On Number Conservation of Non-uniform Cellular Automata

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.