Cellular Automata with Symmetric Local Rules

TL;DR

This paper explores cellular automata with symmetric local rules, showing their relation to the generalized Game of Life, counting rule classes, and analyzing rule compatibility on symmetric lattices.

Contribution

It characterizes symmetric local rules in cellular automata, relates them to the Game of Life, and provides methods for counting and generating rule classes efficiently.

Findings

Symmetric local rules coincide with the generalized Game of Life.

The number of rule equivalence classes is counted and can be generated efficiently.

Compatibility analysis of Life family rules on symmetric lattices is presented.

Abstract

The cellular automata with local permutation invariance are considered. We show that in the two-state case the set of such automata coincides with the generalized Game of Life family. We count the number of equivalence classes of the rules under consideration with respect to permutations of states. This reduced number of rules can be efficiently generated in many practical cases by our C program. Since a cellular automaton is a combination of a local rule and a lattice, we consider also maximally symmetric two-dimensional lattices. In addition, we present the results of compatibility analysis of several rules from the Life family.