Effective complexity of stationary process realizations

revised version: November 2011
Nihat Ay, Markus Mueller, and Arleta Szkola
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Submission date: 11. Jan. 2010
Pages: 16
published in: Entropy, 13 (2011) 6, p. 1200-1211 
DOI number (of the published article): 10.3390/e13061200
MSC-Numbers: 60, 94
Keywords and phrases: Effective complexity, Kolmogorov complexity, stochastic processes
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Abstract:
The concept of effective complexity of an object as the minimal description length of its regularities has been initiated by Gell-Mann and Lloyd. The regularities are modeled by means of ensembles, that is probability distributions on finite binary strings. In our previous paper [1] we propose a definition of effective complexity in precise terms of algorithmic information theory. Here we investigate the effective complexity of binary strings generated by stationary, in general not computable, processes. We show that under not too strong conditions long typical process realizations are effectively simple. Our results become most transparent in the context of coarse effective complexity which is a modification of the original notion of effective complexity that uses less parameters in its definition. A similar modification of the related concept of sophistication has been suggested by Antunes and Fortnow.