Universality of 8-State Reversible and Conservative Triangular Partitioned Cellular Automata

Abstract

A partitioned cellular automaton (PCA) is a subclass of a standard CA such that each cell is divided into several parts, and the next state of a cell is determined only by the adjacent parts of its neighbor cells. This framework is useful for designing reversible CAs. Here, we investigate isotropic three-neighbor 8-state triangular PCAs where a cell has three parts, and each part has two states. They are called elementary triangular PCAs (ETPCAs). There are 256 ETPCAs, and they are extremely simple since each of their local transition functions is described by only four local rules. In this paper, we study computational universality of nine kinds of reversible and conservative ETPCAs. It has already been shown that one of them is universal. Here, we newly show universality of another. It is proved by showing that a Fredkin gate, a universal reversible logic gate, can be simulated in it. From these results and by dualities, we can conclude six among the nine are universal. We also show the remaining three are non-universal. Thus, universality of all the reversible and conservative ETPCAs is clarified.