this paper, we study the relationship between the call-by-name and call-by-value paradigms for Parigot's -calculus. The -calculus is an extension of the simply-typed lambda calculus with certain sequential control operators. We show that, in the presence of product and disjunction types, the call-by-name and call-by-value -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

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...

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 non-constructive, 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 λµ-calculus [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 ...

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 double-negation 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 Curry-Howard. This analogy, however, is not sufficient to accept the ¯-calculus

We establish the Curry-Howard isomorphism between constructive classical logic and CPS-calculus. CPS-calculus 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,CPS-calculus,CPS-transform,CPS-semantics 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 sequent-style classical logic (i.e., LK) and its variants. In this context, above fact is related to the non-deterministic behavior of cut-elimination. Of course, by Gentzen's theorem, LK has a Strongly Normalizable(SN) cut-elimination procedure. The problem is, it is not Church-Rosser(CR). Constructive ...

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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 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...

We show that the SN and CR cut-elimination procedure on Gentzen-style classical logic LKT/LKQ, as presented in Danos et al.(1994), is isomorphic to call-by-name (CBN) and call-by-value (CBV) reduction system respectively. Our method is simple. We assign typed -terms on intuitionistic decoration of LKT/LKQ so as to simulate the cut-elimination procedure by fi-contraction --- 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) cut-elimination procedure. This thread began with Girard's linear logic(LL)[9], followed by LC [10] and the logic of unity (LU) [11]....

. A new proof of strong normalization of Parigot's (second order) -calculus is given by a reduction-preserving embedding into system F (second order polymorphic -calculus). The main idea is to use the least stable supertype for any type. These non-strictly 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 non-interleaving positive xed-point types. As a non-trivial 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...

We show that the Gentzen-style classical logic LKT, as presented in Danos et al.(1993), can be considered as the classical call-by-name (CBN) calculus. First, we present a new term calculus for LKT which faithfully simulates cut-elimination procedure, namely tq-protocol. Then, we give a translation of Parigot's ¯-calculus. This translation can be seen as the classical extension to the Plotkin's CPS translation. Using this translation, one can show that reductions of the ¯-calculus can be simulated by the tq-protocol. Specifically the Strongly Normalizable(SN) and Church-Rosser(CR) property of the ¯-calculus is shown to be a consequence of that of the tq-protocol. Key Words: LKT, lambda-mu calculus, cut-elimination and normalization, classical logic, linear logic. 1 Introduction 1.1 Previous Works Continuation Passing Style (CPS) calculus: Since Griffin's influential work [8] on the Curry-Howard correspondence between classical proofs and CPS programs, there has been much interest ...