  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.
