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DOI: 10.4230/LIPIcs.FSTTCS.2021.47

Normal Sequences with Non-Maximal Automatic Complexity

Authors: Liam Jordon and Philippe Moser

Published in: LIPIcs, Volume 213, 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)

Abstract

This paper examines Automatic Complexity, a complexity notion introduced by Shallit and Wang in 2001 [Jeffrey O. Shallit and Ming-wei Wang, 2001]. We demonstrate that there exists a normal sequence T such that I(T) = 0 and S(T) ≤ 1/2, where I(T) and S(T) are the lower and upper automatic complexity rates of T respectively. We furthermore show that there exists a Champernowne sequence C, i.e. a sequence formed by concatenating all strings of length one followed by concatenating all strings of length two and so on, such that S(C) ≤ 2/3.

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Liam Jordon and Philippe Moser. Normal Sequences with Non-Maximal Automatic Complexity. In 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 213, pp. 47:1-47:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{jordon_et_al:LIPIcs.FSTTCS.2021.47,
  author =	{Jordon, Liam and Moser, Philippe},
  title =	{{Normal Sequences with Non-Maximal Automatic Complexity}},
  booktitle =	{41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)},
  pages =	{47:1--47:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-215-0},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{213},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Chekuri, Chandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2021.47},
  URN =		{urn:nbn:de:0030-drops-155580},
  doi =		{10.4230/LIPIcs.FSTTCS.2021.47},
  annote =	{Keywords: Automatic Complexity, finite-state complexity, normal sequences, Champernowne sequences, de Bruijn strings, Kolmogorov complexity}
}

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Normal Sequences with Non-Maximal Automatic Complexity

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