Two Graphs: Resolving the Periodic Reversibility of One-dimensional Finite Cellular Automata

Abstract: Finite cellular automata (FCA) are widely used in simulating nonlinear complex systems, and their reversibility is closely related to information loss during the evolution. However, only a relatively small portion of their reversibility problems has been solved. In this paper, we perform calculations on two graphs and discover that the reversibility of any one-dimensional FCA exhibits periodicity as the number of cells increases. We also successfully provide a method to compute the reversibility sequence that encompasses the reversibility of one-dimensional FCA with any number of cells. Additionally, the calculations in this paper are applicable to FCA with various types of boundaries. This means that we will have an efficient method to determine the reversibility of almost all one-dimensional FCA, with a complexity independent of cell number.