Abstract

We show how the charaterization of the polytime functions by Bellantoni and Cook [1] can be extended to characterize any stage of the Grzegorzyk hierarchy above the second, thus proposing an answer to a problem posted by Clote [3]. This is done by allowing an arbitrary fixed number of distinct positions for variables instead of only two as in the original work of Bellantoni and Cook. As turned out after writing down this paper, comparable results were also proved by Bellantoni and Niggl [2]. Keywords: Recursion theory, Complexity theory, Grzegorzcyk. 1 Introduction Bellantoni and Cook [1] characterized the polytime functions by distinguishing between two sorts of arguments of functions, called normal and safe arguments. Recursion is only allowed over normal arguments, whereas the recursively computed values must be inserted in a safe position, and function composition is defined accordingly. Related tiering notions also appeared elsewhere, e.g. in Leivant [5] or Simmons [10]. Clote [...

Citations