A computation-universal two-dimensional 8-state triangular reversible cellular automaton

A reversible cellular automaton (RCA) is a cellular automaton (CA) whose global function is injective and every configuration has at most one predecessor. Margolus showed that there is a computation-universal two-dimensional 2-state RCA. But his RCA has nonuniform neighbor, so Morita and Ueno proposed 16-state computation-universal RCA using partitioned cellular automata (PCA). Because PCA can be regarded as a subclass of standard CA, their models has standard neighbor. In this paper, we show that the number of states of Morita and Ueno’s models can be reducible. To decrease the number of states from their models with preserving isotropic and bit-preserving properties, we used triangular 3neighbor, and thus 8-state RCA can be possible. This is the smallest state two-dimensional RCA under the condition of isotropic property on the framework of PCA. We show that our model can simulate basic circuit elements such as unit wires, delay elements, crossing wires, switch gates and inverse switch gates. And it is possible to construct a Fredkin gate by combining these elements. Since Fredkin gate is known to be a universal logic gate, our model has computation-universality.