  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.
