Intrinsic complexity of recursive functions on natural numbers with standard order

TL;DR

This paper advances the understanding of the intrinsic complexity spectra of recursive functions on natural numbers, providing criteria for their classification and expanding the known spectra beyond previous limitations.

Contribution

It offers a more complete classification of the intrinsic complexity spectra of unary recursive functions, introducing criteria and broadening the known spectra to include new cases.

Findings

Identifies criteria for functions to have spectra equal to computable, c.e., or Δ₂ degrees.

Shows existence of functions with spectra containing all c.e. degrees but not all Δ₂ degrees.

Expands the classification of intrinsic complexity spectra for recursive functions.

Abstract

Intrinsic complexity of a relation on a given computable structure is captured by the notion of its degree spectrum - the set of Turing degrees of images of the relation in all computable isomorphic copies of that structure. We investigate the intrinsic complexity of unary total recursive functions on nonnegative integers with standard order. According to existing results, possible spectra of such functions include three sets consisting of precisely: the computable degree, all c.e. degrees and all $\Delta_2$ degrees. These results, however, fall far short of the full classification. In this paper, we obtain a more complete picture by giving a few criteria for a function to have intrinsic complexity equal to one of the three candidate sets of degrees. Our investigations are based on the notion of block functions and a broader class of quasi-block functions beyond which all functions of interest have intrinsic complexity equal to the c.e. degrees. We also answer the questions raised by Wright and Harrison-Trainor by showing that the division between computable, c.e. and $\Delta_2$ degrees is insufficient in this context as there is a unary total recursive function whose spectrum contains all c.e. degrees but is strictly contained in the $\Delta_2$ degrees.