František Mráz

Ordered Restarting Automata for Picture Languages

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.

Clearing Restarting Automata

Restarting automata were introduced as a model for analysis by reduction, which is a linguistically motivated method for checking correctness of a sentence. We propose a new restricted version of restarting automata called clearing restarting automata with a very simple definition but simultaneously with interesting properties with respect to their possible applications. The new model can be learned very efficiently from positive examples and its stronger version can be used to learn effectively a large class of languages. We relate the class of languages recognized by clearing restarting automata to the Chomsky hierarchy.

A Taxonomy of Forgetting Automata

Forgetting automata are nondeterministic linear bounded automata whose rewriting capability is restricted as follows: each cell of the tape can only be “erased” (rewritten by a special symbol) or completely “deleted”.

Degrees of non-monotonicity for restarting automata

In the literature various notions of monotonicity for restarting automata have been studied. Here we introduce two new variants of monotonicity for restarting automata and for two-way restarting automata: left-monotonicity and right-left-monotonicity. It is shown that for the various types of deterministic and nondeterministic (two-way) restarting automata without auxiliary symbols, these notions yield infinite hierarchies, and we compare these hierarchies to each other. Further, as a tool used to simplify some of the proofs, the shrinking restarting automaton is introduced, which is a generalization of the standard (length-reducing) restarting automaton to the weight-reducing case. Some of the consequences of this generalization are also discussed.

Forgetting automata and context-free languages

It is shown that context-free languages can be characterized by linear bounded automata with the following restriction: the head can either move right without rewriting or move left with erasing the current cell (i.e. rewriting it with a special, nonrewriteable, symbol). If, instead of erasing, we consider deleting (complete removing of the cell), the corresponding automata are less powerful.

Restarting Automata with Rewriting

Motivated by natural language analysis we introduce restarting automata with rewriting. They are acceptors on the one hand, and (special) regulated rewriting systems on the other hand. The computation of a restarting automaton proceeds in cycles: in each cycle, a bounded substring of the input word is rewritten by a shorter string, and the computation restarts on the arising shorter word.

Monotone deterministic RL-Automata don't need auxiliary symbols

It is known that for monotone deterministic one-way restarting automata, the use of auxiliary symbols does not increase the expressive power. Here we show that the same is true for deterministic two-way restarting automata that are right- or left-monotone. Actually in these cases it suffices to admit delete operations instead of the more general rewrite operations. In addition, we characterize the classes of languages that are accepted by these types of two-way restarting automata by certain combinations of deterministic pushdown automata and deterministic transducers.

Characterization of Context-Free Languages by Erasing Automata

It is shown that context-free languages are recognizable by (non-deterministic) erasing automata; thereby a hypothesis of [1] is denied. In addition, the class of context-free languages is characterized by means of the automata which erase each cell at the second visit at latest.

Two-dimensional sgraffito automata

We present a new model of a two-dimensional computing device called sgraffito automaton and demonstrate its significance. In general, the model is simple, allows a clear design of important computations and defines families exhibiting good properties. It does not exceed the power of finite-state automata when working over one-dimensional inputs. On the other hand, it induces a family of picture languages that strictly includes REC and the deterministic variant recognizes languages in DREC as well as those accepted by four-way automata.

Different Types of Monotonicity for Restarting Automata

We consider several classes of rewriting automata with a restart operation and the monotonicity property of computations by such automata. It leads to three natural definitions of (right) monotonicity of automata. Besides the former monotonicity, two new types, namely a-monotonicity and g-monotonicity, for such automata are introduced. We provide a taxonomy of the relevant language classes, and answer the (un)decidability questions concerning these properties.