Spector bar recursion over finite partial functions

Abstract

We introduce a new, demand-driven variant of Spector's bar recursion in the spirit of the Berardi-Bezem-Coquand functional. The recursion takes place over finite partial functions $u$, where the control parameter $\varphi$, used in Spector's bar recursion to terminate the computation at sequences $s$ satisfying $\varphi(\hat{s})<|s|$, now acts as a guide for deciding exactly where to make bar recursive updates, terminating the computation whenever $\varphi(\hat{u})\in\mbox{dom}(u)$. We begin by exploring theoretical aspects of this new form of recursion, then in the main part of the paper we show that demand-driven bar recursion can be directly used to give an alternative functional interpretation of classical countable choice. We provide a short case study as an illustration, in which we extract a new bar recursive program from the proof that there is no injection from $\mathbb{N}\to\mathbb{N}$ to $\mathbb{N}$, and compare this to the program that would be obtained using Spector's original variant. We conclude by formally establishing that our new bar recursor is primitive recursively equivalent to the original Spector bar recursion, and thus defines the same class of functionals when added to G\"odel's system $\sf T$.