The equivalence of bar recursion and open recursion

Several extensions of Gödel’s system T with new forms of recursion have been designed for the purpose of giving a computational interpretation to classical analysis. One can organise many of these extensions into two groups: those based on bar recursion, which include Spector’s original bar recursion, modified bar recursion and the more recent products of selections functions, or those based on open recursion which in particular include the symmetric Berardi-Bezem-Coquand (BBC) functional. We relate these two groups by showing that both open recursion and the BBC functional are primitive recursively equivalent to a variant of modified bar recursion. Our results, in combination with existing research, essentially complete the classification up to primitive recursive equivalence of those extensions of system T used to give a direct computational interpretation to choice principles.