Theory of Computing Systems

, Volume 39, Issue 5, pp 707–721 | Cite as

Some Results on Effective Randomness

Article

Abstract

We investigate the characterizations of effective randomness in terms of Martin-Lof tests and martingales. First, we address a question of Ambos-Spies and Kucera, who asked for a characterization of computable randomness in terms of tests. We argue that computable randomness can be characterized in terms of Martin-Lof tests and effective probability distributions on Cantor space. Second, we show that the class of Martin-Lof random sequences coincides with the class of sequences that are random with respect to computable martingale processes; the latter randomness notion was introduced by Hitchcock and Lutz. Third, we analyze the sequence of measures of the components of a universal Martin-Lof test. Kucera and Slaman showed that any component of a universal Martin-Lof test defines a class of Martin-Lof random measure. Further, since the sets in a Martin-Lof 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 r1, r2,... of reals such that each rn is Martin-Lof random and less than 2-n there is a universal Martin-Lof test U1, U2,... such that Un{0,1} has measure rn.

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Copyright information

© Springer 2005

Authors and Affiliations

  1. 1.Fakultat fur Mathematik und Informatik, Ruprecht-Karls-Universitat Heidelberg, Im Neuenheimer Feld 294, D-69120 HeidelbergGermany
  2. 2.Department of Mathematics, University of California, Berkeley, CA 94720USA

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