Irreducible Rules and Equivalence Classes of One-dimensional Cellular Automata

TL;DR

This paper analyzes the symmetry-induced equivalence classes of one-dimensional cellular automata, introducing the concept of irreducible rules and examining how various transformations affect these classes.

Contribution

It introduces the concept of irreducible local rules and studies how symmetry transformations partition cellular automata into equivalence classes, including effects of neighborhood scaling.

Findings

Number of equivalence classes of irreducible binary local rules determined

Impact of symmetry transformations on class partitioning analyzed

Scaling of neighborhoods alters the number of equivalence classes

Abstract

One-dimensional cellular automata are discrete dynamical systems that operate on an infinite lattice of sites and are characterized by the locality and uniformity of their update rule. Permutations of the state set and isometric transformations of the lattice induce symmetry transformations on the set of local rules and the set of global maps of cellular automata, resulting in a partitioning of the set of cellular automata into equivalence classes. The concept of an irreducible local rule that depends on all its coordinates is used to analyse the equivalence classes and results on the number of equivalence classes of irreducible binary local rules and binary global maps are presented. Finally, another symmetry operator based on the scaling of neighbourhoods is introduced and the change in the number of equivalence classes is analysed.