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

The row projection (resp., column projection) of a given two-dimensional language L is the one-dimensional language consisting of first rows (resp., first columns) of all two-dimensional words in L. The operation of row projection has previously been studied under the name "frontier language" , and previous work in this area has focused primarily 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.