Realizability Interpretation and Normalization of Typed Call-by-Need λ-calculus With Control

Étienne Miquey 1, 2 Hugo Herbelin 2
1 GALLINETTE - Gallinette : vers une nouvelle génération d'assistant à la preuve
Inria Rennes – Bretagne Atlantique , LS2N - Laboratoire des Sciences du Numérique de Nantes
2 PI.R2 - Design, study and implementation of languages for proofs and programs
Inria de Paris, CNRS - Centre National de la Recherche Scientifique, UPD7 - Université Paris Diderot - Paris 7, PPS - Preuves, Programmes et Systèmes
Abstract : We define a variant of realizability where realizers are pairs of a term and a substitution. This variant allows us to prove the normalization of a simply-typed call-by-need λ-calculus with control due to Ariola et al. Indeed, in such call-by-need calculus, substitutions have to be delayed until knowing if an argument is really needed. In a second step, we extend the proof to a call-by-need λ-calculus equipped with a type system equivalent to classical second-order predicate logic, representing one step towards proving the normalization of the call-by-need classical second-order arithmetic introduced by the second author to provide a proof-as-program interpretation of the axiom of dependent choice.
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Étienne Miquey, Hugo Herbelin. Realizability Interpretation and Normalization of Typed Call-by-Need λ-calculus With Control. FOSSACS 18 - 21st International Conference on Foundations of Software Science and Computation Structures, Apr 2018, Thessalonique, Greece. pp.276-292, ⟨10.1007/978-3-319-89366-2_15⟩. ⟨hal-01624839v2⟩

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