We present a new model of a two-dimensional computing device called Sgraffito automaton. In general, the model is quite simple, which allows a clear design of computations. When restricted to one-dimensional inputs, that is, strings, the Sgraffito automaton does not exceed the power of finite-state automata. On the other hand, for two-dimensional inputs, it yields a family of picture languages with good closure properties that strictly includes the class REC  of recognizable picture languages. The deterministic Sgraffito automata define a class of picture languages that includes the class of deterministic recognizable picture languages DREC, the class of picture languages that are accepted by four-way alternating automata, those that are accepted by deterministic one-marker automata, and the sudoku-deterministically recognizable picture languages, but the membership problem for the accepted languages is still decidable in polynomial time. In addition, the deterministic Sgraffito automata accept some unary picture languages that are outside of the class REC.

Several types of systems of parallel communicating restarting automata are introduced and studied. The main result shows that, for all types of restarting automata X, centralized systems of restarting automata of type X have the same computational power as non-centralized systems of restarting automata of the same type and the same number of components. This result is proved by a direct simulation. In addition, it is shown that for one-way restarting automata without auxiliary symbols, systems of degree at least two are more powerful than the component automata themselves. Finally, a lower and an upper bound are given for the expressive power of systems of parallel communicating restarting automata.

In the literature various types of restarting automata have been studied that are based on contextual rewriting. A word w is accepted by such an automaton if, starting from the initial configuration that corresponds to input w, the word w is reduced to the empty word by a finite number of applications of these contextual rewritings. This approach is reminiscent of the notion of McNaughton families of languages. Here we put the aforementioned types of restarting automata into the context of McNaughton families of languages, relating the classes of languages accepted by these automata in particular to the class GCSL of growing context-sensitive languages and to the class CRL of Church–Rosser languages.

We study cooperating distributed systems (CD-systems) of stateless deterministic restarting automata with window size 1 that are equipped with an external pushdown store. In this way we obtain an automata-theoretical characterization for the class of word languages that are linearizations of context-free trace languages.

It is shown that the class of left-to-right regular languages coincides with the class of languages that are accepted by monotone deterministic RL-automata, in this way establishing a close correspondence between a classical parsing algorithm and a certain restricted type of analysis by reduction.

As already 2-monotone R-automata accept NP-complete languages, we introduce a restricted variant of j-monotonicity for restarting automata, called sequential j-monotonicity. For restarting automata without auxiliary symbols, this restricted variant still yields infinite hierarchies. However, for restarting automata with auxiliary symbols, all degrees of sequential monotonicity collapse to the first level, implying that RLWW-automata that are sequentially monotone of degree j for any j ≥ 1 only accept context-free languages.

It is known that the weakly monotone restarting automata accept exactly the growing context-sensitive languages. We introduce a measure on the degree of weak monotonicity and show that the language classes obtained in this way form strict hierarchies for the various types of deterministic and nondeterministic restarting automata without auxiliary symbols.

A finite rewriting system is presented that does not satisfy the homotopical finiteness condition FDT, although it satisfies the homological finiteness condition FHT. This system is obtained from a group $G$ and a finitely generated subgroup $H$ of $G$ through a monoid extension that is almost an HNN extension. The FHT property of the extension is closely related to the ${\rm FP}_2$ property for the subgroup $H$, while the FDT property of the extension is related to the finite presentability of $H$. The example system separating the FDT property from the FHT property is then obtained by applying this construction to an example group.