THE HERBRAND FUNCTIONAL INTERPRETATION OF THE DOUBLE NEGATION SHIFT

Abstract

AbstractThis paper considers a generalisation of selection functions over an arbitrary strong monad T, as functionals of type $J_R^T X = (X \to R) \to TX$. It is assumed throughout that R is a T-algebra. We show that $J_R^T$ is also a strong monad, and that it embeds into the continuation monad $K_R X = (X \to R) \to R$. 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 powerset monad ${\cal P}_{\rm{f}} \left( \cdot \right)$. These are used to show that when $TX = {\cal P}_{\rm{f}} \left( X \right)$ the explicitly controlled product of T-selection functions calculates a witness to the Herbrand functional interpretation of the double negation shift.