Reduction of behavior of additive cellular automata on groups

TL;DR

This paper investigates additive cellular automata on various groups, revealing simplified behaviors on abelian groups and complex, diverse behaviors on non-commutative groups, including self-similar structures and glider-like phenomena.

Contribution

It introduces new results on ACA behaviors on infinite abelian and non-commutative groups, including classes like DHC, and constructs automata with glider gun-like dynamics.

Findings

States on abelian groups simplify over certain time sequences.

Behavioral patterns include gliders and self-similar structures.

Non-commutative groups exhibit more diverse automaton behaviors.

Abstract

A class of additive cellular automata (ACA) on a finite group is defined by an index-group $\m g$ and a finite field $\m F_p$ for a prime modulus $p$ \cite{Bul_arch_1}. This paper deals mainly with ACA on infinite commutative groups and direct products of them with some non commutative $p$-groups. It appears that for all abelian groups, the rules and initial states with finite supports define behaviors which being restricted to some infinite regular series of time moments become significantly simplified. In particular, for free abelian groups with $n$ generators states $V^{[t]}$ of ACA with a rule $R$ at time moments $t=p^k,k>k_0,$ can be viewed as $||R||$ copies of initial state $V^{[0]}$ moving through an $n$-dimensional Euclidean space. That is the behavior is similar to gliders from J.Conway's automaton {\sl Life}. For some other special infinite series of time moments the automata states approximate self-similar structures and the approximation becomes better with time. An infinite class $\mathrm{DHC}(\mbf S,\theta)$ of non-commutative $p$-groups is described which in particular includes quaternion and dihedral $p$-groups. It is shown that the simplification of behaviors takes place as well for direct products of non-commutative groups from the class $\mathrm{DHC}(\mbf S,\theta)$ with commutative groups. Finally, an automaton on a non-commutative group is constructed such that its behavior at time moments $2^k,k\ge2,$ is similar to a glider gun. It is concluded that ACA on non-commutative groups demonstrate more diverse variety of behaviors comparing to ACA on commutative groups.