Rectangles and Squares Recognized by Two-Dimensional Automata

We consider sets of rectangles and squares recognized by deterministic and non-deterministic two-dimensional finite-state automata. We show that NFAs are strictly more powerful than DFAs, even for pictures over a one-symbol alphabet. In the process, we show that the pitcure languages recognized by NFAs are not closed under complement, resolving a long-standing open question. We also show that NFAs can only recognize sets of rectangles from the outside that correspond to simple regular languages. Finally, we show that sets of squares recognized by DFAs can be as sparse as any recursively enumerable set.