Computational Hierarchy of Elementary Cellular Automata

TL;DR

This paper introduces a framework for analyzing elementary cellular automata based on their ability to emulate each other, leading to insights into their complexity, chaos, and potential for Turing-completeness.

Contribution

It proposes a novel emulation-based approach to classify cellular automata and defines a new notion of chaos for discrete computational systems.

Findings

Chaotic automata cannot emulate any other automata non-trivially.

A graph of emulation relationships among elementary cellular automata was constructed.

The approach can inform the design of efficient, Turing-complete parallel computational systems.

Abstract

The complexity of cellular automata is traditionally measured by their computational capacity. However, it is difficult to choose a challenging set of computational tasks suitable for the parallel nature of such systems. We study the ability of automata to emulate one another, and we use this notion to define such a set of naturally emerging tasks. We present the results for elementary cellular automata, although the core ideas can be extended to other computational systems. We compute a graph showing which elementary cellular automata can be emulated by which and show that certain chaotic automata are the only ones that cannot emulate any automata non-trivially. Finally, we use the emulation notion to suggest a novel definition of chaos that we believe is suitable for discrete computational systems. We believe our work can help design parallel computational systems that are Turing-complete and also computationally efficient.