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Effective Complexity of Stationary Process Realizations
Max Planck Institute for Mathematics in the Sciences, Inselstr. 22, Leipzig 04103, Germany
Institute of Mathematics 7-2, Technical University Berlin, Straße des 17. Juni 136, Berlin 10623, Germany
Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA
* Author to whom correspondence should be addressed.
Received: 8 May 2011; in revised form: 15 June 2011 / Accepted: 17 June 2011 / Published: 22 June 2011
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, which is the probability distributions on finite binary strings. In our previous paper  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 needs less parameters in its definition. A similar modification of the related concept of sophistication has been suggested by Antunes and Fortnow.
Keywords: effective complexity; Kolmogorov complexity; Shannon entropy; computable stationary processes; coarse effective complexity
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Ay, N.; Müller, M.; Szkoła, A. Effective Complexity of Stationary Process Realizations. Entropy 2011, 13, 1200-1211.
Ay N, Müller M, Szkoła A. Effective Complexity of Stationary Process Realizations. Entropy. 2011; 13(6):1200-1211.
Ay, Nihat; Müller, Markus; Szkoła, Arleta. 2011. "Effective Complexity of Stationary Process Realizations." Entropy 13, no. 6: 1200-1211.