A Full Computation-relevant Topological Dynamics Classification of Elementary Cellular Automata

TL;DR

This paper provides a comprehensive classification of elementary cellular automata based on their dynamical behaviors, highlighting that complex ECAs are sensitive but not chaotic, and proposing a link to computational universality.

Contribution

It offers the first complete dynamical classification of elementary cellular automata using fundamental system notions, connecting complexity to the edge of chaos.

Findings

Complex ECAs are sensitive but not chaotic.

Complex ECAs are not eventually weakly periodic.

Complex ECAs are conjectured to be at the edge of chaos.

Abstract

Cellular automata are both computational and dynamical systems. We give a complete classification of the dynamic behaviour of elementary cellular automata (ECA) in terms of fundamental dynamic system notions such as sensitivity and chaoticity. The "complex" ECA emerge to be sensitive, but not chaotic and not eventually weakly periodic. Based on this classification, we conjecture that elementary cellular automata capable of carrying out complex computations, such as needed for Turing-universality, are at the "edge of chaos".