Lempel-Ziv complexity analysis of one dimensional cellular automata

Abstract

Cellular automata (CA) have long attracted attention as dynamical systems with local updating rules and yet can exhibit, for certain rules, complex, long space and time correlated patterns. This contrast with other rules which results in trivial patterns being homogeneous or periodic. In this article we approach CA from two related angles: we analyze the information transfer in the time evolution of CA driven sequences and; we revisit the sensibility of the initial configuration on sequence evolution. In order to do so, we borrow a recently reported information distance based on Kolmogorov algorithmic complexity. The normalized information distance has been used previously to find a hierarchical clustering of CA rules. What is different in our approach, is the temporal analysis of the sequence evolutions by correlating different calculated distances with entropy density. Entropy rate, is a length invariant measure of the amount of irreducible randomness in a dynamical process. In order to perform our analysis, we incorporate to the practical calculation of the entropy rate and the distance measure, the use of Lempel-Ziv complexity. Lempel-Ziv complexity carries a number of practical advantages while avoiding the uncomputable nature of Kolmogorov randomness. The reduction of entropy density during time evolution can be related to energy dissipation through Landauer principle. Related to the last fact, is the computational capabilities of CA as information processing rules, were the performed analysis could be used to select CA rules amiable for simulating different physical process. The tools developed in this article for the analysis of the CA are easily extendible to the study of other one dimensional dynamical systems.