Decision algorithms for reversibility of 1D cellular automata under reflective boundary conditions

Reversibility is one of the most significant properties of cellular automata (CA). In this paper, we focus on the reversibility of one-dimensional finite CA under reflective boundary conditions (RBC). We present two algorithms for deciding the reversibility of one-dimensional CA under RBC. Both algorithms work for not only linear rules but also non-linear rules. The first algorithm is to determine what we call the "strict reversibility" of CA. The second algorithm is to compute what we call the "reversibility function" of CA. Reversibility functions are proved to be periodic. Based on the algorithms, we list some experiment results of one-dimensional CA under RBC and analyse some features of this family of CA.