Ranking Primitive Recursions: The Low Grzegorczyk Classes Revisited

Abstract

Traditional results in subrecursion theory are integrated with the recent work in "predicative recursion" by defining a simple ranking $\rho$ of all primitive recursive functions. The hierarchy defined by this ranking coincides with the Grzegorczyk hierarchy at and above the linear-space level. Thus, the result is like an extension of the Schwichtenberg--Müller theorems. When primitive recursion is replaced by recursion on notation, the same series of classes is obtained except with the polynomial time computable functions at the first level.