Existential witness extraction in classical realizability and via a negative translation

TL;DR

This paper demonstrates methods for extracting existential witnesses from classical proofs using Krivine's realizability, extending the framework with primitive numerals and comparing it to negative translation techniques for efficiency and implementation benefits.

Contribution

It introduces an extension of classical realizability with primitive numerals and compares witness extraction methods, highlighting advantages of call/cc over negative translation.

Findings

Witness extraction in Sigma01 formulas reduces to Friedman's method via negative translation.

Extending realizability with primitive numerals improves computational efficiency.

Using call/cc offers implementation advantages over negative translation.

Abstract

We show how to extract existential witnesses from classical proofs using Krivine's classical realizability---where classical proofs are interpreted as lambda-terms with the call/cc control operator. We first recall the basic framework of classical realizability (in classical second-order 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 Sigma01-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.