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Normal Sequences with Non-Maximal Automatic Complexity

Authors Liam Jordon , Philippe Moser

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ACM Subject Classification
  • Theory of computation → Formal languages and automata theory
Keywords
  • Automatic Complexity
  • finite-state complexity
  • normal sequences
  • Champernowne sequences
  • de Bruijn strings
  • Kolmogorov complexity

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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) https://doi.org/10.4230/LIPIcs.FSTTCS.2021.47

Author Details

Liam Jordon
  • Department of Computer Science, Maynooth University, Maynooth, Ireland
Philippe Moser
  • Department of Computer Science, Maynooth University, Maynooth, Ireland

Funding

  • Jordon, Liam: Supported by the Irish Research Council’s Government of Ireland Postgraduate Scholarship Programme. Grant number: GOIPG/2017/1200.

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