Reduction of behavior of additive cellular automata on groups

Details Reduction of behavior of additive cellular automata on groups Reduction of behavior of additive cellular automata on groups

2010-04-25

arxiv.org/abs/1004.4361v1

Nonlinear Sciences

Cellular Automata and Lattice Gases

32 pages, 20 figures

Scientific paper

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.

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