Non-uniform number-conserving elementary cellular automata

Abstract

In this paper, we investigate non-uniform elementary cellular automata (i.e., one-dimensional cellular automata whose cells can use different Wolfram rules to update their states) in the context of number conservation. As a result, we obtain an exhaustive characterization of such number-conserving cellular automata on all finite grids both with periodic and null boundary conditions. The characterization obtained allows, inter alia, to enumerate all number-conserving non-uniform elementary cellular automata, in particular those that are reversible. Surprisingly, the numbers obtained are closely related to the Fibonacci sequence.