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### Intrinsic Complexity of Recursive Functions on Natural Numbers with Standard Order

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### Abstract

The 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, the possible spectra of such functions include three sets consisting of precisely: the computable degree, all c.e. degrees and all Δ₂ 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 [Wright, 2018] and Harrison-Trainor [Harrison-Trainor, 2018] by showing that the division between computable, c.e. and Δ₂ 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 Δ₂ degrees.

### BibTeX - Entry

@InProceedings{bazhenov_et_al:LIPIcs.STACS.2022.8,
author =	{Bazhenov, Nikolay and Kaloci\'{n}ski, Dariusz and Wroc{\l}awski, Micha{\l}},
title =	{{Intrinsic Complexity of Recursive Functions on Natural Numbers with Standard Order}},
booktitle =	{39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages =	{8:1--8:20},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-222-8},
ISSN =	{1868-8969},
year =	{2022},
volume =	{219},
editor =	{Berenbrink, Petra and Monmege, Benjamin},
publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
}