### 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 language | English (US) |
---|---|

Pages (from-to) | 422-433 |

Number of pages | 12 |

Journal | Lecture Notes in Computer Science |

Volume | 3404 |

State | Published - Sep 12 2005 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Lecture Notes in Computer Science*,

*3404*, 422-433.

}

*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.

Research output: Contribution to journal › Conference article

TY - JOUR

T1 - Kolmogorov-Loveland randomness and stochasticity

AU - Merkle, Wolfgang

AU - Miller, Joseph

AU - Nies, André

AU - Reimann, Jan Severin

AU - Stephan, Frank

PY - 2005/9/12

Y1 - 2005/9/12

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=24144458881&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=24144458881&partnerID=8YFLogxK

M3 - Conference article

VL - 3404

SP - 422

EP - 433

JO - Lecture Notes in Computer Science

JF - Lecture Notes in Computer Science

SN - 0302-9743

ER -