Nonlinear cellular automata are extensively used in simulations, image processing, cryptography, and so on. The determination of their fundamental properties, injectivity and surjectivity, related to information loss during the evolution, is necessary in various applications. Currently, people still use Amoroso's algorithms for injectivity and surjectivity determinations, but this incurs significant computational costs when applied to complex nonlinear cellular automata. We have optimized Amoroso's surjectivity algorithm, improving its operational efficiency greatly and extended its applicability to various boundaries. Furthermore, we have introduced new theorems and algorithms for determining injectivity, which offer substantial improvements over Amoroso's algorithm in both time and space. With these new algorithms, we are equipped to determine the properties of larger and more complex cellular automata, thereby employing more advanced cellular automata to achieve increasingly complex functionalities.

翻译：非线性元胞自动机广泛应用于模拟、图像处理、密码学等领域。确定其与演化过程中信息损失相关的基本性质——单射性与满射性——在多种应用中至关重要。目前，学界仍采用Amoroso算法进行单射性与满射性判定，但将其应用于复杂非线性元胞自动机时会产生巨大的计算开销。我们优化了Amoroso的满射性判定算法，大幅提升了其运算效率，并将其适用范围扩展至多种边界条件。此外，我们提出了新的单射性判定定理与算法，在时间与空间效率上均较Amoroso算法有显著提升。借助这些新算法，我们能够判定更大规模、更复杂结构的元胞自动机性质，从而运用更先进的元胞自动机实现日益复杂的功能。