We discovered that certain patterns called injective patterns remain stable during the revolution process, allowing us to create many reversible CA simply by using them to design the revolution rules. By examining injective patterns, we investigated their structural stability during revolutions. This led us to discover extended patterns and pattern mixtures that can create more reversible cellular automata. Furthermore, our research proposed a new way to study the reversibility of CA by observing the structure of local rule $f$. In this paper, we will explicate our study and propose an efficient method for finding the injective patterns. Our algorithms can find injective rules and generate local rule $f$ by traversing $2^{N}$, instead of $2^{2^{N}}$ to check all injective rules and pick the injective ones.

翻译：我们发现，某些被称为“单射模式”的模式在演化过程中保持稳定，这使我们能够直接利用它们设计演化规则，从而构造大量可逆元胞自动机。通过研究单射模式，我们探讨了它们在演化过程中的结构稳定性，进而发现了可构造更多可逆元胞自动机的扩展模式与模式混合体。此外，我们的研究提出了一种通过观察局部规则$f$的结构来研究元胞自动机可逆性的新方法。本文将详细阐述我们的研究，并提出一种高效寻找单射模式的算法。该算法通过遍历$2^{N}$种模式（而非$2^{2^{N}}$种）来发现单射规则并生成局部规则$f$，从而避免了穷举所有单射规则并筛选的步骤。