懒惰元胞自动机的幂等性准则
Idempotence criteria for lazy cellular automata
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中文总结 AI 辅助
该研究提出懒惰元胞自动机非幂等性的相容性准则，证明其在阿贝尔群上等价于非幂等性，并构造反例说明不可省略可交换性假设。
中文摘要 AI 辅助
懒惰元胞自动机τ:A^G→A^G由包含群单位元的有限邻域S⊆G、模式p∈A^S和写入符号a∈A\{p(e)}确定：当邻域模式等于p时，其局部规则将中心符号改为a。对于p取值为a的非空位置集T，我们引入p沿右平移St（t∈T）的两个相容性条件。我们证明，相容集的存在性是τ在任意群上非幂等的必要条件；当p取值为a的每个位置与p取值为p(e)的每个位置都可交换时，该条件也是充分的。特别地，相容性刻画了阿贝尔群上的非幂等性，将对SS={st:s,t∈S}上模式的搜索替换为对p取值为a的位置集的搜索。最后，对每个n≥4的二面体群D_n，我们构造了一个承认相容集的幂等懒惰元胞自动机，表明不能简单省略可交换性假设。
英文摘要
A lazy cellular automaton $τ:A^G\to A^G$ is determined by a finite neighborhood $S\subseteq G$ containing the group identity, a pattern $p\in A^S$, and a writing symbol $a\in A\setminus\{p(e)\}$: its local rule changes the central symbol to $a$ precisely when the neighborhood pattern equals $p$. For a nonempty set $T$ of positions where $p$ takes the value $a$, we introduce two compatibility conditions on $p$ along the right translates $St$, $t\in T$. We show that the existence of a compatible set is necessary for $τ$ to be non-idempotent over every group, and that it is also sufficient whenever every position where $p$ takes the value $a$ commutes with every position where $p$ takes the value $p(e)$. In particular, compatibility characterizes non-idempotence over abelian groups, replacing a search over patterns on $SS=\{st:s,t\in S\}$ by a search over sets of positions where $p$ takes the value $a$. Finally, for every dihedral group $D_n$ with $n\geq 4$, we construct an idempotent lazy cellular automaton that admits a compatible set, showing that the commutation hypothesis cannot simply be dropped.
发表机构
- Universidad de Guadalajara, Centro Universitario de Ciencias Exactas e Ingenierías（瓜达拉哈拉大学，精确科学与工程学院）
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