Pure iteration and periodicity: A note on some small sub-recursive classes

Abstract

We define a hierarchy of small sub-recursive classes, based on the schema of pure iteration. is compared with a similar hierarchy, based on primitive recursion, for which a collapse is equivalent to a collapse of the small Grzegorczyk-classes. Our hierarchy does collapse, and the induced relational class is shown to have a highly periodic structure; indeed a unary predicate is decidable in iff it is definable in Presburger Arithmetic. The concluding discussion contrasts our findings to those of Kutyłowski [12]. © 2008 Springer-Verlag Berlin Heidelberg.