The property of reversibility is quite meaningful for the classic theoretical computer science model, cellular automata. For the reversibility problem for a CA under null boundary conditions, while linear rules have been studied a lot, the non-linear rules remain unexplored at present. The paper investigates the reversibility problem of general one-dimensional CA on a finite field $\mathbb{Z}_p$, and proposes an approach to optimize the Amoroso's infinite CA surjectivity detection algorithm. This paper proposes algorithms for deciding the reversibility of one-dimensional CA under null boundary conditions. We propose a method to decide the strict reversibility of one-dimensional CA under null boundary conditions. We also provide a bucket chain based algorithm for calculating the reversibility function of one-dimensional CA under null boundary conditions. These decision algorithms work for not only linear rules but also non-linear rules. In addition, it has been confirmed that the reversibility function always has a period, and its periodicity is related to the periodicity of the corresponding bucket chain. Some of our experiment results of reversible CA are presented in the paper, complementing and validating the theoretical aspects, and thereby further supporting the research conclusions of this paper.

翻译：可逆性是经典理论计算机科学模型元胞自动机的一个重要性质。对于零边界条件下元胞自动机的可逆性问题，线性规则已得到大量研究，但非线性规则目前仍未被探索。本文研究有限域 $\mathbb{Z}_p$ 上一维一般元胞自动机的可逆性问题，并提出一种优化Amoroso无穷元胞自动机满射性检测算法的方法。本文提出了在零边界条件下判定一维元胞自动机可逆性的算法。我们提出了一种在零边界条件下判定一维元胞自动机严格可逆性的方法，还提出了一种基于桶链的算法来计算零边界条件下二维元胞自动机的可逆性函数。这些判定算法不仅适用于线性规则，也适用于非线性规则。此外，我们证实可逆性函数始终具有周期性，且其周期性与对应桶链的周期性相关。文中展示了一些可逆元胞自动机的实验结果，这些结果补充并验证了理论部分，从而进一步支持了本文的研究结论。