Existential witness extraction in classical realizability and via a negative translation

Abstract. We show how to extract existential witnesses from classical proofs using Krivine’s classical realizability—where classical proofs are interpreted as λ-terms with the call/cc control operator. We first recall the basic framework of classical realizability (in classical secondorder arithmetic) and show how to extend it with primitive numerals for faster computations. Then we show how to perform witness extraction in this framework, by discussing several techniques depending on the shape of the existential formula. In particular, we show that in the Σ0 1 -case, Krivine’s witness extraction method reduces to Friedman’s through a well-suited negative translation to intuitionistic second-order arithmetic. Finally we discuss the advantages of using call/cc rather than a negative translation, especially from the point of view of an implementation.

• Type: Journal
• Year: 2011
• Where: CORR
• Authors: Alexandre Miquel
• Keywords: CORR 2011, Education, Negative Translation, Well-suited Negative Translation, Witness Extraction