# A split-and-perturb decomposition of number-conserving cellular automata

## TL;DR

This paper introduces a novel decomposition method for analyzing number-conserving cellular automata across any dimension and state set, simplifying their characterization and enabling the discovery of all such automata in complex cases.

## Contribution

A new, unique split-and-perturb decomposition approach for number-conserving cellular automata applicable in any dimension and state set.

## Key findings

- Decomposition simplifies the structure of number-conserving automata.
- The set of all split functions has a simple structure.
- The perturbation set forms a linear space, facilitating analysis.

## Abstract

This paper concerns $d$-dimensional cellular automata with the von Neumann neighborhood that conserve the sum of the states of all their cells. These automata, called number-conserving or density-conserving cellular automata, are of particular interest to mathematicians, computer scientists and physicists, as they can serve as models of physical phenomena obeying some conservation law. We propose a new approach to study such cellular automata that works in any dimension $d$ and for any set of states $Q$. Essentially, the local rule of a cellular automaton is decomposed into two parts: a split function and a perturbation. This decomposition is unique and, moreover, the set of all possible split functions has a very simple structure, while the set of all perturbations forms a linear space and is therefore very easy to describe in terms of its basis. We show how this approach allows to find all number-conserving cellular automata in many cases of $d$ and $Q$. In particular, we find all three-dimensional number-conserving CAs with three states, which until now was beyond the capabilities of computers.