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DOI: 10.4230/LIPIcs.STACS.2020.46

Information Distance Revisited

Authors: Bruno Bauwens

Published in: LIPIcs, Volume 154, 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)

Abstract

We consider the notion of information distance between two objects x and y introduced by Bennett, Gács, Li, Vitanyi, and Zurek [C. H. Bennett et al., 1998] as the minimal length of a program that computes x from y as well as computing y from x, and study different versions of this notion. In the above paper, it was shown that the prefix version of information distance equals max (K(x|y),K(y|x)) up to additive logarithmic terms. It was claimed by Mahmud [Mahmud, 2009] that this equality holds up to additive O(1)-precision. We show that this claim is false, but does hold if the distance is at least logarithmic. This implies that the original definition provides a metric on strings that are at superlogarithmically separated.

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Bruno Bauwens. Information Distance Revisited. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 46:1-46:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bauwens:LIPIcs.STACS.2020.46,
  author =	{Bauwens, Bruno},
  title =	{{Information Distance Revisited}},
  booktitle =	{37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
  pages =	{46:1--46:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-140-5},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{154},
  editor =	{Paul, Christophe and Bl\"{a}ser, Markus},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.46},
  URN =		{urn:nbn:de:0030-drops-119071},
  doi =		{10.4230/LIPIcs.STACS.2020.46},
  annote =	{Keywords: Kolmogorov complexity, algorithmic information distance}
}

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DOI: 10.4230/LIPIcs.STACS.2014.125

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

Authors: Bruno Bauwens

Published in: LIPIcs, Volume 25, 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)

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.

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Bruno Bauwens. Asymmetry of the Kolmogorov complexity of online predicting odd and even bits. In 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 25, pp. 125-136, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{bauwens:LIPIcs.STACS.2014.125,
  author =	{Bauwens, Bruno},
  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 =	{Mayr, Ernst W. and Portier, Natacha},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2014.125},
  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}
}

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Information Distance Revisited

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

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