License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
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DOI: 10.4230/LIPIcs.STACS.2020.55
URN: urn:nbn:de:0030-drops-119168
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2020/11916/
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Nies, André ; Stephan, Frank

Randomness and Initial Segment Complexity for Probability Measures

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Abstract

We study algorithmic randomness properties for probability measures on Cantor space. We say that a measure μ on the space of infinite bit sequences is Martin-Löf absolutely continuous if the non-Martin-Löf random bit sequences form a null set with respect to μ. We think of this as a weak randomness notion for measures. We begin with examples, and a robustness property related to Solovay tests. Our main work connects our property to the growth of the initial segment complexity for measures μ; the latter is defined as a μ-average over the complexity of strings of the same length. We show that a maximal growth implies our weak randomness property, but also that both implications of the Levin-Schnorr theorem fail. We briefly discuss K-triviality for measures, which means that the growth of initial segment complexity is as slow as possible. We show that full Martin-Löf randomness of a measure implies Martin-Löf absolute continuity; the converse fails because only the latter property is compatible with having atoms. In a final section we consider weak randomness relative to a general ergodic computable measure. We seek appropriate effective versions of the Shannon-McMillan-Breiman theorem and the Brudno theorem where the bit sequences are replaced by measures.

BibTeX - Entry

@InProceedings{nies_et_al:LIPIcs:2020:11916,
  author =	{Andr{\'e} Nies and Frank Stephan},
  title =	{{Randomness and Initial Segment Complexity for Probability Measures}},
  booktitle =	{37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
  pages =	{55:1--55:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-140-5},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{154},
  editor =	{Christophe Paul and Markus Bl{\"a}ser},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/11916},
  URN =		{urn:nbn:de:0030-drops-119168},
  doi =		{10.4230/LIPIcs.STACS.2020.55},
  annote =	{Keywords: algorithmic randomness, probability measure on Cantor space, Kolmogorov complexity, statistical superposition, quantum states}
}

Keywords: algorithmic randomness, probability measure on Cantor space, Kolmogorov complexity, statistical superposition, quantum states
Collection: 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)
Issue Date: 2020
Date of publication: 04.03.2020

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