Cellular automata are synchronous discrete dynamical systems used to describe complex dynamic behaviors. The dynamic is based on local interactions between the components, these are defined by a finite graph with an initial node coloring with two colors. In each step, all nodes change their current color synchronously to the least/most frequent color in their neighborhood and in case of a tie, keep their current color. After a finite number of rounds these systems either reach a fixed point or enter a 2-cycle. The problem of counting the number of fixed points for cellular automata is #P-complete. In this paper we consider cellular automata defined by a tree. We propose an algorithm with run-time $O(n\Delta)$ to count the number of fixed points, here $\Delta$ is the maximal degree of the tree. We also prove upper and lower bounds for the number of fixed points. Furthermore, we obtain corresponding results for pure cycles, i.e., instances where each node changes its color in every round. We provide examples demonstrating that the bounds are sharp. The results are proved for the minority and the majority model.

翻译：细胞自动机是一种同步离散动力学系统，用于描述复杂动态行为。其动力学基于组件间的局部相互作用，这些组件由有限图定义，初始节点使用两种颜色染色。在每一步中，所有节点同时将其当前颜色更新为其邻域中最少/最常见的颜色，若出现平局则保持当前颜色。经过有限轮次后，这些系统要么达到不动点，要么进入2-循环。细胞自动机不动点数量的计数问题属于#P-完全问题。本文考虑由树定义的细胞自动机，提出一个运行时间为$O(n\Delta)$的算法来计算不动点数量，其中$\Delta$为树的最大度数。同时，我们证明了不动点数量的上下界。此外，针对纯循环（即每个节点每轮均改变颜色的情形）得到相应结论。通过实例表明这些边界是紧的。上述结果针对少数模型和多数模型均成立。