Recognition and Complexity Results for Projection Languages of Two-Dimensional Automata

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.