In this paper, we completely solve the reversibility of one-dimensional finite cellular automata (FCA). This means that we will have an efficient method to determine the reversibility of any FCA with all numbers (n) of cells. The complexity of this algorithm is independent of n. We perform calculations on two new kinds of graphs and discover that the reversibility of any FCA exhibits periodicity as n increases. We successfully provide a method to compute the reversibility sequence that encompasses the reversibility of FCA with any number of cells. Additionally, the calculations in this paper are applicable to FCA with various types of boundaries.

翻译：本文完整解决了一维有限元胞自动机（FCA）的可逆性问题。这意味着我们将提出一种高效方法，能够判定任意包含所有细胞数量n的FCA的可逆性，且该算法复杂度与n无关。通过构建两类新型图结构进行计算，我们发现任意FCA的可逆性随n增大呈现周期性规律。我们成功提出一种可逆序列计算方法，该序列涵盖任意细胞数量下FCA的可逆性。此外，本文计算方法适用于各类边界条件的FCA。