Hybrid realizability for intuitionistic and classical choice

Abstract : In intuitionistic realizability like Kleene's or Kreisel's, the axiom of choice is trivially realized. It is even provable in Martin-Löf's intu-itionistic type theory. In classical logic, however, even the weaker axiom of countable choice proves the existence of non-computable functions. This logical strength comes at the price of a complicated computational interpretation which involves strong recursion schemes like bar recursion. We take the best from both worlds and define a realizability model for arithmetic and the axiom of choice which encompasses both intuitionistic and classical reasoning. In this model two versions of the axiom of choice can co-exist in a single proof: intuitionistic choice and classical countable choice. We interpret intuitionistic choice efficiently, however its premise cannot come from classical reasoning. Conversely, our version of classical choice is valid in full classical logic, but it is restricted to the countable case and its realizer involves bar recursion. Having both versions allows us to obtain efficient extracted programs while keeping the provability strength of classical logic.
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Communication dans un congrès
31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS '16, Jul 2016, New York, United States. ACM Press, 2016, Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS '16, New York, NY, USA, July 5-8, 2016. 〈10.1145/2933575.2934511〉
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Contributeur : Valentin Blot <>
Soumis le : samedi 14 avril 2018 - 14:29:34
Dernière modification le : mardi 17 avril 2018 - 15:03:20

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Valentin Blot. Hybrid realizability for intuitionistic and classical choice. 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS '16, Jul 2016, New York, United States. ACM Press, 2016, Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS '16, New York, NY, USA, July 5-8, 2016. 〈10.1145/2933575.2934511〉. 〈hal-01766881〉

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