The row projection (resp., column projection) of a two-dimensional language $L$ is the one-dimensional language consisting of all first rows (resp., first columns) of each two-dimensional word in $L$. The operation of row projection has previously been studied under the name "frontier language", and previous work has focused on one- and two-dimensional language classes. In this paper, we study projections of languages recognized by various two-dimensional automaton classes. We show that both the row and column projections of languages recognized by (four-way) two-dimensional automata are exactly context-sensitive. We also show that the column projections of languages recognized by unary three-way two-dimensional automata can be recognized using nondeterministic logspace. Finally, we study the state complexity of projection languages for two-way two-dimensional automata, focusing on the language operations of union and diagonal concatenation.

中文翻译：

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二维自动机投影语言的识别和复杂度结果

二维语言$L$的行投影（resp., columnprojection）是由$L$中每个二维单词的所有第一行（分别为第一列）组成的一维语言。行投影的操作之前曾以“前沿语言”为名进行研究，之前的工作主要集中在一维和二维语言类上。在本文中，我们研究了各种二维自动机类识别的语言的投影。我们表明（四向）二维自动机识别的语言的行和列投影都完全是上下文敏感的。我们还表明，可以使用非确定性对数空间识别由一元三向二维自动机识别的语言的列投影。最后，