Surjectivity and injectivity are the most fundamental problems in cellular automata (CA). We simplify and modify Amoroso's algorithm into optimum and make it compatible with fixed, periodic and reflective boundaries. A new algorithm (injectivity tree algorithm) for injectivity is also proposed. After our theoretic analysis and experiments, our algorithm for injectivity can save much space and 90\% or even more time compared with Amoroso's algorithm for injectivity so that it can support the decision of CA with larger neighborhood sizes. At last, we prove that the reversibility with the periodic boundary and global injectivity of one-dimensional CA is equivalent.

翻译：单射性与满射性是元胞自动机（CA）中最基础的问题。我们对Amoroso算法进行了简化与优化，使其兼容固定边界、周期边界和反射边界。此外，提出了一种用于判定单射性的新算法（单射性树算法）。经理论分析与实验验证，我们的单射性算法相比Amoroso算法可大幅节省存储空间，并节省90%甚至更多的计算时间，从而支持更大邻域半径的元胞自动机判定。最后，我们证明了一维元胞自动机在周期边界下的可逆性与全局单射性是等价的。