Two graphs: Resolving the periodic reversibility of one-dimensional finite cellular automata

Abstract

Finite cellular automata (FCA), as discrete dynamical systems, are widely used in simulations, coding theory, information theory, and so on. Its reversibility, related to information loss in system evolution, is one of the most important problems. In this paper, we perform calculations on two elaborate graphs — the reversibility graph and the circuit graph and discover that the reversibility of one-dimensional FCA exhibits periodicity as the number of cells increases. We provide a method to compute the reversibility sequence that encompasses the reversibility of one-dimensional FCA with any number of cells in $O(q^{m}V+qVk+V{k}^2)$ where $q,m,V,k$ are the number of states, neighborhood, vertices, and elementary circuits in the reversibility graph of FCA. The calculations in this paper are applicable to FCA with two mainstream boundaries, null boundary and periodic boundary. This means we have an efficient method to determine the reversibility of almost all one-dimensional FCA, with a complexity independent of cell number.