On Various Negative Translations

Ferreira, Gilda; Oliva, Paulo

doi:10.4204/eptcs.47.4

Cited by 13 publications

(11 citation statements)

References 22 publications

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“…We can for instance add one constant for each natural number. 9 The reader might recognize the rule (ς ) of Wadler's sequent calculus [38].…”

Section: Reduction Rulesmentioning

confidence: 99%

“…Continuation-passing style translations are indeed very useful to embed languages with classical control into purely functional ones [7,15]. From a logical point of view, they generally amount to negative translations that allow us to embed classical logic into intuitionistic logic [9]. Yet, we know that removing classical control (i.e.…”

Section: A Continuation-passing Style Translationmentioning

confidence: 99%

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A Classical Sequent Calculus with Dependent Types

Miquey

2019

ACM Trans. Program. Lang. Syst.

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Dependent types are a key feature of the proof assistants based on the Curry-Howard isomorphism. It is well-known that this correspondence can be extended to classical logic by enriching the language of proofs with control operators. However, they are known to misbehave in the presence of dependent types, unless dependencies are restricted to values. Moreover, while sequent calculi naturally support continuation-passing style interpretations, there is no such presentation of a language with dependent types. The main achievement of this paper is to give a sequent calculus presentation of a call-by-value language with a control operator and dependent types, and to justify its soundness through a continuation-passing style translation.We start from the call-by-value version of the λµμ-calculus. We design a minimal language with a value restriction and a type system that includes a list of explicit dependencies to maintains type safety. We then show how to relax the value restriction and introduce delimited continuations to directly prove the consistency by means of a continuation-passing-style translation. Finally, we relate our calculus to a similar system by Lepigre, and present a methodology to transfer properties from this system to our own. c b n a This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 Unported License. 1:2 Étienne MiqueyNevertheless, dependent types are known to misbehave in the presence of control operators, and lead to logical inconsistencies [17]. Since the same problem arises with a wider class of effects, it seems that we are facing the following dilemma: either we choose an effectful language (allowing us to write more programs) while accepting the lack of dependent types, or we choose a dependently typed language (allowing us to write finer specifications) and give up effects.Many works have tried to fill the gap between effectful programming languages and logic, by accommodating weaker forms of dependent types with computational effects (e.g., divergence, I/O, local references, exceptions). Amongst other works, we can cite the recent works by Ahman et al.[1], by Vákár [35,36] or by Pédrot and Tabareau who proposed a systematical way to add effects to type theory [33]. Side effects-that are impure computations in functional programming-are often interpreted by means of monads. Interestingly, control operators can be interpreted similarly through the continuation monad, but the continuation monad generally lacks the properties necessary to fit these frameworks.Although dependent types and classical logic have been deeply studied separately, the problem of accommodating both features 1 in one and the same system has not found a completely satisfying answer yet. Recent works from Herbelin [18] and Lepigre [22] proposed some restrictions on dependent types to make them compatible with a classical proof system, while Blot [5] designed a hybrid realizability model where dependent types are restricted to an intuitionistic fragment. Call-by-value and value restrictionIn lang...

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“…We can for instance add one constant for each natural number. 9 The reader might recognize the rule (ς ) of Wadler's sequent calculus [38].…”

Section: Reduction Rulesmentioning

confidence: 99%

Section: A Continuation-passing Style Translationmentioning

confidence: 99%

A Classical Sequent Calculus with Dependent Types

Miquey

2019

ACM Trans. Program. Lang. Syst.

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“…In this section, we briefly recall four existing translations [7,8]. In 1925, the first translation is published by Kolmogorov [2].…”

Section: Negative Translationsmentioning

confidence: 99%

Polarizing Double-Negation Translations

Boudard

Hermant

2013

Logic for Programming, Artificial Intelligence, and Reasoning

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Abstract. Double-negation translations are used to encode and decode classical proofs in intuitionistic logic. We show that, in the cut-free fragment, we can simplify the translations and introduce fewer negations. To achieve this, we consider the polarization of the formulae and adapt those translation to the different connectives and quantifiers. We show that the embedding results still hold, using a customized version of the focused classical sequent calculus. We also prove the latter equivalent to more usual versions of the sequent calculus. This polarization process allows lighter embeddings, and sheds some light on the relationship between intuitionistic and classical connectives.

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“…This is an extended version of our Classical Logic and Computation (CL&C) workshop 2010 paper, which appeared in [6]. The main differences to the workshop version are that here all proofs are included, and the analysis of the negative translations is first done over the weaker setting of minimal logic (rather than intuitionistic logic).…”

Section: Introductionmentioning

confidence: 97%

On the Relation Between Various Negative Translations

Ferreira¹,

Oliva²

2012

Logic, Construction, Computation

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Several proof translations of classical mathematics into intuitionistic (or even minimal) mathematics have been proposed in the literature over the past century. These are normally referred to as negative translations or doublenegation translations. Amongst those, the most commonly cited are translations due to Kolmogorov, Gödel, Gentzen, Kuroda and Krivine (in chronological order). In this paper we propose a framework for explaining how these different translations are related to each other. More precisely, we define a notion of a (modular) simplification starting from Kolmogorov translation, which leads to a partial order between different negative translations. In this derived ordering, Kuroda, Krivine and Gödel-Gentzen are minimal elements. A new minimal translation is introduced.

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On Various Negative Translations

Cited by 13 publications

References 22 publications

A Classical Sequent Calculus with Dependent Types

A Classical Sequent Calculus with Dependent Types

Polarizing Double-Negation Translations

On the Relation Between Various Negative Translations

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