保数元胞自动机的分裂扰动分解
A split-and-perturb decomposition of number-conserving cellular automata
摘要：
本文研究了$d$维元胞自动机，其vonNeumann邻域保持所有细胞状态之和。这些自动机，称为数量守恒或密度守恒的元胞自动机，是数学家，计算机科学家和物理学家特别感兴趣的，因为它们可以作为遵守某些守恒定律的物理现象的模型。我们提出了一种新的方法来研究这样的元胞自动机，工作在任何维度$d$和任何一组状态$Q$。本质上，元胞自动机的局部规则被分解为两部分：分裂函数和扰动。这种分解是唯一的，而且，所有可能的分裂函数的集合具有非常简单的结构，而所有扰动的集合形成线性空间，因此很容易用它的基来描述。我们展示了这种方法如何允许找到所有的数量守恒的元胞自动机在许多情况下的$d$和$Q$。特别是，我们发现所有的三维数守恒CA与三个国家，这到目前为止是超出了计算机的能力。
英文摘要：
This paper concerns $d$-dimensional cellular automata with the von Neumann neighborhood that conserve the sum of the states of all their cells. These automata, called number-conserving or density-conserving cellular automata, are of particular interest to mathematicians, computer scientists and physicists, as they can serve as models of physical phenomena obeying some conservation law. We propose a new approach to study such cellular automata that works in any dimension $d$ and for any set of states $Q$. Essentially, the local rule of a cellular automaton is decomposed into two parts: a split function and a perturbation. This decomposition is unique and, moreover, the set of all possible split functions has a very simple structure, while the set of all perturbations forms a linear space and is therefore very easy to describe in terms of its basis. We show how this approach allows to find all number-conserving cellular automata in many cases of $d$ and $Q$. In particular, we find all three-dimensional number-conserving CAs with three states, which until now was beyond the capabilities of computers.
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