Abstract
We introduce a two-dimensional variant of the restarting automaton with window size three-by-three for processing rectangular pictures. In each rewrite step such an automaton can only replace the symbol in the middle position of its window by a symbol that is smaller with respect to a fixed ordering on the tape alphabet.
When restricted to one-dimensional inputs (that is, words) the deterministic variant of these ordered restarting automata only accepts regular languages, while the nondeterministic one can accept some languages that are not even context-free. We then concentrate on the deterministic two-dimensional ordered restarting automaton, showing that it is quite expressive as it can simulate the deterministic sgraffito automaton, and we present some closure and non-closure properties for the class of picture languages accepted by these automata.