Typed realizability for first-order classical analysis

Valentin Blot

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We describe a realizability framework for classical first-order logic in which realizers live in (a model of) typed λμ-calculus. This allows a direct interpretation of classical proofs, avoiding the usual negative translation to intuitionistic logic. We prove that the usual terms of Godel’s system T realize the axioms of Peano arithmetic, and that under some assumptions on the computational model, the bar recursion operator realizes the axiom of dependent choice. We also perform a proper analysis of relativization, which allows for less technical proofs of adequacy. Extraction of algorithms from proofs of (formula presented) formulas relies on a novel implementation of Friedman’s trick exploiting the control possibilities of the language. This allows to have extracted programs with simpler types than in the case of negative translation followed by intuitionistic realizability.

Original languageEnglish
JournalLogical Methods in Computer Science
Issue number4
Publication statusPublished - 31 Dec 2015


  • Axiom of choice
  • Bar-recursion
  • Classical realizability
  • Lambda-mu calculus


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