Surprisingly complex number-conserving binary Moore neighborhood cellular automata

Abstract

Until recently, only few number-conserving binary Moore neighborhood cellular automata had been known. Last year we presented the $(\Omega, \Lambda)$ framework that allows one to generate hundreds of thousands of such rules, with the additional advantage of providing a simple and consistent way to interpret changes in the cellular space as movements of individually identifiable ones that could represent physical particles. Since then, using a recursive algorithm, we managed to generate all number-conserving binary 2-dimensional Moore neighborhood cellular automata. In this paper, we present examples of number-conserving rules that fall outside of the $(\Omega, \Lambda)$ framework and exhibit surprisingly complex behavior.