The Herbrand Functional Interpretation of the Double Negation Shift

This paper considers a generalisation of selection functions over an arbitrary strong monad T , as functionals of type J RX = (X → R) → TX. It is assumed throughout that R is a T -algebra. We show that J R is also a strong monad, and that it embeds into the continuation monad KRX = (X → R) → R. We use this to derive that the explicitly controlled product of T -selection functions is definable from the explicitly controlled product of quantifiers, and hence from Spector’s bar recursion. We then prove several properties of this product in the special case when T is the finite power set monad Pf(·). These are used to show that when TX = Pf(X) the explicitly controlled product of T -selection functions calculates a witness to the Herbrand functional interpretation of the double negation shift.