The complexity of stochastic sequences

We review and slightly strengthen known results on the Kolmogorov complexity of prefixes of effectively random sequences. First, there are recursively random random sequences such that for any computable, nondecreasing and unbounded function f and for almost all n, the uniform complexity of the length n prefix of the sequence is bounded by f(n). Second, a similar result with bounds of the form f(n) log n holds for partial-recursively random sequences. Furthermore, we demonstrate that there is no Mises-Wald-Church stochastic sequence such that all nonempty prefixes of the sequence have Kolmogorov complexity O(log n). This implies a sharp bound for the complexity of the prefixes of MisesWald-Church stochastic and of partial-recursively random sequences. As an immediate corollary to these results, we obtain the known separation of the classes of recursively random and of Mises-Wald-Church stochastic sequences.

• Type: Conference
• Year: 2003
• Where: COCO
• Authors: Wolfgang Merkle
• Keywords: COCO 2003, Kolmogorov Complexity, Mises-wald-church Stochastic Sequences, Random Sequences