Some Results on Effective Randomness

  • Wolfgang Merkle
  • Nenad Mihailović
  • Theodore A. Slaman
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3142)

Abstract

We investigate the characterizations of effective randomness in terms of Martin-Löf covers and martingales. First, we address a question of Ambos-Spies and Kučera [1], who asked for a characterization of computable randomness in terms of tests. We argue that computable randomness can be characterized in term of Martin-Löf tests and effective probability distributions on Cantor space.

Second, we show that the class of Martin-Löf random sets coincides with the class of sets of reals that are random with respect to computable martingale processes. This improves on results of Hitchcock and Lutz [8], who showed that the latter class is contained in the class of Martin-Löf random sets and is a strict superset of the class of rec-random sets.

Third, we analyze the sequence of measures of the components of a universal Martin-Löf test. Kučera and Slaman [12] showed that any component of a universal Martin-Löf test defines a class of Martin-Löf random measure. Further, since the sets in a Martin-Löf test are uniformly computably enumerable, so is the corresponding sequence of measures. We prove an exact converse and hence a characterization. For any uniformly computably enumerable sequence r 1,r 2,... of reals such that each r i is Martin-Löf random and less than 2 − − i there is a universal Martin-Löf test U 1, U 2,... such that U i { 0,1} ∞  has measure r i .

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Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Wolfgang Merkle
    • 1
  • Nenad Mihailović
    • 1
  • Theodore A. Slaman
    • 2
  1. 1.Institut für InformatikRuprecht-Karls-Universität Heidelberg 
  2. 2.Department of MathematicsUniversity of California 

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