Mathematical Physics of Cellular Automata

TL;DR

This paper derives a universal mathematical map for deterministic 1D cellular automata, classifies rules based on symmetries, and links complexity to symmetry breaking, providing insights into Wolfram's CA classification.

Contribution

It introduces a universal map for CA, extends it to multiple dimensions, and connects symmetry breaking to complexity in CA dynamics.

Findings

Universal map for 1D CA derived

Classification of CA rules via symmetry invariances

Complexity linked to symmetry breaking of modular addition

Abstract

A universal map is derived for all deterministic 1D cellular automata (CA) containing no freely adjustable parameters. The map can be extended to an arbitrary number of dimensions and topologies and its invariances allow to classify all CA rules into equivalence classes. Complexity in 1D systems is then shown to emerge from the weak symmetry breaking of the addition modulo an integer number p. The latter symmetry is possessed by certain rules that produce Pascal simplices in their time evolution. These results elucidate Wolfram's classification of CA dynamics.