arXiv 1004.43612010-04-25nlin.CGcs.NE

群上加法元胞自动机行为的约化

Reduction of behavior of additive cellular automata on groups

Valeriy Bulitko

备注：: 32 pages, 20 figures

摘要：

定义了一类有限群上的加法元胞自动机，它是由一个指标群$\m g$和一个有限域$\m F_p$定义的，其中$p$是素数模\cite{Bul_arch_1} .本文主要研究无限交换群及其与某些非交换$p$ -群的直积上的蚁群算法。看来，对于所有的阿贝尔群，规则和初始状态与有限的支持定义的行为被限制到一些无限的正则序列的时刻变得显着简化。特别地，对于具有$n$生成元的自由阿贝尔群，ACA的状态$V^{[t]}$在时刻$t=p^k,k>k_0,$具有规则$R$，可以被视为初始状态$V^{[0]}$的$||R||$副本移动通过$n$维欧几里得空间。也就是说，它的行为类似于J.Conway的自动机{ \sl Life}中的滑翔机。对于其他一些特殊的无穷时间矩序列，该自动机状态近似自相似结构，并且这种近似随着时间的推移而变得更好。本文描述了一类非交换$p$ -群的无限类$\mathrm{DHC}(\mbf S,θ)$，它特别包括四元数和二面体$p$ -群.它表明，简化的行为发生，以及从类$\mathrm{DHC}(\mbf S,θ)$与交换群的非交换群的直积。最后，构造了一个非交换群上的自动机，使得它在时刻$2^k,k\ge2,$的行为类似于滑翔枪。结果表明，与交换群上的蚁群算法相比，非交换群上的蚁群算法表现出更多样的行为。

英文摘要：

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 $\mathrm{DHC}(\mbf S,θ)$ 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 $\mathrm{DHC}(\mbf S,θ)$ 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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英文摘要：

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