Elementary Cellular Automata as Multiplicative Automata

TL;DR

This work addresses extending elementary cellular automata (ECA) into multiplicative automata and complex-valued outputs by mapping Wolfram-code truth tables through permuted $n$-dimensional Galois-field structures with octonion-based pointers. The approach uses Cayley-Dickson and Fano automorphisms to generate multiplication tables and brute-force neighborhood permutations to realize each Wolfram code as a pointer array, yielding a polynomial and a complex vector field output. It demonstrates identity solutions for $5+4n$ factors for codes up to $32$ bits and provides a Java implementation with associated automorphism libraries and validation tests. The results suggest a framework for complex-CA analysis (e.g., Bloch-sphere representations and Fourier-like techniques) and outline future translations to C++ or Python.

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.