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

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

Cellular Automata and Discrete Complex Systems 20th International Workshop, AUTOMATA 2014, Himeji, Japan, July 7-9, 2014, Revised Selected Papers

We construct a one-dimensional reversible cellular automaton that is computationally universal in a rather strong sense while being highly non-sensitive to initial conditions as a dynamical system. The cellular automaton has no sensitive subsystems. The construction is based on a simulation of a reversible Turing machine, where a bouncing signal activates the Turing machine to make single steps whenever the signal passes over the machine.</p