Streamlined subrecursive degree theory

The study of honest elementary degrees has its roots in subrecursion theory from the nineteen seventies. Some relevant papers are Meyer & Ritchie [12] and Machtey [9–11]. These papers deal with subrecursive function classes being generated by so-called honest functions where an honest function is defined relative to a subrecursive class S and a model of computation: a function f : N → N is honest if the number of steps in a computation of f is bounded by ψ(x, f(x)) for some ψ ∈ S. Influenced by subrecursion theory from the seventies, Kristiansen introduces the honest elementary degrees about twenty years later. These degrees are the equivalence classes induced on the honest functions by the reducibility relation “being (Kalmar) elementary in”, but now a function is regarded to be honest if it is monotone, dominates 2 and has (Kalmar) elementary graph. This notion of honesty is in certain respects equivalent to the one from the seventies, but not in every respect, and the combination of the particular reducibility relation “being elementary in” and the novel definition of an honest function is the basis for the following pivotal theorem: