Effective complexity of stationary process realizations

TL;DR

This paper investigates the effective complexity of binary strings generated by stationary processes, showing that typical long realizations are effectively simple, especially when using a modified, coarse version of effective complexity.

Contribution

It extends the concept of effective complexity to stationary process realizations and introduces a coarse version that simplifies the analysis.

Findings

Long typical process realizations are effectively simple.

Coarse effective complexity provides clearer insights.

Results apply to non-computable stationary processes.

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 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.