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DOI: 10.4230/LIPIcs.CSL.2023.11

Degree Spectra, and Relative Acceptability of Notations

Authors: Nikolay Bazhenov and Dariusz Kalociński

Published in: LIPIcs, Volume 252, 31st EACSL Annual Conference on Computer Science Logic (CSL 2023)

Abstract

We investigate the interplay between the degree spectrum of a computable relation R on the computable structure (ω, <), i.e., natural numbers with the standard order, and the computability of the successor, relativized to that relation, in all computable copies of the structure - the property we dub successor’s recoverability. In computable structure theory, this property is used to show that of all possible Turing degrees that could belong to the spectrum of R (namely, of all Δ₂ degrees), in fact only the computably enumerable degrees are contained in the spectrum. Interestingly, successor’s recoverability (in the unrelativized form) appears also in philosophy of computing where it is used to distinguish between acceptable and deviant encodings (notations) of natural numbers. Since Shapiro’s notations are rarely seen through the lens of computable structure theory, we first lay the elementary conceptual groundwork to understand notations in terms of computable structures and show how results pertinent to the former can fundamentally inform our understanding of the latter. Secondly, we prove a new result which shows that for a large class of computable relations (satisfying a certain effectiveness condition), having all c.e. degrees as a spectrum implies that the successor is recoverable from the relation. The recoverability of successor may be otherwise seen as relativized acceptability of every notation for the structure. We end with remarks about connections of our result to relative computable categoricity and to a similar direction pursued by Matthew Harrison-Trainor in [Harrison-Trainor, 2018], and with an open question.

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Nikolay Bazhenov and Dariusz Kalociński. Degree Spectra, and Relative Acceptability of Notations. In 31st EACSL Annual Conference on Computer Science Logic (CSL 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 252, pp. 11:1-11:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bazhenov_et_al:LIPIcs.CSL.2023.11,
  author =	{Bazhenov, Nikolay and Kaloci\'{n}ski, Dariusz},
  title =	{{Degree Spectra, and Relative Acceptability of Notations}},
  booktitle =	{31st EACSL Annual Conference on Computer Science Logic (CSL 2023)},
  pages =	{11:1--11:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-264-8},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{252},
  editor =	{Klin, Bartek and Pimentel, Elaine},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2023.11},
  URN =		{urn:nbn:de:0030-drops-174725},
  doi =		{10.4230/LIPIcs.CSL.2023.11},
  annote =	{Keywords: Computable structure theory, Degree spectra, \omega-type order, C.e. degrees, \Delta₂ degrees, Acceptable notation, Successor, Learnability}
}

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DOI: 10.4230/LIPIcs.STACS.2022.8

Intrinsic Complexity of Recursive Functions on Natural Numbers with Standard Order

Authors: Nikolay Bazhenov, Dariusz Kalociński, and Michał Wrocławski

Published in: LIPIcs, Volume 219, 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)

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.

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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)

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@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},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.8},
  URN =		{urn:nbn:de:0030-drops-158185},
  doi =		{10.4230/LIPIcs.STACS.2022.8},
  annote =	{Keywords: Computable Structure Theory, Degree Spectra, \omega-Type Order, c.e. Degrees, d.c.e. Degrees, \Delta₂ Degrees, Learnability}
}

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Degree Spectra, and Relative Acceptability of Notations

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Intrinsic Complexity of Recursive Functions on Natural Numbers with Standard Order

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