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In intuitionistic realizability like Kleene's or Kreisel's, the axiom of choice is trivially realized. It is even provable in Martin-Löf's intuitionistic 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.
|Title of host publication||LICS '16, Proceedings of the 31st Annual ACM-IEEE Symposium on Logic in Computer Science|
|Place of Publication||New York, U. S. A.|
|Number of pages||10|
|Publication status||Published - 5 Jul 2016|
|Event||31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2016 - New York, USA United States|
Duration: 5 Jul 2016 → 8 Jul 2016
|Conference||31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2016|
|Country/Territory||USA United States|
|Period||5/07/16 → 8/07/16|
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- 1 Finished
28/02/14 → 31/07/17
Project: Research council