初等元胞自动机的对角基与对角周期
Diagonal Bases and Diagonal Periods of Elementary Cellular Automata
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中文总结 AI 辅助
本文研究初等元胞自动机对角族在有限窗口中的基性质，通过多项式提升和三角性分析，确定24条通用二元基规则，并揭示Pascal变换与规则30的周期特性。
中文摘要 AI 辅助
哪些元胞自动机对角族在每个有限窗口中都能构成基？对于初等规则的规范多项式提升，两个真值表位决定三角性，矩阵对角线上的单位决定可逆性。恰好有24条规则给出通用二元基；它们在所有模数下保持通用性。在三角二元坐标映射中，Pascal变换通过将OR卷积转换为逐点乘法而唯一表征，而增量则变为严格前缀求和。显式逆与坐标比较区分了稀疏性与求值成本。规则30的多项式构造给出了插值阶和素数模周期的斐波那契界。精确的加法周期锚定了模二和模三的有限普查。这些结果将全窗口基分类与表示的优化以及有限窗口中观察到的周期模式区分开来。
英文摘要
Which cellular-automaton diagonal families form bases in every finite window? For canonical polynomial lifts of elementary rules, two truth-table bits determine triangularity, and units on the matrix diagonal determine invertibility. Exactly 24 rules give universal binary bases; all remain universal over every modulus. Among triangular binary coordinate maps, the Pascal transform is uniquely characterized by converting OR convolution into pointwise multiplication, while increment becomes strict prefix summation. Explicit inverses and coordinate comparisons distinguish sparsity from evaluation cost. A Rule 30 polynomial construction gives Fibonacci bounds on interpolation order and prime-modulus periods. Exact additive periods anchor a finite census modulo two and three. These results separate all-window basis classification from optimization of a representation and from period patterns observed in finite windows.
