Elementary Cellular Automata as Multiplicative Automata

TL;DR

This paper introduces a novel method to extend elementary cellular automata into the complex number domain using Galois fields and octonion multiplication, enabling new analytical and computational capabilities.

Contribution

It presents a new approach to convert ECA into multiplicative automata using algebraic structures, expanding their mathematical and computational framework.

Findings

Identity solutions are identified within the new automata model.

A polynomial representation of the automata is derived.

Implementation in Java demonstrates practical applicability.

Abstract

Elementary cellular automata (ECA) are converted into multiplicative versions by using permuted n-dim Galois fields and octonion multiplication tables as binary pointers to each rule's Wolfram code truth table. This enables an extension of the binary ECA to complex numbers, identity solutions are found, produces a polynomial, and is implemented in Java.