Matthias Wendlandt

Parameterized Prefix Distance between Regular Languages

We investigate the parameterized prefix distance between regular languages. The prefix distance between words is extended to languages in such a way that the distances of all words up to length n to the mutual other language are summed up. Tight upper bounds for the distance between unary as well as non-unary regular languages are derived. It is shown that there are pairs of languages having a constant, degree k polynomial, and exponential distance. Moreover, for every constant and every polynomial, languages over a binary alphabet are constructed that have exactly that distance. From the density and census functions of regular languages the orders of possible distances between languages are derived and are shown to be decidable.

On Simulation Cost of Unary Limited Automata

A \(k\)-limited automaton is a linear bounded automaton that may rewrite each tape cell only in the first \(k\) visits, where \(k\ge 0\) is a fixed constant. It is known that these automata accept context-free languages only. We investigate the descriptional complexity of deterministic limited automata accepting unary languages. Since these languages are necessarily regular, we study the cost in the number of states when a \(k\)-limited automaton is simulated by finite automata. For the conversion of a \(4n\)-state \(1\)-limited automaton into one-way or two-way deterministic or nondeterministic finite automata a lower bound of \(n\cdot F(n)\) states is shown, where \(F\) denotes Landau’s function. So, even the ability deterministically to rewrite any cell only once gives an enormous descriptional power. For the simulation cost for removing the ability to rewrite each cell \(k\ge 1\) times, that is, the cost for the simulation of (sweeping) \(k\)-limited automata by deterministic finite automata, we obtain a lower bound of \(n\cdot F(n)^{k}\). A polynomial upper bound is shown for the simulation by two-way deterministic finite automata, where the degree of the polynomial is quadratic in \(k\). If the \(k\)-limited automaton is rotating, the upper bound reduces to \(O(n^{k+1})\). A lower bound of \(\varOmega (n^{k+1})\) is derived even for nondeterministic two-way finite automata. So, for rotating \(k\)-limited automata, the trade-off for the simulation is tight in the order of magnitude.

Tinput-Driven Pushdown Automata

In input-driven pushdown automata (\(\text {IDPDA}\)) the input alphabet is divided into three distinct classes and the actions on the pushdown store (push, pop, nothing) are solely governed by the input symbols. Here, this model is extended in such a way that the input of an \(\text {IDPDA}\) is preprocessed by a deterministic sequential transducer. These automata are called tinput-driven pushdown automata (\(\text {TDPDA}\)) and it turns out that \(\text {TDPDA}\)s are more powerful than \(\text {IDPDA}\)s but still not as powerful as real-time deterministic pushdown automata. Nevertheless, even this stronger model has still good closure and decidability properties. In detail, it is shown that \(\text {TDPDA}\)s are closed under the Boolean operations union, intersection, and complementation. Furthermore, decidability procedures for the inclusion problem as well as for the questions of whether a given automaton is a \(\text {TDPDA}\) or an \(\text {IDPDA}\) are developed. Finally, representation theorems for the context-free languages using \(\text {IDPDA}\)s and \(\text {TDPDA}\)s are established.

Deterministic input-driven queue automata

We introduce and study the model of deterministic input-driven queue automata. On such devices, the input letters uniquely determine the operations on the memory store which is organized as a queue. In particular, we consider the case where only a finite number of turns on the queue is allowed. The resulting language families share with regular languages many desirable properties. We show that emptiness and several other problems are decidable. Furthermore, we investigate closure under Boolean operations. The existence of an infinite and tight hierarchy depending on the number of turns is also proved.

Deterministic Set Automata

We consider the model of deterministic set automata which are basically deterministic finite automata equipped with a set as an additional storage medium. The basic operations on the set are the insertion of elements, the removing of elements, and the test whether an element is in the set. We investigate the computational power of deterministic set automata and compare the language class accepted with the context-free languages and classes of languages accepted by queue automata. As results the incomparability to all classes considered is obtained. In the second part of the paper, we examine the closure properties of the class of DSA languages under Boolean operations. Finally, we show that deterministic set automata may be an interesting model from a practical point of view by proving that their emptiness problem is decidable.

