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Randomness on Computable Probability Spaces - A Dynamical Point of View

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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.

BibTeX - Entry

@InProceedings{gacs_et_al:LIPIcs:2009:1828,
  author =	{Peter Gacs and Mathieu Hoyrup and Cristobal Rojas},
  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 =	{Susanne Albers and Jean-Yves Marion},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2009/1828},
  URN =		{urn:nbn:de:0030-drops-18280},
  doi =		{http://dx.doi.org/10.4230/LIPIcs.STACS.2009.1828},
  annote =	{Keywords: Schnorr randomness, Birkhoff's ergodic theorem, Computable measures}
}

Keywords: Schnorr randomness, Birkhoff's ergodic theorem, Computable measures
Seminar: 26th International Symposium on Theoretical Aspects of Computer Science
Issue date: 2009
Date of publication: 2009


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