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DOI: 10.4230/LIPIcs.STACS.2009.1828

Randomness on Computable Probability Spaces - A Dynamical Point of View

Authors: Peter Gacs, Mathieu Hoyrup, and Cristobal Rojas

Published in: LIPIcs, Volume 3, 26th International Symposium on Theoretical Aspects of Computer Science (2009)

Abstract

We extend the notion of randomness (in the version introduced by Schnorr) to computable Probability Spaces and compare it to a \emph{dynamical} notion of randomness: typicality. Roughly, a point is \emph{typical} for some dynamic, if it follows the statistical behavior of the system (Birkhoff's pointwise ergodic theorem). We prove that a point is Schnorr random if and only if it is typical for every \emph{mixing} computable dynamics. To prove the result we develop some tools for the theory of computable probability spaces (for example, morphisms) that are expected to have other applications.

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Peter Gacs, Mathieu Hoyrup, and Cristobal Rojas. Randomness on Computable Probability Spaces - A Dynamical Point of View. In 26th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 3, pp. 469-480, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{gacs_et_al:LIPIcs.STACS.2009.1828,
  author =	{Gacs, Peter and Hoyrup, Mathieu and Rojas, Cristobal},
  title =	{{Randomness on Computable Probability Spaces - A Dynamical Point of View}},
  booktitle =	{26th International Symposium on Theoretical Aspects of Computer Science},
  pages =	{469--480},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-09-5},
  ISSN =	{1868-8969},
  year =	{2009},
  volume =	{3},
  editor =	{Albers, Susanne and Marion, Jean-Yves},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2009.1828},
  URN =		{urn:nbn:de:0030-drops-18280},
  doi =		{10.4230/LIPIcs.STACS.2009.1828},
  annote =	{Keywords: Schnorr randomness, Birkhoff's ergodic theorem, Computable measures}
}

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Randomness on Computable Probability Spaces - A Dynamical Point of View

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