A two-dimensional automaton operates on arrays of symbols. While a standard (four-way) two-dimensional automaton can move its input head in four directions, restricted two-dimensional automata are only permitted to move their input heads in three or two directions; these models are called three-way and two-way two-dimensional automata, respectively. In two dimensions, we may extend the notion of concatenation in multiple ways, depending on the words to be concatenated. We may row-concatenate (resp., column-concatenate) a pair of two-dimensional words when they have the same number of columns (resp., rows). In addition, the diagonal concatenation operation combines two words at their lower-right and upper-left corners, and is not dimension-dependent. In this paper, we investigate closure properties of restricted models of two-dimensional automata under various concatenation operations. We give non-closure results for two-way two-dimensional automata under row and column concatenation in both the deterministic and nondeterministic cases. We further give positive closure results for the same concatenation operations on unary nondeterministic two-way two-dimensional automata. Finally, we study closure properties of diagonal concatenation on both two- and three-way two-dimensional automata.

中文翻译：

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二维自动机的连接操作和受限变体

二维自动机对符号数组进行操作。标准（四向）二维自动机可以在四个方向上移动其输入头，而受限二维自动机只能在三个或两个方向上移动其输入头；这些模型分别称为三向和双向二维自动机。在二维中，我们可以通过多种方式扩展连接的概念，这取决于要连接的单词。当一对二维单词具有相同的列数（或行）时，我们可以行连接（或列连接）一对二维单词。另外，对角线连接操作将两个单词在它们的右下角和左上角合并，并且不依赖于维度。在本文中，我们研究了在各种连接操作下二维自动机受限模型的闭合特性。在确定性和非确定性情况下，我们在行和列连接下给出了二维二维自动机的非闭合结果。我们进一步给出了一元非确定性二维二维自动机上相同连接操作的正闭包结果。最后，我们研究了二维和三向二维自动机上对角连接的闭包特性。我们进一步给出了一元非确定性二维二维自动机上相同连接操作的正闭包结果。最后，我们研究了二维和三向二维自动机上对角连接的闭包特性。我们进一步给出了一元非确定性二维二维自动机上相同连接操作的正闭包结果。最后，我们研究了二维和三向二维自动机上对角连接的闭包特性。