The Invertibility of Cellular Automata with Menory: Correcting Errors and New Conclusions

TL;DR

This paper corrects a critical error in the criteria for invertibility of cellular automata with memory, providing accurate conditions and linking them to a related automaton type for easier invertibility determination.

Contribution

It offers the correct necessary and sufficient conditions for invertibility of one-dimensional CAM and connects CAM to an isomorphic automaton for practical analysis.

Findings

Corrected the invertibility criteria for CAM, including identity cases.

Linked CAM to an isomorphic automaton with identical behavior.

Provided a practical method for determining CAM invertibility.

Abstract

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.