On Number Conservation of Non-uniform Cellular Automata

TL;DR

This paper investigates the number conservation property in 1D non-uniform cellular automata, providing characterization, decision algorithms, and synthesis methods to identify and construct such automata.

Contribution

It introduces a reachability tree-based characterization and decision algorithm for number conservation in non-uniform CAs, along with a synthesis scheme for constructing them.

Findings

A reachability tree effectively characterizes number conservation.

A decision algorithm determines if a non-uniform CA is number conserving.

A synthesis scheme generates number conserving non-uniform CAs.

Abstract

This paper studies the number conservation property of 1-dimensional non-uniform cellular automata (CAs). In a non-uniform cellular automaton (CA), different cells may follow different rules. The present work considers that the cells follow Wolfram's CAs rules. A characterization tool, named Reachability tree is used to discover the number conservation property of non-uniform CAs. Then a decision algorithm is reported to conclude whether a given non-uniform CA with $n$ cells is number conserving or not. Finally, a synthesis scheme is developed to get an $n$-cell number conserving non-uniform CA.