Operator representation and logistic extension of elementary cellular automata

TL;DR

This paper introduces an operator-based framework for elementary cellular automata (ECA), revealing symmetries, creating a rule classification table, and extending the rules with a logistic parameter that induces diverse phase transitions.

Contribution

It presents a novel operator representation of ECA, constructs a periodic table of rules, and develops a logistic extension that explores complex dynamical behaviors.

Findings

Mirror and complementary rules are connected via operator transformations.

A periodic table clusters rules with similar behaviors.

Tuning the logistic parameter induces phase transitions in ECA behaviors.

Abstract

We redefine the transition function of elementary cellular automata (ECA) in terms of discrete operators. The operator representation provides a clear hint about the way systems behave both at the local and the global scale. We show that mirror and complementary symmetric rules are connected to each other via simple operator transformations. It is possible to decouple the representation into two pairs of operators which are used to construct a periodic table of ECA that maps all unique rules in such a way that rules having similar behavior are clustered together. Finally, the operator representation is used to implement a generalized logistic extension to ECA. Here a single tuning parameter scales the pace with which operators iterate the rules. We show that, as this parameter is tuned, many rules of ECA undergo multiple phase transitions between periodic, locally chaotic, chaotic and complex (Class 4) behavior.