The reversibility problem for a family of two-dimensional cellular automata

In this paper the reversibility problem of a family of two-dimensional cellular automata is completely resolved. It is well known that the reversibility problem is a very difficult one in general. In order to determine whether a cellular automaton is reversible or not the reversibility of its rule matrix is studied via linear algebraic tools. However, in this particular study the authors consider a novel approach. By observing the algebraic structures of rule matrices that represent these families and associating them with polynomials in two variables in a quotient ring, the solution to the reversibility problem is simplified greatly. Hence, this approach not only drastically decreases the computational time for determining the reversibility of these families but also provides an explicit construction of reverse cellular automata in the case of the existence of their inverses. The paper concludes with a consideration of the rule matrices of these families in obtaining linear codes over group rings, which are referred to as zero-divisor codes.