Temporally Non-Uniform Cellular Automata (t-NUCA): Reversibility and Cyclic behavior

Abstract: In this work, we propose a variant of non-uniform cellular automata, named as Temporally Non-Uniform Cellular Automata (t-NUCAs), which temporally use two rules, $f$ and $g$ in a sequence $\mathcal{R}$. To observe reversibility in t-NUCAs, we study their injectivity and surjectivity properties. Unlike classical CAs, some irreversible t-NUCAs show the behavior similar to reversible t-NUCAs. To study this behavior, we define restricted surjectivity of t-NUCA and introduce restricted reversibility which shows reversibility of t-NUCA for a set of initial configurations. By further investigating the remaining irreversible t-NUCAs, some t-NUCAs are found which have many-to-one mapping in their configuration space, but do not have non-reachable (Garden-of-Eden) configurations. We name these t-NUCAs as weakly reversible t-NUCAs. Under finite lattice size, a t-NUCA, like any classical CA, shows cyclic behavior. We explore this cyclic behavior and discuss its relation with rule sequence. Finally, we note down the possible longest cycle length of a t-NUCA, based on the lattice size and rule sequence.