Real-Time Reversible One-Way Cellular Automata

Real-time one-way cellular automata (\({\text {OCA}}\)) are investigated towards their ability to perform reversible computations with regard to formal language recognition. It turns out that the standard model with fixed boundary conditions is quite weak in terms of reversible information processing, since it is shown that in this case exactly the regular languages can be accepted reversibly. We then study a modest extension which allows that information may flow circularly from the leftmost cell into the rightmost cell. It is shown that this extension does not increase the computational power in the general case, but does increase it for reversible computations. On the other hand, the model is less powerful than real-time reversible two-way cellular automata. Additionally, we obtain that the corresponding language class is closed under Boolean operations, and we prove the undecidability of several decidability questions. Finally, it is shown that the reversibility of an arbitrary real-time circular one-way cellular automaton is undecidable as well.

Size of Unary One-Way Multi-head Finite Automata

We investigate the descriptional complexity of deterministic one-way multi-head finite automata accepting unary languages. It is known that in this case the languages accepted are regular. Thus, we study the increase of the number of states when an n-state k-head finite automaton is simulated by a classical (one-head) deterministic or nondeterministic finite automaton. In the former case an upper bound of O(n·F(t·n) k − 1) and a lower bound of n·F(n) k − 1 states is shown, where t is a constant and F denotes Landau’s function. Since both bounds are of order \(e^{\Theta(\sqrt{n \cdot \ln(n)})}\), the trade-off for the simulation is tight in the order of magnitude. For the latter case we obtain an upper bound of O(n 2k ) and a lower bound of Ω(n k ) states. We investigate also the costs for the conversion of one-head nondeterministic finite automata to deterministic k-head finite automata, that is, we trade nondeterminism for heads. Finally, as an application of the simulation results, we show that decidability problems for unary deterministic k-head finite automata such as emptiness or equivalence are LOGSPACE-complete.

States and heads do count for unary multi-head finite automata

Unary deterministic one-way multi-head finite automata characterize the unary regular languages. Here they are studied with respect to the existence of head and state hierarchies. It turns out that for any fixed number of states, there is an infinite proper head hierarchy. In particular, the head hierarchy for stateless deterministic one-way multi-head finite automata is obtained using unary languages. On the other hand, it is shown that for a fixed number of heads, m+1 states are more powerful than m states. Finally, the open question of whether emptiness is undecidable for stateless one-way two-head finite automata is addressed and, as a partial answer, undecidability can be shown if at least four states are provided.

Reversible Limited Automata

A k-limited automaton is a linear bounded automaton that may rewrite each tape square only in the first k visits, where \(k\ge 0\) is a fixed constant. It is known that these automata accept context-free languages only. We investigate deterministic k-limited automata towards their ability to perform reversible computations, that is, computations in which every configuration has at most one predecessor. A first result is that, for all \(k\ge 0\), sweeping k-limited automata accept regular languages only. In contrast to reversible finite automata, all regular languages are accepted by sweeping 0-limited automata. Then we study the computational power gained in the number k of possible rewrite operations. It is shown that the reversible 2-limited automata accept regular languages only and, thus, are strictly weaker than general 2-limited automata. Furthermore, a proper inclusion between reversible 3-limited and 4-limited automata languages is obtained. The next levels of the hierarchy are separated between every k and \(k+3\) rewrite operations. Finally, it turns out that all k-limited automata accept Church-Rosser languages only, that is, the intersection between context-free and Church-Rosser languages contains an infinite hierarchy of language families beyond the deterministic context-free languages.

Reversible Ordered Restarting Automata

Stateless deterministic ordered restarting automata characterize the class of regular languages. Here we introduce a notion of reversibility for these automata and show that each regular language is accepted by such a reversible stateless deterministic ordered restarting automaton. We study the descriptional complexity of these automata, showing that they are exponentially more succinct than nondeterministic finite-state acceptors. We also look at the case of unary input alphabets.