Non-definability of the Ackermann function with type 1 partial primitive recursion

Abstract.

The paper builds on a simply typed term system ${\cal PR}^\omega$ providing a notion of partial primitive recursive functional on arbitrary Scott domains $D_\sigma$ that includes a suitable concept of parallelism. Computability on the partial continuous functionals seems to entail that Kleene's schema of higher type simultaneous course-of-values recursion (SCVR) is not reducible to partial primitive recursion. So an extension ${\cal PR}^{\omega e}$ is studied that is closed under SCVR and yet stays within the realm of subrecursiveness. The twist are certain type 1 Gödel recursors ${\cal R}_k$ for simultaneous partial primitive recursion. Formally, the value ${\cal R}_k\vec{g}\vec{H} x i$ is a function $f_i$ of type $\iota \to \iota$, however, ${\cal R}_k$ is modelled such that $f_i$ is a finite element of $D_{\iota\to\iota}$ or in other words, a partial sequence. It is shown that the Ackermann function is not definable in ${\cal PR}^{\omega e}$.