Cooperating systems of four-dimensional finite automata

Abstract

M. Blum and C. Hewitt first proposed two-dimensional automata as a computational model of two-dimensional pattern processing in 1967, and investigated their pattern recognition abilities. Since then, many researchers in this field have investigated many properties of automata on a two- or three-dimensional tape. However, the question of whether processing four-dimensional digital patterns is much more difficult than processing two- or three-dimensional ones is of great interest from both theoretical and practical standpoints. Thus, the study of four-dimensional automata as a computational model of four-dimensional pattern processing has been meaningful. This article introduces a cooperating system of four-dimensional finite automata as one model of four-dimensional automata. A cooperating system of four-dimensional finite automata consists of a finite number of four-dimensional finite automata and a four-dimensional input tape, where these finite automata work independently (in parallel). The finite automata whose input heads scan the same cell of the input tape can communicate with each other, i.e., every finite automaton is allowed to know the internal states of other finite automata on the cell it is scanning at the moment. In this article we mainly investigate the accepting powers of a cooperating system of seven-way four-dimensional finite automata. The seven-way four-dimensional finite automaton is a four-dimensional finite automaton whose input head can move east, west, south, north, up, down, or in the future, but not in the past, on a four-dimensional input tape.