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Documents authored by Wrocławski, Michał

Document
DOI: 10.4230/LIPIcs.MFCS.2024.65
Punctual Presentability in Certain Classes of Algebraic Structures

Authors: Dariusz Kalociński, Luca San Mauro, and Michał Wrocławski

Published in: LIPIcs, Volume 306, 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)

Abstract
Punctual structure theory is a rapidly emerging subfield of computable structure theory which aims at understanding the primitive recursive content of algebraic structures. A structure with domain ℕ is punctual if its relations and functions are (uniformly) primitive recursive. One of the fundamental problems of this area is to understand which computable members of a given class of structures admit a punctual presentation. We investigate such a problem for a number of familiar classes of algebraic structures, paying special attention to the case of trees, presented both in a relational and functional signature.
Cite as

Dariusz Kalociński, Luca San Mauro, and Michał Wrocławski. Punctual Presentability in Certain Classes of Algebraic Structures. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 65:1-65:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{kalocinski_et_al:LIPIcs.MFCS.2024.65,
  author =	{Kaloci\'{n}ski, Dariusz and San Mauro, Luca and Wroc{\l}awski, Micha{\l}},
  title =	{{Punctual Presentability in Certain Classes of Algebraic Structures}},
  booktitle =	{49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
  pages =	{65:1--65:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-335-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{306},
  editor =	{Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.65},
  URN =		{urn:nbn:de:0030-drops-206212},
  doi =		{10.4230/LIPIcs.MFCS.2024.65},
  annote =	{Keywords: fully primitive recursive structures, punctual presentability, trees, injection structures}
}
Document
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.
Cite as

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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