Parametrised bar recursion: A unifying framework for realizability interpretations of classical dependent choice

Abstract

During the last twenty years or so a wide range of realizability interpretations of classical analysis have been developed. In many cases, these are achieved by extending the base interpreting system of primitive recursive functionals with some form of bar recursion, which realizes the negative translation of either countable or countable dependent choice. In this work we present the many variants of bar recursion used in this context as instantiations of a general, parametrised recursor, and give a uniform proof that under certain conditions this recursor realizes a corresponding family of parametrised dependent choice principles. From this proof, the soundness of most of the existing bar recursive realizability interpretations of choice, including those based on the Berardi-Bezem-Coquand functional, modified realizability and the more recent products of selection functions of Escardó and Oliva, follows as a simple corollary. We achieve not only a uniform framework in which familiar realizability interpretations of choice can be compared, but show that these represent just simple instances of a large family of potential interpretations of dependent choice principles.