Thomas Powell

TL;DR: The 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.

TL;DR: It is shown that Spector's “restricted” form of bar recursion is sufficient (over system T) to define spector's search functional, and it follows that this iterated product of selection functions is T-equivalent to the corresponding iterated products of quantifiers.

TL;DR: It is proved that modified bar recursion (in the form of the implicitly controlled product of selection functions), open recursion, update recursion and the Berardi-BezemCoquand realizer for countable choice are all primitive recursively equivalent in the model of continuous functionals.

TL;DR: It is shown that the finite product of selection functions (for all finite types) is primitive recursively equivalent to Goedel's higher-type recursor and that other classical arithmetical principles such as bounded collection and finite choice are more naturally interpreted via the product ofselection functions.

TL;DR: It is argued that the result that a functional interpretation of proofs in analysis can be given by the product of selection functions has genuine practical value by interpreting some well-known theorems of mathematics and demonstrating that the product gives these theoresms a natural computational interpretation that can be clearly understood in game theoretic terms.