A two-dimensional automaton has a read-only input head that moves in four directions on a finite array of cells labeled by symbols of the input alphabet. A three-way two-dimensional automaton is prohibited from making upward moves, while a two-way two-dimensional automaton can only move downward and rightward.

We show that the language emptiness problem for unary three-way nondeterministic two-dimensional automata is NP-complete, and is in P for general-alphabet two-way nondeterministic two-dimensional automata. We also show that the language equivalence and inclusion problems for two-way deterministic two-dimensional automata are decidable, while the language universality, equivalence, and inclusion problems for two-way nondeterministic two-dimensional automata are undecidable. The deterministic case is the first known positive decidability result for a language equivalence problem on two-dimensional automata over a general alphabet.

Finally, we discuss the notion of row and column projection languages. We show that the row projection language of a unary three-way nondeterministic two-dimensional automaton is always regular, and that there exists a unary three-way deterministic two-dimensional automaton with a nonregular column projection language. For two-way nondeterministic two-dimensional automata, on the other hand, both the row and column projection languages are always regular.

二维自动机具有只读输入头，该输入头在由输入字母的符号标记的单元格的有限阵列上沿四个方向移动。禁止双向二维自动机向上移动，而双向二维自动机只能向下和向右移动。

我们表明，一元三路不确定的二维自动语言空虚问题是NP -完整，并在P一般字母表双向不确定的二维自动机。我们还表明，双向确定性二维自动机的语言等效性和包含性问题是可以确定的，而双向不确定性二维自动机的语言通用性，等效性和包含性问题是不确定的。确定性情况是在通用字母表上二维自动机上的语言等效问题的第一个已知的肯定可确定性结果。

最后，我们讨论行和列投影语言的概念。我们表明，一元三向不确定二维自动机的行投影语言始终是规则的，并且存在具有一列不规则投影语言的一元三向确定二维自动机。另一方面，对于双向非确定性二维自动机，行和列投影语言始终都是规则的。