Cellular automata with memory (CAM) are widely used in fields such as image processing, pattern recognition, simulation, and cryptography. The invertibility of CAM is generally considered to be chaotic. Paper [Invertible behavior in elementary cellular automata with memory, Juan C. Seck-Tuoh-Mora et al., Information Sciences, 2012] presented necessary and sufficient conditions for the invertibility of elementary CAM, but it contains a critical error: it classifies identity CAM as non-invertible, whereas identity CAM is undoubtedly invertible. By integrating Amoroso's algorithm and cycle graphs, we provide the correct necessary and sufficient conditions for the invertibility of one-dimensional CAM. Additionally, we link CAM to a specific type of cellular automaton that is isomorphic to CAM, behaves identically, and has easily determinable invertibility. This makes it a promising alternative tool for CAM applications.

翻译：带记忆元胞自动机（CAM）广泛应用于图像处理、模式识别、仿真和密码学等领域。CAM的可逆性通常被认为是混沌的。论文《Invertible behavior in elementary cellular automata with memory》（Juan C. Seck-Tuoh-Mora 等人，Information Sciences，2012）提出了基本CAM可逆性的充要条件，但其中存在一个关键错误：该文将恒等CAM归类为不可逆，而恒等CAM无疑是可逆的。通过整合Amoroso算法与循环图，我们为一维CAM的可逆性提供了正确的充要条件。此外，我们将CAM与一类特定的元胞自动机联系起来，这类自动机与CAM同构、行为一致，且其可逆性易于判定。这使其成为CAM应用中一种有前景的替代工具。