Separations of Non-monotonic Randomness Notions

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-L¨of 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-L¨of 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-m...

• Type: Conference
• Year: 2009
• Where: CCA
• Authors: Laurent Bienvenu, Rupert Hölzl, Thorsten Kräling, Wolfgang Merkle
• Keywords: CCA 2009, Martin-L¨of Randomness, Non-monotonic Randomness Notions, Random Sequences, Theoretical Computer Science