Measuring Behavioural Similarity of Cellular Automata

TL;DR

This paper introduces a high-dimensional vector space model to organize and analyze the behavioral similarities among the vast family of semi-totalistic cellular automata, facilitating discovery and understanding of interesting automata.

Contribution

It proposes a novel continuous vector space representation for cellular automata behaviors, moving beyond traditional class-based categorizations.

Findings

Automata with similar behaviors are close in the vector space.

The space reveals structure and relationships among automata.

It aids in identifying interesting or unique automata.

Abstract

Conway's Game of Life is the best-known cellular automaton. It is a classic model of emergence and self-organization, it is Turing-complete, and it can simulate a universal constructor. The Game of Life belongs to the set of semi-totalistic cellular automata, a family with 262,144 members. Many of these automata may deserve as much attention as the Game of Life, if not more. The challenge we address here is to provide a structure for organizing this large family, to make it easier to find interesting automata, and to understand the relations between automata. Packard and Wolfram (1985) divided the family into four classes, based on the observed behaviours of the rules. Eppstein (2010) proposed an alternative four-class system, based on the forms of the rules. Instead of a class-based organization, we propose a continuous high-dimensional vector space, where each automaton is represented by a point in the space. The distance between two automata in this space corresponds to the differences in their behavioural characteristics. Nearest neighbours in the space have similar behaviours. This space should make it easier for researchers to see the structure of the family of semi-totalistic rules and to find the hidden gems in the family.