Strong normalization proof with CPS-translation for second order classical natural deduction

Nakazawa, Koji; Tatsuta, Makoto

doi:10.2178/jsl/1058448444

Cited by 12 publications

(19 citation statements)

References 5 publications

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“…Unlike the failed strict simulation by CPS reported in [29] that only occurred with the closure rules, the need for garbage in our translation is already clearly visible in the subcase E = [] for π and the case ǫ. But the garbage is also effective for the closure rules, where the most delicate rule is the translation of t(u :: l) that mentions l and u only in the continuation argument K to t's translation.…”

Section: X]t : [T/x]g [T/x]k) For T Any U L or C And In Particularmentioning

confidence: 57%

“…It is very much inspired from a "continuation and garbage passing style" translation for Parigot's λµ-calculus, proposed by Ikeda and Nakazawa [20]. While they use garbage to overcome the problems of earlier CPS translations that did not carry β-steps to at least one β-step if they were under a vacuous µ-binding, as reported in [29], we ensure strict simulation of ǫ, π and µ. Therefore, we can avoid the separate proof of strong normalisation of permutation steps alone that is used in addition to the CPS in [6] (there in order to treat disjunction and not for sequent calculi as we do).…”

Section: Cgps Translation For λJmentioning

confidence: 99%

“…In order to achieve strict simulation, we define continuation-and-garbage passing style (CGPS) translations, following an idea due to Ikeda and Nakazawa [20]. Garbage will provide room for observing reduction where continuation-passing alone would inevitably produce an identification, leading to failure of strict simulation in several published proofs for variants of operationalized classical logic, noted by [29] (the problem being β-reductions under vacuous µ-abstractions). As opposed to [20], in our intuitionistic setting garbage can be reduced to "units", and garbage reduction is simply erasing a garbage unit.…”

Section: Introductionmentioning

confidence: 99%

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Continuation-Passing Style and Strong Normalisation for Intuitionistic Sequent Calculi

Santo¹,

Matthes²,

Pinto³

2009

Logical Methods in Computer Science

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Abstract. The intuitionistic fragment of the call-by-name version of Curien and Herbelin's λµμ-calculus is isolated and proved strongly normalising by means of an embedding into the simply-typed λ-calculus. Our embedding is a continuation-and-garbagepassing style translation, the inspiring idea coming from Ikeda and Nakazawa's translation of Parigot's λµ-calculus. The embedding strictly simulates reductions while usual continuation-passing-style transformations erase permutative reduction steps. For our intuitionistic sequent calculus, we even only need "units of garbage" to be passed. We apply the same method to other calculi, namely successive extensions of the simply-typed λ-calculus leading to our intuitionistic system, and already for the simplest extension we consider (λ-calculus with generalised application), this yields the first proof of strong normalisation through a reduction-preserving embedding. The results obtained extend to second and higher-order calculi.

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Section: X]t : [T/x]g [T/x]k) For T Any U L or C And In Particularmentioning

confidence: 57%

Section: Cgps Translation For λJmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Continuation-Passing Style and Strong Normalisation for Intuitionistic Sequent Calculi

Santo¹,

Matthes²,

Pinto³

2009

Logical Methods in Computer Science

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“…The notion of augmentations for λµ-calculus was introduced in [13] to correct the error of the strong normalization proof by a CPS-translation in [18]. The error in the proof of [5] is due to the same sort of problem, which is called erasing-continuations in [13].…”

Section: Introductionmentioning

confidence: 99%

Strong normalization of classical natural deduction with disjunctions

Nakazawa

Tatsuta

2008

Annals of Pure and Applied Logic

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This paper proves the strong normalization of classical natural deduction with disjunction and permutative conversions, by using CPS-translation and augmentations. Using them, this paper also proves the strong normalization of classical natural deduction with general elimination rules for implication and conjunction, and their permutative conversions. This paper also proves that natural deduction can be embedded into natural deduction with general elimination rules, strictly preserving proof normalization.

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“…In particular, strong normalization results of λµ2 [27,23] and its extensions, e.g. with inductive types [21], have been a central research topic, because of the proof-theoretical importance of strong normalization.…”

Section: Introductionmentioning

confidence: 99%

Relational Parametricity and Control

Hasegawa¹

2006

Logical Methods in Computer Science

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Abstract. We study the equational theory of Parigot's second-order λµ-calculus in connection with a call-by-name continuation-passing style (CPS) translation into a fragment of the second-order λ-calculus. It is observed that the relational parametricity on the target calculus induces a natural notion of equivalence on the λµ-terms. On the other hand, the unconstrained relational parametricity on the λµ-calculus turns out to be inconsistent. Following these facts, we propose to formulate the relational parametricity on the λµ-calculus in a constrained way, which might be called "focal parametricity".

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Strong normalization proof with CPS-translation for second order classical natural deduction

Abstract: This paper points out an error of Parigot's proof of strong normalization of second order classical natural deduction by the CPS-translation, discusses erasing-continuation of the CPS-translation, and corrects that proof by using the notion of augmentations.

Cited by 12 publications

References 5 publications

Continuation-Passing Style and Strong Normalisation for Intuitionistic Sequent Calculi

Continuation-Passing Style and Strong Normalisation for Intuitionistic Sequent Calculi

Strong normalization of classical natural deduction with disjunctions

Relational Parametricity and Control

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