Intrinsic Complexity of Recursive Functions on Natural Numbers with Standard Order

Authors Nikolay Bazhenov , Dariusz Kalociński , Michał Wrocławski

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Author Details

Nikolay Bazhenov
  • Sobolev Institute of Mathematics, Novosibirsk, Russia
Dariusz Kalociński
  • Institute of Computer Science, Polish Academy of Sciences, Warsaw, Poland
Michał Wrocławski
  • Faculty of Philosophy, University of Warsaw, Poland


We would like to thank Matthew Harrison-Trainor, Aleksander Iwanow, Mars Yamaleev for helpful discussions and anonymous reviewers for comments.

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Nikolay Bazhenov, Dariusz Kalociński, and Michał Wrocławski. Intrinsic Complexity of Recursive Functions on Natural Numbers with Standard Order. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 8:1-8:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computability
  • Computable Structure Theory
  • Degree Spectra
  • ω-Type Order
  • c.e. Degrees
  • d.c.e. Degrees
  • Δ₂ Degrees
  • Learnability


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