Kolmogorov-Loveland Randomness and Stochasticity

An infinite binary sequence X is Kolmogorov-Loveland (or KL) random if there is no computable non-monotonic betting strategy that succeeds on X in the sense of having an unbounded gain in the limit while betting successively on bits of X. A sequence X is KL-stochastic if there is no computable non-monotonic selection rule that selects from X an infinite, biased sequence. One of the major open problems in the field of effective randomness is whether Martin-L¨of randomness is the same as KL-randomness. Our first main result states that KL-random sequences are close to Martin-L¨of random sequences in so far as every KL-random sequence has arbitrarily dense subsequences that are Martin-L¨of 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-L¨of random. However, this splitting property does not characterize KLrandomness; we construct a sequence that is not even computably random such that...

• Type: Conference
• Year: 2005
• Where: STACS
• Authors: Wolfgang Merkle, Joseph S. Miller, André Nies, Jan Reimann, Frank Stephan
• Keywords: Effective Split, Kl-random Sequence, Martin-L¨of Random, STACS 2005, Theoretical Computer Science