstract)
 IN PROCEEDINGS OF LICS '96
, 1996
"... Classical logic is one of the best examples of a mathematical theory that is truly useful to computer science. Hardware and software engineers apply the theory routinely. Yet from a foundational standpoint, there are aspects of classical logic that are problematic. Unlike intuitionistic logic, class ..."
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Cited by 30 (2 self)
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Classical logic is one of the best examples of a mathematical theory that is truly useful to computer science. Hardware and software engineers apply the theory routinely. Yet from a foundational standpoint, there are aspects of classical logic that are problematic. Unlike intuitionistic logic, classical logic is often held to be nonconstructive, and so, is said to admit no proof semantics. To draw an analogy in the proofsas programs paradigm, it is as if we understand well the theory of manipulation between equivalent specifications (which we do), but have comparatively little foundational insight of the process of transforming one program to another that implements the same specification. This extended abstract outlines a semantic theory of classical proofs based on a variant of Parigot's λµcalculus [24], but presented here as a type theory. After reviewing the conceptual problems in the area and the potential benefits of such a theory, we sketch the key steps of our approach in ...
An Environment Machine for the λμCalculus
, 1998
"... this paper is to demonstrate how Parigot's ¯calculus (Parigot 1992) may act as a correct foundation for functional programming enriched with control operators. The ¯calculus is an extension of the calculus that provides classical logic with an algorithmic interpretation. The extension is twofold. ..."
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Cited by 11 (0 self)
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this paper is to demonstrate how Parigot's ¯calculus (Parigot 1992) may act as a correct foundation for functional programming enriched with control operators. The ¯calculus is an extension of the calculus that provides classical logic with an algorithmic interpretation. The extension is twofold. On the one hand, new syntactic constructs (¯abstraction and naming) are given in order to encode classical proofs. On the other hand, the calculus is supplied with new notions of reduction in order to give algorithmic content to the doublenegation rule of classical logic. Since Griffin's pioneering work (Griffin 1990), based on Felleisen's theory (Felleisen et al. 1987; Felleisen and Hieb 1992), it is known how the notion of control in functional programming is related to classical logic through a correspondance akin to the isomorphism of CurryHoward. This analogy, however, is not sufficient to accept the ¯calculus
Towards Logical Understanding of Delimited Continuations (Extended Abstract)
 IN CONTINUATIONS WORKSHOP
, 2000
"... ..."
Constructive Classical Logic as CPScalculus
, 1999
"... We establish the CurryHoward isomorphism between constructive classical logic and CPScalculus. CPScalculus exactly means the target language of Continuation Passing Style(CPS) transforms. Constructive classical logic we refer to are LKT and LKQ introduced by Danos et al.(1993). Keywords: Const ..."
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Cited by 5 (5 self)
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We establish the CurryHoward isomorphism between constructive classical logic and CPScalculus. CPScalculus exactly means the target language of Continuation Passing Style(CPS) transforms. Constructive classical logic we refer to are LKT and LKQ introduced by Danos et al.(1993). Keywords: Constructive Classical Logic,CPScalculus,CPStransform,CPSsemantics 1. Introduction 1.1. What is Constructive Classical Logic? It has long been thought that classical logic cannot be put to use for computational purposes. It is because, in general, the normalization process for the the proof of classical logic has a lot of critical pairs. Classical logic we consider in this paper is Gentzen's sequentstyle classical logic (i.e., LK) and its variants. In this context, above fact is related to the nondeterministic behavior of cutelimination. Of course, by Gentzen's theorem, LK has a Strongly Normalizable(SN) cutelimination procedure. The problem is, it is not ChurchRosser(CR). Constructive ...
Cut Elimination for Classical Proofs as Continuation Passing Style Computation
 In Proceedings of the Asian Computing Science Conference 98
, 1998
"... . We show that the one can consider proof of the Gentzen's LK as the continuation passing style(CPS) programs; and the cutelimination procedure for LK as computation. To be more precise, we observe that Strongly Normalizable(SN) and ChurchRosser(CR) cutelimination procedure for (intuitionistic d ..."
