Universality of One-Dimensional Reversible and Number-Conserving Cellular Automata

TL;DR

This paper demonstrates that increasing neighborhood size in one-dimensional reversible and number-conserving cellular automata enables complex, universal behavior, contrasting with trivial dynamics observed in smaller neighborhoods.

Contribution

It constructs a 4-neighbor RNCCA from any 2-neighbor RPCA, establishing universality for RNCCA with larger neighborhoods and states.

Findings

96-state 4-neighbor RNCCA is computationally universal.

2-neighbor RNCCA with 2 states exhibits trivial behavior.

Complex RNCCAs emerge with larger neighborhoods.

Abstract

We study one-dimensional reversible and number-conserving cellular automata (RNCCA) that have both properties of reversibility and number-conservation. In the case of 2-neighbor RNCCA, Garc\'ia-Ramos proved that every RNCCA shows trivial behavior in the sense that all the signals in the RNCCA do not interact each other. However, if we increase the neighborhood size, we can find many complex RNCCAs. Here, we show that for any one-dimensional 2-neighbor reversible partitioned CA (RPCA) with s states, we can construct a 4-neighbor RNCCA with 4s states that simulates the former. Since it is known that there is a computationally universal 24-state 2-neighbor RPCA, we obtain a universal 96-state 4-neighbor RNCCA.