New aspects of symmetry of elementary cellular automata

TL;DR

This paper introduces a new classification method for elementary cellular automata based on the symmetry of their state transition networks, revealing equivalences among rules and providing insights into their structural properties.

Contribution

It proposes a novel classification based on state network symmetry, identifying rule equivalences and extending the understanding of automaton structures beyond known groupings.

Findings

Most rules align with the known 88-group classification.

Certain rules are shown to be equivalent through the new classification.

The structure of state networks offers additional insights into automaton behavior.

Abstract

We present a new classification of elementary cellular automata. It is based on the structure of the network of states, connected with the transitions between them; the latter are determined by the automaton rule. Recently an algorithm has been proposed to compress the network of states (M. J. Krawczyk, Physica A 390 (2011) 2181). In this algorithm, states are grouped into classes, according to the local symmetry of the network. In the new classification, an automaton is described by the number of classes #(N) as dependent on the system size N. In most cases, the results reflect the known classification into 88 groups. However, the function #(N) also appears to be the same for some rules which have not been grouped together yet. In this way, the automaton 23 is equivalent to 232, 77 to 178, 105 to 150, the pair (43, 113) to the pair (142, 212) and the group (12, 68, 207, 221) to the group (34, 48, 187, 243). Furthermore, automata 51, 204, the pair (15, 85) and the pair (170,240) are all mutually equivalent. Results are also presented on the structure of networks of states.