This paper studies three classes of cellular automata from a computational point of view: freezing cellular automata where the state of a cell can only decrease according to some order on states, cellular automata where each cell only makes a bounded number of state changes in any orbit, and finally cellular automata where each orbit converges to some fixed point. Many examples studied in the literature fit into these definitions, in particular the works on cristal growth started by S. Ulam in the 60s. The central question addressed here is how the computational power and computational hardness of basic properties is affected by the constraints of convergence, bounded number of change, or local decreasing of states in each cell. By studying various benchmark problems (short-term prediction, long term reachability, limits) and considering various complexity measures and scales (LOGSPACE vs. PTIME, communication complexity, Turing computability and arithmetical hierarchy) we give a rich and nuanced answer: the overall computational complexity of such cellular automata depends on the class considered (among the three above), the dimension , and the precise problem studied. In particular, we show that all settings can achieve universality in the sense of Blondel-Delvenne-Kurka, although short term predictability varies from NLOGSPACE to P-complete. Besides, the computability of limit configurations starting from computable initial configurations separates bounded-change from convergent cellular automata in dimension 1, but also dimension 1 versus higher dimensions for freezing cellular automata. Another surprising dimension-sensitive result obtained is that nilpotency becomes decidable in dimension 1 for all the three classes, while it stays undecidable even for freezing cellular automata in higher dimension.

中文翻译：

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冻结、有界变化和收敛元胞自动机

本文从计算的角度研究了三类元胞自动机：冻结元胞自动机，其中一个元胞的状态只能根据状态的某种顺序减少，元胞自动机，其中每个元胞只在任何轨道上进行有限数量的状态变化，最后是每个轨道收敛到某个固定点的元胞自动机。文献中研究的许多例子都符合这些定义，特别是 S. Ulam 在 60 年代开始的关于水晶生长的工作。这里解决的核心问题是基本属性的计算能力和计算难度如何受到收敛约束、有界变化数量或每个单元中状态的局部减少的影响。通过研究各种基准问题（短期预测、长期可达性、限制）并考虑到各种复杂性度量和尺度（LOGSPACE 与 PTIME、通信复杂性、图灵可计算性和算术层次结构），我们给出了一个丰富而细致入微的答案：这种元胞自动机的整体计算复杂性取决于所考虑的类（在上述三个中)、维数和所研究的精确问题。特别是，我们表明所有设置都可以实现 Blondel-Delvenne-Kurka 意义上的通用性，尽管短期可预测性从 NLOGSPACE 到 P-complete 不等。此外，从可计算初始配置开始的极限配置的可计算性将有界变化与维度 1 中的收敛元胞自动机分开，但也将维度 1 与更高维度的冻结元胞自动机分开。