The firing squad synchronization problem (FSSP, for short) on cellular automata has been studied extensively for more than fifty years, and a rich variety of FSSP algorithms has been proposed. Here we study the classical FSSP on a model of fault-tolerant cellular automata that might have possibly some defective cells and present the first state-efficient implementations of fault-tolerant FSSP algorithms for one-dimensional (1D) and two-dimensional (2D) cellular arrays. It is shown that, under some constraints on the length and distribution of defective cells, any 1D cellular array of length n with p defective cell segments can be synchronized in \(2n-2+p\) steps and the algorithm is realized on a 1D cellular automaton of length \(n, 2 \le n \le 50\), having 164 states and 4792 transition rules. In addition, we give by far a smaller-state implementation of a 2D FSSP algorithm that can synchronize any 2D rectangular array of size \(m \times n\), possibly including at most O(mn) isolated defective zones, exactly in \(2(m+n)-4\) steps on a cellular automaton with only 6 states and 935 transition rules.

中文翻译：

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容错FSSP算法的状态有效实现

五十多年来，关于元胞自动机的发射小队同步问题（简称FSSP）已经得到了广泛的研究，并且提出了各种各样的FSSP算法。在这里，我们在容错细胞自动机模型上研究经典的FSSP，该模型可能有一些缺陷单元，并介绍了针对一维（1D）和二维（2D）的容错FSSP算法的第一个状态有效实现。细胞阵列。结果表明，在对缺陷单元的长度和分布有一定约束的情况下，任何具有p个缺陷单元段的长度为n的一维单元阵列都可以以\（2n-2 + p \）步进行同步，并且该算法是在a一维元胞自动机的长度\（n，2 \ le n \ le 50 \），具有164个状态和4792个转换规则。另外，到目前为止，我们给出了2D FSSP算法的较小状态实现，该算法可以同步大小为\ {m \ times n \}的任何2D矩形数组，可能最多包含O（mn）个隔离的缺陷区域，恰好在\ （2（m + n）-4 \）步入只有6个状态和935个过渡规则的元胞自动机。