Hierarchies based on the number of cooperating systems of three-dimensional finite automata

The question of whether processing three-dimensional digital patterns is much more diffi cult than twodimensional ones is of great interest from both theoretical and practical standpoints. Recently, owing to advances in many application areas, such as computer vision, robotics, and so forth, it has become increasingly apparent that the study of three-dimensional pattern processing is of crucial importance. Thus, the study of three-dimensional automata as a computational model of three-dimensional pattern processing has become meaningful. This article introduces a cooperating system of three-dimensional finite automata as one model of three-dimensional automata. A cooperating system of three-dimensional fi nite automata consists of a fi nite number of three-dimensional fi nite automata and a three-dimensional input tape where these fi nite automata work independently (in parallel). Those fi nite automata whose input heads scan the same cell of the input tape can communicate with each other, i.e., every fi nite automaton is allowed to know the internal states of other fi nite automata on the cell it is scanning at the moment. In this article, we continue the study of cooperating systems of three dimensional fi nite automata, and mainly investigate hierarchies based on the number of their cooperating systems.