Separations of non-monotonic randomness notions (Q2907050)

There are several definitions of algorithmic randomness. One of the (seemingly) natural definitions was introduced by Schnorr: an infinite sequence is random if no computable strategy succeeds on it by betting on bits in order. The problem with this definition is that some such ``random'' sequences are highly compressible -- which contradicts to our intuition that random sequences should not be compressible. Muchnik conjectured that we can get the usual incompressibility-based Martin-Löf randomness if we allow non-monotonic strategies, in which bets do not have to be on bits in order. This conjecture is still open. To get a better understanding of this situation, \textit{J. S. Miller} and \textit{A. Nies} [Bull. Symb. Log. 12, No. 3, 390--410 (2006; Zbl 1169.03033)] introduced several intermediate concepts of randomness in which we can bet out of order, but the selection of bits must be fixed from the very beginning (and cannot adaptively change depending on the sequence). The authors provide a full classification of these intermediate concepts: for every ordered pair of concepts \((C,C')\), they either prove that \(C\)-randomness implies \(C'\)-randomness, or prove the existence of a \(C\)-random sequence which is not \(C'\)-random.