Randomness on Computable Probability Spaces—A Dynamical Point of View
Identifieur interne : 002504 ( Main/Exploration ); précédent : 002503; suivant : 002505Randomness on Computable Probability Spaces—A Dynamical Point of View
Auteurs : Peter Gacs [États-Unis] ; Mathieu Hoyrup [France] ; Cristobal Rojas [Canada]Source :
- Theory of Computing Systems [ 1432-4350 ] ; 2011.
English descriptors
Abstract
We extend the notion of randomness (in the version introduced by Schnorr) to computable probability spaces and compare it to a dynamical notion of randomness: typicality. Roughly, a point is 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 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.
Url:
DOI: 10.1007/s00224-010-9263-x
Affiliations:
- Canada, France, États-Unis
- Grand Est, Lorraine (région), Ontario
- Metz, Nancy, Toronto
- Université de Lorraine, Université de Toronto
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Roughly, a point is 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 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.</div></front></TEI><affiliations><list><country><li>Canada</li><li>France</li><li>États-Unis</li></country><region><li>Grand Est</li><li>Lorraine (région)</li><li>Ontario</li></region><settlement><li>Metz</li><li>Nancy</li><li>Toronto</li></settlement><orgName><li>Université de Lorraine</li><li>Université de Toronto</li></orgName></list><tree><country name="États-Unis"><noRegion><name sortKey="Gacs, Peter" sort="Gacs, Peter" uniqKey="Gacs P" first="Peter" last="Gacs">Peter Gacs</name></noRegion></country><country name="France"><region name="Grand Est"><name sortKey="Hoyrup, Mathieu" sort="Hoyrup, Mathieu" uniqKey="Hoyrup M" first="Mathieu" last="Hoyrup">Mathieu Hoyrup</name></region></country><country name="Canada"><region name="Ontario"><name sortKey="Rojas, Cristobal" sort="Rojas, Cristobal" uniqKey="Rojas C" first="Cristobal" last="Rojas">Cristobal Rojas</name></region></country></tree></affiliations></record>Pour manipuler ce document sous Unix (Dilib)
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