Separations of non-monotonic randomness notions

TL;DR

This paper investigates various non-monotonic and non-adaptive randomness notions in algorithmic randomness, providing a comprehensive classification and exploring their relationships with classical randomness concepts.

Contribution

It introduces and classifies weak versions of Kolmogorov-Loveland randomness with non-adaptive strategies, clarifying their position relative to established randomness notions.

Findings

Full classification of non-monotonic randomness notions.

Identification of relationships between weak Kolmogorov-Loveland and Martin-Loef randomness.

Insights into the structure of non-adaptive randomness classes.

Abstract

In the theory of algorithmic randomness, several notions of random sequence are defined via a game-theoretic approach, and the notions that received most attention are perhaps Martin-Loef randomness and computable randomness. The latter notion was introduced by Schnorr and is rather natural: an infinite binary sequence is computably random if no total computable strategy succeeds on it by betting on bits in order. However, computably random sequences can have properties that one may consider to be incompatible with being random, in particular, there are computably random sequences that are highly compressible. The concept of Martin-Loef randomness is much better behaved in this and other respects, on the other hand its definition in terms of martingales is considerably less natural. Muchnik, elaborating on ideas of Kolmogorov and Loveland, refined Schnorr's model by also allowing non-monotonic strategies, i.e. strategies that do not bet on bits in order. The subsequent ``non-monotonic'' notion of randomness, now called Kolmogorov-Loveland randomness, has been shown to be quite close to Martin-Loef randomness, but whether these two classes coincide remains a fundamental open question. As suggested by Miller and Nies, we study in this paper weak versions of Kolmogorov-Loveland randomness, where the betting strategies are non-adaptive (i.e., the positions of the bits to bet on should be decided before the game). We obtain a full classification of the different notions we consider.