Bar recursion as primitive recursion with nonstandard numbers

Sam Sanders

Published 2015-02-12Version 1

Bar recursion is a form of higher-type recursion, originally introduced by Spector, which allows for the extraction of constructive information from proofs involving the axiom of dependent choice. Alternative approaches with similar aim are: open recursion, update recursion, the so-called BBC-functional, and selection functions. Unfortunately, none of these forms of recursion, including \emph{modified} bar recursion, are Kleene-S1-S9-computable over the total continuous functionals, and a more direct or effective approach has been called for. To this end, we show in this paper that bar recursion becomes primitive recursive inside Nelson's syntactic framework of Nonstandard Analysis. In particular, we show that modified bar recursion of type zero, which corresponds to the Gandy-Hyland functional, equals a primitive recursive functional involving nonstandard numbers. Similar results for more general bar recursion are discussed.