Classifying the phase transition threshold for Ackermannian functions

Abstract

It is well known that the Ackermann function can be defined via diagonalization from an iteration hierarchy (of Grzegorczyk type) which is built on a start function like the successor function. In this paper we study for a given start function g iteration hierarchies with a sub-linear modulus h of iteration. In terms of g and h we classify the phase transition for the resulting diagonal function from being primitive recursive to being Ackermannian.

Phase Transitions Related to the Pigeonhole Principle

“…Then F f is Ackermannian, due to Theorem 1 in [OW09]. If f would be non decreasing then Lemma 2 would yield…”