License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.STACS.2014.125
URN: urn:nbn:de:0030-drops-44520
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2014/4452/
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Bauwens, Bruno

Asymmetry of the Kolmogorov complexity of online predicting odd and even bits

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Abstract

Symmetry of information states that C(x)+C(y|x)=C(x,y)+O(log(C(x))). In [Chernov, Shen, Vereshchagin, and Vovk, 2008] an online variant of Kolmogorov complexity is introduced and we show that a similar relation does not hold. Let the even (online Kolmogorov) complexity of an n-bitstring x_1 x_2...x_n be the length of a shortest program that computes x_2 on input x_1, computes x_4 on input x_1 x_2 x_3, etc; and similar for odd complexity. We show that for all n there exists an n-bit x such that both odd and even complexity are almost as large as the Kolmogorov complexity of the whole string. Moreover, flipping odd and even bits to obtain a sequence x_2 x_1 x_4 x_3..., decreases the sum of odd and even complexity to C(x). Our result is related to the problem of inferrence of causality in timeseries.

BibTeX - Entry

@InProceedings{bauwens:LIPIcs:2014:4452,
  author =	{Bruno Bauwens},
  title =	{{Asymmetry of the Kolmogorov complexity of online predicting odd and even bits}},
  booktitle =	{31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)},
  pages =	{125--136},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-65-1},
  ISSN =	{1868-8969},
  year =	{2014},
  volume =	{25},
  editor =	{Ernst W. Mayr and Natacha Portier},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2014/4452},
  URN =		{urn:nbn:de:0030-drops-44520},
  doi =		{10.4230/LIPIcs.STACS.2014.125},
  annote =	{Keywords: (On-line) Kolmogorov complexity, (On-line) Algorithmic Probability, Philosophy of Causality, Information Transfer}
}

Keywords: (On-line) Kolmogorov complexity, (On-line) Algorithmic Probability, Philosophy of Causality, Information Transfer
Collection: 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)
Issue Date: 2014
Date of publication: 05.03.2014

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