Strong normalization proofs by CPS-translations

Ikeda, Satoshi; Nakazawa, Koji

doi:10.1016/j.ipl.2006.03.009

Cited by 10 publications

(20 citation statements)

References 18 publications

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“…The technique is a variation of continuation-passing style, called continuation-and-garbage-passing style [8]. CPS translations alone do not suffice for a strict simulation of all reductions.…”

Section: Monadic Lambda-calculusmentioning

confidence: 99%

“…The first issue is whether the technique of "garbage-passing", as used in the translation of λ-calculus with control operators in [8] and later for translation of intuitionistic sequent calculus [5,6], can be captured through some monad. Less ambitiously, one would hope for a precise comparison that allows to see why there is no need for extra rules such as assoc in the target of the garbage-passing translation.…”

Section: Final Remarksmentioning

confidence: 99%

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Monadic Translation of Intuitionistic Sequent Calculus

Santo

Matthes²,

Pinto

2009

Lecture Notes in Computer Science

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Abstract. This paper proposes and analyses a monadic translation of an intuitionistic sequent calculus. The source of the translation is a typed λ-calculus previously introduced by the authors, corresponding to the intuitionistic fragment of the call-by-name variant of λµμ of Curien and Herbelin, and the target is a variant of Moggi's monadic meta-language, where the rewrite relation includes extra permutation rules that may be seen as variations of the "associativity" of bind (the Kleisli extension operation of the monad). The main result is that the monadic translation simulates reduction strictly, so that strong normalisation (which is enjoyed at the target, as we show) can be lifted from the target to the source. A variant translation, obtained by adding an extra monad application in the translation of types, still enjoys strict simulation, while requiring one fewer extra permutation rule from the target. Finally we instantiate, for the cases of the identity monad and the continuations monad, the meta-language into the simply-typed λ-calculus. By this means, we give a generic account of translations of sequent calculus into natural deduction, which encompasses the traditional mapping studied by Zucker and Pottinger, and CPS translations of intuitionistic sequent calculus.

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Section: Monadic Lambda-calculusmentioning

confidence: 99%

Section: Final Remarksmentioning

confidence: 99%

Monadic Translation of Intuitionistic Sequent Calculus

Santo

Matthes²,

Pinto

2009

Lecture Notes in Computer Science

View full text Add to dashboard Cite

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“…One of them is an interpretation between languages with different type systems or logical infra-structure, possibly with corresponding differences at the level of program constructors and computational behavior. Examples are when the source language (but not the target language): (i) allows permutative conversions, possibly related to connectives like disjunction [6]; (ii) is a language for classical logic, usually with control operators [13,16,20]; (iii) is a language for type theory [1,2] (extending (ii) to variants of pure type systems that have dependent types and polymorphism).…”

Section: Introductionmentioning

confidence: 99%

“…But it is, nevertheless, an eminently useful requirement if one wants to infer strong normalisation of the source calculus from strong normalisation of the simply-typed λ-calculus, as we do. 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).…”

Section: Introductionmentioning

confidence: 99%

“…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

Santo

Matthes

Pinto

2007

Lecture Notes 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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An Isomorphism Between Cut-Elimination Procedure and Proof Reduction

Nakazawa

2007

Lecture Notes in Computer Science

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Strong normalization proofs by CPS-translations

Cited by 10 publications

References 18 publications

Monadic Translation of Intuitionistic Sequent Calculus

Monadic Translation of Intuitionistic Sequent Calculus

Continuation-Passing Style and Strong Normalisation for Intuitionistic Sequent Calculi

An Isomorphism Between Cut-Elimination Procedure and Proof Reduction

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