Deterministic ordered restarting automata for picture languages

The ordered restarting automaton (processing strings) is introduced, and it is shown that its nondeterministic variant is very expressive, as it accepts some languages that are not even context-free, while the deterministic ordered restarting automata just accept the regular languages. Then three different extensions of the deterministic ordered restarting automaton to two-dimensional inputs are defined that differ in the way in which they can move their read/write windows. We compare the classes of picture languages that these types of automata accept to each other and to some well studied classes of picture languages from the literature, and we present some closure and non-closure properties for them.