Do stronger definitions of randomness exist?

  • Authors:

  • Bruno Durand;Vladimir Kanovei;Vladimir A. Uspensky;Nikolai Vereshchagin

  • Affiliations:

  • L.I.P., Ecole Normale Supérieure de Lyon, 46 Allée d'Italie, 69364 Lyon Cedex 07, France;Department of Mathematical Logic and Theory of Algorithms, Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Vorobjyovih Gorih, Moscow 119899, Russia;Department of Mathematical Logic and Theory of Algorithms, Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Vorobjyovih Gorih, Moscow 119899, Russia;Department of Mathematical Logic and Theory of Algorithms, Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Vorobjyovih Gorih, Moscow 119899, Russia

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2003

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Abstract

In this paper, we investigate refined definition of random sequences. Classical definitions (Martin-Löf tests of randomness, uncompressibility, unpredictability, or stochasticness) make use of the notion of algorithm. We present alternative definitions based on set theory and explain why they depend on the model of ZFC that is considered. We also present a possible generalization of the definition when small infinite regularities are allowed.