Reversible Elementary Triangular Partitioned Cellular Automata

Abstract

A three-neighbor triangular partitioned cellular automaton (TPCA) is a CA such that each cell is triangular-shaped and has three parts. A TPCA is called an elementary TPCA (ETPCA), if it is rotation-symmetric, and each part of a cell has only two states. The class of ETPCAs is one of the simplest subclasses of twodimensional CAs, since the local function of each ETPCA is described by only four local rules. Here, we investigate the universality of reversible ETPCAs (RETPCAs). First, nine kinds of conservative RETPCAs, in which the total number of particles is conserved in their evolution processes, are studied. It is proved that six conservative RETPCAs among nine are Turing universal and intrinsically universal by showing any reversible logic circuit composed of Fredkin gates can be simulated in them. It is also shown that three conservative RETPCAs among nine are non-universal. Next, we study a specific non-conservative RETPCA T0347, where 0347 is its identification number in the class of 256 ETPCAs. In spite of its simplicity it exhibits complex behavior like the Game of Life CA. Using gliders to represent signals, Turing universality and intrinsic universality of T0347 are also proved.