A Note on Decision Problems for Three-Way Two-Dimensional Finite Automata

In [2], Blum and Hewitt introduced a two-dimensional finite automaton which can be considered as a natural extension of the one-dimensional two-way finite automaton [4] to two dimensions. On the other hand, in [7,8], Rosenfeld introduced a new type of automaton on a two-dimensional tape (called the three-way two-dimensional finite automaton) which can be considered as a natural extension of the onedimensional one-way finite automaton [4] to two dimensions. It is well known [S] that one-dimensional one-way finite automata are no less powerful than one-dimensional two-way finite automata. For the two-dimensional case, the similar fact to this does not hold. That is, three-way two-dimensional finite automata are less powerful than (four-way) two-dimensional finite automata [7,8]. In [9], Taniguchi and Kasami investigated the decision problems for (fourway) two-dimensional finite automata, and showed, for example, that the emptiness problem for those automata is undecidable. It is natural to ask what results one can obtain :, about the decision problems for three-way two-dimensional finite automata. In this note, we are mainly concerned with presenting some answers about this question. Let C be a finite set of symbols. A two-dimensional tape over X is a twodimensional rectangular array of elements of X. The set of all two-dimensional tapes over % is denoted by zt2). Given a tape x in Zt2), we let Q,(x) be the number of rows of x and Q,(x) be the number of columns of x. If 1 < i < Qr(x) and 1 <j < Q,(x), we let x(i, j) denote the symbol in x with coordinates (i, j). Furthermore, we define