On the order of lazy cellular automata
Edgar Alcalá-Arroyo, Alonso Castillo-Ramirez
TL;DR
This work introduces lazy cellular automata defined over a group universe and an alphabet, focusing on the order ord(tau) of such automata. It proves that finite order forces a period-1 dynamics and derives a general upper bound for ord(tau) in terms of the unique active transition p and writing symbol a, with idempotence guaranteed under certain conditions. The authors then analyze the case where p is quasi-constant, obtaining a precise classification of ord(tau) depending on whether the non-constant element r in p is equal to the group identity and on the relation between the writing symbol a and p. They show that all orders n≥2 occur (via appropriate constructions), and pose open problems on idempotency characterization and the algebraic structure of lazy versus invertible cellular automata within the full CA monoid.
Abstract
We study the most elementary family of cellular automata defined over an arbitrary group universe $G$ and an alphabet $A$: the lazy cellular automata, which act as the identity on configurations in $A^G$, except when they read a unique active transition $p \in A^S$, in which case they write a fixed symbol $a \in A$. As expected, the dynamical behavior of lazy cellular automata is relatively simple, yet subtle questions arise since they completely depend on the choice of $p$ and $a$. In this paper, we investigate the order of a lazy cellular automaton $τ: A^G \to A^G$, defined as the cardinality of the set $\{ τ^k : k \in \mathbb{N} \}$. In particular, we establish a general upper bound for the order of $τ$ in terms of $p$ and $a$, and we prove that this bound is attained when $p$ is a quasi-constant pattern.