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DOI: 10.4230/LIPIcs.STACS.2019.57

Random Noise Increases Kolmogorov Complexity and Hausdorff Dimension

Authors: Gleb Posobin and Alexander Shen

Published in: LIPIcs, Volume 126, 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)

Abstract

Consider a bit string x of length n and Kolmogorov complexity alpha n (for some alpha<1). It is always possible to increase the complexity of x by changing a small fraction of bits in x [Harry Buhrman et al., 2005]. What happens with the complexity of x when we randomly change each bit independently with some probability tau? We prove that a linear increase in complexity happens with high probability, but this increase is smaller than in the case of arbitrary change considered in [Harry Buhrman et al., 2005]. The amount of the increase depends on x (strings of the same complexity could behave differently). We give exact lower and upper bounds for this increase (with o(n) precision). The same technique is used to prove the results about the (effective Hausdorff) dimension of infinite sequences. We show that random change increases the dimension with probability 1, and provide an optimal lower bound for the dimension of the changed sequence. We also improve a result from [Noam Greenberg et al., 2018] and show that for every sequence omega of dimension alpha there exists a strongly alpha-random sequence omega' such that the Besicovitch distance between omega and omega' is 0. The proofs use the combinatorial and probabilistic reformulations of complexity statements and the technique that goes back to Ahlswede, Gács and Körner [Ahlswede et al., 1976].

Cite as

Gleb Posobin and Alexander Shen. Random Noise Increases Kolmogorov Complexity and Hausdorff Dimension. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 57:1-57:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{posobin_et_al:LIPIcs.STACS.2019.57,
  author =	{Posobin, Gleb and Shen, Alexander},
  title =	{{Random Noise Increases Kolmogorov Complexity and Hausdorff Dimension}},
  booktitle =	{36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)},
  pages =	{57:1--57:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-100-9},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{126},
  editor =	{Niedermeier, Rolf and Paul, Christophe},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2019.57},
  URN =		{urn:nbn:de:0030-drops-102969},
  doi =		{10.4230/LIPIcs.STACS.2019.57},
  annote =	{Keywords: Kolmogorov complexity, effective Hausdorff dimension, random noise}
}

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DOI: 10.4230/LIPIcs.MFCS.2018.2

Plain Stopping Time and Conditional Complexities Revisited

Authors: Mikhail Andreev, Gleb Posobin, and Alexander Shen

Published in: LIPIcs, Volume 117, 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)

Abstract

In this paper we analyze the notion of "stopping time complexity", the amount of information needed to specify when to stop while reading an infinite sequence. This notion was introduced by Vovk and Pavlovic [Vovk and Pavlovic, 2016]. It turns out that plain stopping time complexity of a binary string x could be equivalently defined as (a) the minimal plain complexity of a Turing machine that stops after reading x on a one-directional input tape; (b) the minimal plain complexity of an algorithm that enumerates a prefix-free set containing x; (c) the conditional complexity C(x|x*) where x in the condition is understood as a prefix of an infinite binary sequence while the first x is understood as a terminated binary string; (d) as a minimal upper semicomputable function K such that each binary sequence has at most 2^n prefixes z such that K(z)<n; (e) as maxC^X(x) where C^X(z) is plain Kolmogorov complexity of z relative to oracle X and the maximum is taken over all extensions X of x. We also show that some of these equivalent definitions become non-equivalent in the more general setting where the condition y and the object x may differ, and answer an open question from Chernov, Hutter and Schmidhuber [Alexey V. Chernov et al., 2007].

Cite as

Mikhail Andreev, Gleb Posobin, and Alexander Shen. Plain Stopping Time and Conditional Complexities Revisited. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 2:1-2:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{andreev_et_al:LIPIcs.MFCS.2018.2,
  author =	{Andreev, Mikhail and Posobin, Gleb and Shen, Alexander},
  title =	{{Plain Stopping Time and Conditional Complexities Revisited}},
  booktitle =	{43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)},
  pages =	{2:1--2:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-086-6},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{117},
  editor =	{Potapov, Igor and Spirakis, Paul and Worrell, James},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2018.2},
  URN =		{urn:nbn:de:0030-drops-95842},
  doi =		{10.4230/LIPIcs.MFCS.2018.2},
  annote =	{Keywords: Kolmogorov complexity, stopping time complexity, structured conditional complexity, algorithmic information theory}
}

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Random Noise Increases Kolmogorov Complexity and Hausdorff Dimension

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Plain Stopping Time and Conditional Complexities Revisited

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