Digital calculus: Cellular automata dynamics in closed form

TL;DR

This paper derives a closed-form mathematical expression for the universal map of cellular automata using a digit function, enabling deeper theoretical understanding and efficient implementation of cellular automaton rules.

Contribution

It introduces a novel closed-form expression for cellular automata dynamics and establishes foundational properties of the digit function, facilitating analysis and simulation.

Findings

Derived a closed-form universal map for cellular automata

Established properties of the digit function used in the expression

Provided a fast algorithm for cellular automaton rule implementation

Abstract

A simple mathematical expression for the universal map for cellular automata is found in closed form with the help of a digit function, whose most basic properties are established. This result is found after proving a theorem on the composition of functions on finite sets. The expression (and the technique used to obtain it) opens the possibility of gaining mathematical insight in any cellular automaton rule since it constitutes at the same time a simple and fast algorithm to implement any such rule.