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Cited by 4 (4 self)
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. We show that the one can consider proof of the Gentzen's LK as the continuation passing style(CPS) programs; and the cutelimination procedure for LK as computation. To be more precise, we observe that Strongly Normalizable(SN) and ChurchRosser(CR) cutelimination procedure for (intuitionistic decoration of) LKT and LKQ, as presented in Danos et al.(1993), precisely corresponds to callbyname(CBN) and callbyvalue(CBV) CPS calculi, respectively. This can also be seen as an extension to classical logic of ZuckerPottingerMints investigation of the relations between cutelimination and normalization. 1 Introduction Continuation Passing Style(CPS): Since Griffin's influential work [12] on the CurryHoward 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 nonlocal exit or exception handling. In Griffin's result, Plotkin...
Classical Proofs as Programs, Cut Elimination as Computation
, 1998
"... We show that the SN and CR cutelimination procedure on Gentzenstyle classical logic LKT/LKQ, as presented in Danos et al.(1994), is isomorphic to callbyname (CBN) and callbyvalue (CBV) reduction system respectively. Our method is simple. We assign typed terms on intuitionistic decoration of ..."
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Cited by 1 (1 self)
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We show that the SN and CR cutelimination procedure on Gentzenstyle classical logic LKT/LKQ, as presented in Danos et al.(1994), is isomorphic to callbyname (CBN) and callbyvalue (CBV) reduction system respectively. Our method is simple. We assign typed terms on intuitionistic decoration of LKT/LKQ so as to simulate the cutelimination procedure by ficontraction  i.e. we simulate cutelimination by normalization. As a consequence we revealed that these term assignments are precisely the one which are known as continuation passing style (CPS). We also establish the isomorphism between ¯calculus and our CPS calculus. 1 Introduction Proof theory: There is a long line of proof theoretical approaches to understanding "deconstructive " classical logic. That is, classical logic that has Strongly Normalizing (SN) and confluent (ChurchRosser or CR) cutelimination procedure. This thread began with Girard's linear logic(LL)[9], followed by LC [10] and the logic of unity (LU) [11]....
Parigot's Second Order λμCalculus and Inductive Types
, 2001
"... . A new proof of strong normalization of Parigot's (second order) calculus is given by a reductionpreserving embedding into system F (second order polymorphic calculus). The main idea is to use the least stable supertype for any type. These nonstrictly positive inductive types and their associat ..."
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Cited by 1 (0 self)
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. A new proof of strong normalization of Parigot's (second order) calculus is given by a reductionpreserving embedding into system F (second order polymorphic calculus). The main idea is to use the least stable supertype for any type. These nonstrictly positive inductive types and their associated iteration principle are available in system F, and allow to give a translation vaguely related to CPS translations (corresponding to the Kolmogorov embedding of classical logic into intuitionistic logic). However, they simulate Parigot's reductions whereas CPS translations hide them. As a major advantage, this embedding does not use the idea of reducing stability (:: ! ) to that for atomic formulae. Therefore, it even extends to noninterleaving positive xedpoint types. As a nontrivial application, strong normalization of calculus, extended by primitive recursion on monotone inductive types, is established. 1 Introduction calculus [12] essentially is the extension of nat...
On the Relation between the lambdaµCalculus and the Syntactic Theory of Sequential Control
"... . We construct a translation of first order ¯calculus [15] into a subtheory of Felleisen's c calculus [5, 6]. This translation preserves typing and reduction. Then, by constructing the inverse translation, we show that the two calculi are actually isomorphic. 1 Introduction In: F. Pfenning (Ed.), ..."
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. We construct a translation of first order ¯calculus [15] into a subtheory of Felleisen's c calculus [5, 6]. This translation preserves typing and reduction. Then, by constructing the inverse translation, we show that the two calculi are actually isomorphic. 1 Introduction In: F. Pfenning (Ed.), Proceedings of the 5th International Conference on Logic Programming and Automated Reasoning (LPAR'94), Lecture Notes in Computer Science, 822, SpringerVerlag, (1994), pp. 3143. For a long time it has been widely thought that a classical proof, as opposed to an intuitionistic one, did not carry any computational content (see, for instance, [8, App. B, xB.1] and [12, p. 67, Proposition 8.3]). In 1990, however, T. Griffin opened a new research area by introducing a classical formulaeastypes notion of control based on Felleisen's C operator [9]. Since then, various authors have defined different systems that enlighten the constructive content of classical logic [1, 2, 7, 13, 14, 15, 16, 17...