Boundary growth in one-dimensional cellular automata

TL;DR

This paper investigates boundary growth in one-dimensional two-color cellular automata, identifying reducible and irreducible cases, and revealing complex behaviors including non-existent growth exponents.

Contribution

It provides a systematic analysis of boundary growth, characterizes reducible boundaries with morphic words, and introduces empirical methods for studying irreducible boundaries.

Findings

Exact growth rates for reducible boundaries

Morphic words characterize reducible boundaries

Existence of boundaries with non-existent growth exponents

Abstract

We systematically study the boundaries of one-dimensional, 2-color cellular automata depending on 4 cells, begun from simple initial conditions. We determine the exact growth rates of the boundaries that appear to be reducible. Morphic words characterize the reducible boundaries. For boundaries that appear to be irreducible, we apply curve-fitting techniques to compute an empirical growth exponent and (in the case of linear growth) a growth rate. We find that the random walk statistics of irreducible boundaries exhibit surprising regularities and suggest that a threshold separates two classes. Finally, we construct a cellular automaton whose growth exponent does not exist, showing that a strict classification by exponent is not possible.