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Stabilization—An Alternative to DoubleNegation Translation for Classical Natural Deduction
, 2004
"... A new proof of strong normalization of Parigot’s secondorder λµcalculus is given by a reductionpreserving embedding into system F (secondorder polymorphic λcalculus). The main idea is to use the least stable supertype for any type. These nonstrictly positive inductive types and their associate ..."
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A new proof of strong normalization of Parigot’s secondorder λµcalculus is given by a reductionpreserving embedding into system F (secondorder polymorphic λcalculus). 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 Kolmogorov’s doublenegation embedding of classical logic into intuitionistic logic). However, they simulate Parigot’s µreductions whereas CPS translations hide them. As a major advantage, this embedding does not use the idea of reducing stability (¬¬A → A) to that for atomic formulae. Therefore, it even extends to positive fixedpoint types. The article expands on “Parigot’s SecondOrder λµCalculus and Inductive Types ” (Conference Proceedings TLCA 2001, Springer LNCS 2044) by the author. 1
ContinuationPassing Style and Strong Normalisation for Intuitionistic Sequent Calculi
"... Abstract. The intuitionistic fragment of the callbyname version of Curien and Herbelin’s λµ˜µcalculus is isolated and proved strongly normalising by means of an embedding into the simplytyped λcalculus. Our embedding is a continuationandgarbagepassing style translation, the inspiring idea co ..."
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Abstract. The intuitionistic fragment of the callbyname version of Curien and Herbelin’s λµ˜µcalculus is isolated and proved strongly normalising by means of an embedding into the simplytyped λcalculus. Our embedding is a continuationandgarbagepassing style translation, the inspiring idea coming from Ikeda and Nakazawa’s translation of Parigot’s λµcalculus. The embedding simulates reductions while usual continuationpassingstyle 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 simplytyped λcalculus leading to our intuitionistic system, and already for the simplest extension we consider (λcalculus with generalised application), this yields the first proof of strong normalisation through a reductionpreserving embedding. 1
CPS Translating Inductive and Coinductive Types (Extended Abstract)
 In: Proc. of 2002 ACM SIGPLAN Wksh. on Partial Evaluation and SemanticsBased Program Manipulation, PEPM'02
, 2002
"... We investigate CPS translatability of typed lambdacalculi with inductive and coinductive types. We show that tenable Plotkinstyle callbyname CPS translations exist for simply typed lambdacalculi with a natural number type and stream types and, more generally, with arbitrary positive inductive a ..."
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We investigate CPS translatability of typed lambdacalculi with inductive and coinductive types. We show that tenable Plotkinstyle callbyname CPS translations exist for simply typed lambdacalculi with a natural number type and stream types and, more generally, with arbitrary positive inductive and coinductive types. These translations also work in the presence of control operators and generalize for dependently typed calculi where caselike eliminations are only allowed in nondependent forms. No translation is possible along the same lines for small Sigmatypes and sum types with dependent case.
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"... We investigate CPS translatability of typed λcalculi with inductive and coinductive types. We show that tenable Plotkinstyle callbyname CPS translations exist for simply typed λcalculi with a natural number type and stream types and, more generally, with arbitrary positive inductive and coinduc ..."
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We investigate CPS translatability of typed λcalculi with inductive and coinductive types. We show that tenable Plotkinstyle callbyname CPS translations exist for simply typed λcalculi with a natural number type and stream types and, more generally, with arbitrary positive inductive and coinductive types. These translations also work in the presence of control operators and generalize for dependently typed calculi where caselike eliminations are only allowed in nondependent forms. No translation is possible along the same lines for small Σtypes and sum types with dependent case.
CONTINUATIONPASSING STYLE AND STRONG NORMALISATION FOR INTUITIONISTIC SEQUENT CALCULI JOSE ́ ESPÍRITO SANTO, RALPH MATTHES, AND LUÍS PINTO
, 2008
"... Abstract. The intuitionistic fragment of the callbyname version of Curien and Herbelin’s λµµ̃calculus is isolated and proved strongly normalising by means of an embedding into the simplytyped λcalculus. Our embedding is a continuationandgarbagepassing style translation, the inspiring idea ..."
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Abstract. The intuitionistic fragment of the callbyname version of Curien and Herbelin’s λµµ̃calculus is isolated and proved strongly normalising by means of an embedding into the simplytyped λcalculus. Our embedding is a continuationandgarbagepassing style translation, the inspiring idea coming from Ikeda and Nakazawa’s translation of Parigot’s λµcalculus. The embedding strictly simulates reductions while usual continuationpassingstyle 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 simplytyped λcalculus leading to our intuitionistic system, and already for the simplest extension we consider (λcalculus with generalised application), this yields the first proof of strong normalisation through a reductionpreserving embedding. The results obtained extend to second and higherorder calculi.
CERTIFICATE OF APPROVAL
, 2014
"... To my lovely wife, Jenny Eades. ii Program testing can be used to show the presence of bugs, but never to show their absence! –Dijkstra (1970) iii ACKNOWLEDGEMENTS The first person I would like to acknowledge is my advisor Aaron Stump. He is one of the kindest and most intelligent people I have had ..."
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To my lovely wife, Jenny Eades. ii Program testing can be used to show the presence of bugs, but never to show their absence! –Dijkstra (1970) iii ACKNOWLEDGEMENTS The first person I would like to acknowledge is my advisor Aaron Stump. He is one of the kindest and most intelligent people I have had the pleasure to work with, and without his guidance I would have never made it this far. I can only hope to acquire the insight and creativity you have when working on a research problem. Furthermore, I would like to thank him for introducing me to my research area in type theory and the foundations of functional programming languages. Secondly, I would like to thank my wife, Jenny Eades, whose hard work literally made it possible for there to be food on our table and a roof over our heads. She