Kolmogorov-Loveland randomness and stochasticity

Wolfgang Merkle, Joseph Miller, André Nies, Jan Severin Reimann, Frank Stephan

Research output: Contribution to journalConference article

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 < 1 with respect to prefix-free Kolmogorov complexity. This improves on a result by Muchnik, who has shown a similar implication where the premise requires that such compressible prefixes can be found effectively.

Original languageEnglish (US)
Pages (from-to)422-433
Number of pages12
JournalLecture Notes in Computer Science
Volume3404
StatePublished - Sep 12 2005

Fingerprint

Binary sequences
Stochasticity
Randomness
Random Sequence
Prefix
Subsequence
Prefix-free
Kolmogorov Complexity
Selection Rules
Binary Sequences
Biased
Lemma
Open Problems

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Computer Science(all)

Cite this

Merkle, W., Miller, J., Nies, A., Reimann, J. S., & Stephan, F. (2005). Kolmogorov-Loveland randomness and stochasticity. Lecture Notes in Computer Science, 3404, 422-433.
Merkle, Wolfgang ; Miller, Joseph ; Nies, André ; Reimann, Jan Severin ; Stephan, Frank. / Kolmogorov-Loveland randomness and stochasticity. In: Lecture Notes in Computer Science. 2005 ; Vol. 3404. pp. 422-433.
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Merkle, W, Miller, J, Nies, A, Reimann, JS & Stephan, F 2005, 'Kolmogorov-Loveland randomness and stochasticity', Lecture Notes in Computer Science, vol. 3404, pp. 422-433.

Kolmogorov-Loveland randomness and stochasticity. / Merkle, Wolfgang; Miller, Joseph; Nies, André; Reimann, Jan Severin; Stephan, Frank.

In: Lecture Notes in Computer Science, Vol. 3404, 12.09.2005, p. 422-433.

Research output: Contribution to journalConference article

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Merkle W, Miller J, Nies A, Reimann JS, Stephan F. Kolmogorov-Loveland randomness and stochasticity. Lecture Notes in Computer Science. 2005 Sep 12;3404:422-433.