Kolmogorov-Loveland randomness and stochasticity

  • Authors:

  • Wolfgang Merkle;Joseph Miller;André Nies;Jan Reimann;Frank Stephan

  • Affiliations:

  • Universität Heidelberg, Heidelberg, Germany;Indiana University, Bloomington, Indiana;University of Auckland, Auckland, New Zealand;Universität Heidelberg, Heidelberg, Germany;National University of Singapore, Singapore

  • Venue:
  • STACS'05 Proceedings of the 22nd annual conference on Theoretical Aspects of Computer Science
  • Year:
  • 2005

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Abstract

One of the major open problems in the field of effective randomness is whether Martin-Löf randomness is the same as Kolmogorov-Loveland (or KL) randomness, where an infinite binary sequence is KL-random if there is no computable non-monotonic betting strategy that succeeds on the sequence in the sense of having an unbounded gain in the limit while betting successively on bits of the sequence. Our first main result states that every KL-random sequence has arbitrarily dense, easily extractable subsequences that are Martin-Löf random. A key lemma in the proof of this result is that for every effective split of a KL-random sequence at least one of the halves is Martin-Löf random. We show that this splitting property does not characterize KL-randomness by constructing a sequence that is not even computably random such that every effective split yields subsequences that are 2-random, hence are in particular Martin-Löf random. A sequence X is KL-stochastic if there is no computable non-monotonic selection rule that selects from X an infinite, biased sequence. Our second main result asserts that every KL-stochastic sequence has constructive dimension 1, or equivalently, a sequence cannot be KL-stochastic if it has infinitely many prefixes that can be compressed by a factor of α