Automaton Semigroups and Groups: On the Undecidability of Problems Related to Freeness and Finiteness

In this paper, we study algorithmic problems for automaton semigroups and automaton groups related to freeness and finiteness. In the course of this study, we also exhibit some connections between the algebraic structure of automaton (semi)groups and their dynamics on the boundary. First, we show that it is undecidable to check whether the group generated by a given invertible automaton has a positive relation, i.e. a relation p = 1 such that p only contains positive generators. Besides its obvious relation to the freeness of the group, the absence of positive relations has previously been studied and is connected to the triviality of some stabilizers of the boundary. We show that the emptiness of the set of positive relations is equivalent to the dynamical property that all (directed positive) orbital graphs centered at non-singular points are acyclic. Gillibert showed that the finiteness problem for automaton semigroups is undecidable. In the second part of the paper, we show that this undecidability result also holds if the input is restricted to be bi-reversible and invertible (but, in general, not complete). As an immediate consequence, we obtain that the finiteness problem for automaton subsemigroups of semigroups generated by invertible, yet partial automata, so called automaton-inverse semigroups, is also undecidable. Erratum: Contrary to a statement in a previous version of the paper, our approach does not show that that the freeness problem for automaton semigroups is undecidable. We discuss this in an erratum at the end of the paper

On the Complexity of the Word Problem for Automaton Semigroups and Automaton Groups

In this paper, we study the word problem for automaton semigroups and automaton groups from a complexity point of view. As an intermediate concept between automaton semigroups and automaton groups, we introduce automaton-inverse semigroups, which are generated by partial, yet invertible automata. We show that there is an automaton-inverse semigroup and, thus, an automaton semigroup with a PSPACE-complete word problem. We also show that there is an automaton group for which the word problem with a single rational constraint is PSPACE-complete. Additionally, we provide simpler constructions for the uniform word problems of these classes. For the uniform word problem for automaton groups (without rational constraints), we show NL-hardness. Finally, we investigate a question asked by Cain about a better upper bound for the length of a word on which two distinct elements of an automaton semigroup must act differently

The Self-Similarity of Free Semigroups and Groups (Logic, Algebraic system, Language and Related Areas in Computer Science)

We give a survey on results regarding self-similar and automaton presentations of free groups and semigroups and related products. Furthermore, we discuss open problems and results with respect to algebraic decision problems in this area

Kaskadenzerlegung spezieller Automatenklassen

Der Begriff des endlichen Automaten spielt für die Informatik eine große Rolle. Vom Chip-Design über die Progammimplementierung bis hin zur Sprach- und Automatentheorie findet er Anwendung. Dies ist Grund genug sich mit endlichen Automaten genauer zu beschäftigen. Auf algebraischer Seite ist der endliche Automat eng verwandt mit der Halbgruppe oder dem Monoid. Zwar sind diese Konzepte weniger anschaulich als ein endlicher Automat, sie erlauben jedoch einen anderen Blickwinkel und machen die mathematische Betrachtung an einigen Stellen einfacher. Durch das Krohn-Rhodes-Theorem ist bekannt, dass sich eine beliebige endliche Halbgruppe in einfache Gruppen und FlipFlops zerlegen lasst. Die Rückkopplungsfreiheit dieser Zerlegung motiviert den Begriff der "Kaskadenzerlegung". Während die einfachen Gruppen, die dabei auftreten, in der ursprünglichen Halbgruppe selbst enthalten sind, ist dies beim FlipFlop nicht notwendigerweise der Fall. Es stellt sich daher die Frage: Gibt es eine Menge von strukturell möglichst einfachen Halbgruppen, die als Bausteine eine Zerlegung jeder – auch komplexeren – Halbgruppe so ermöglichen, dass jeder verwendete Baustein in der Halbgruppe selbst enthalten ist? Ist die Menge endlich und wie funktioniert die Zerlegung? Angetrieben durch diese Fragestellung werden in dieser Arbeit Zerlegungen von Halbgruppen und Monoiden aus speziellen Klassen genauer untersucht, für die die Frage nach den Bausteinen beantwortet werden kann

On the Structure Theory of Partial Automaton Semigroups

We study automaton structures, i.e. groups, monoids and semigroups generated by an automaton, which, in this context, means a deterministic finite-state letter-to-letter transducer. Instead of considering only complete automata, we specifically investigate semigroups generated by partial automata. First, we show that the class of semigroups generated by partial automata coincides with the class of semigroups generated by complete automata if and only if the latter class is closed under removing a previously adjoined zero, which is an open problem in (complete) automaton semigroup theory stated by Cain. Then, we show that no semidirect product (and, thus, also no direct product) of an arbitrary semigroup with a (non-trivial) subsemigroup of the free monogenic semigroup is an automaton semigroup. Finally, we concentrate on inverse semigroups generated by invertible but partial automata, which we call automaton-inverse semigroups, and show that any inverse automaton semigroup can be generated by such an automaton (showing that automaton-inverse semigroups and inverse automaton semigroups coincide)

Orbit Expandability of Automaton Semigroups and Groups

We introduce the notion of expandability in the context of automaton semigroups and groups: a word is k-expandable if one can append a suffix to it such that the size of the orbit under the action of the automaton increases by at least k. This definition is motivated by the question which {\omega}-words admit infinite orbits: for such a word, every prefix is expandable. In this paper, we show that, on input of a word u, an automaton T and a number k, it is decidable to check whether u is k-expandable with respect to the action of T. In fact, this can be done in exponential nondeterministic space. From this nondeterministic algorithm, we obtain a bound on the length of a potential orbit-increasing suffix x. Moreover, we investigate the situation if the automaton is invertible and generates a group. In this case, we give an algebraic characterization for the expandability of a word based on its shifted stabilizer. We also give a more efficient algorithm to decide expandability of a word in the case of automaton groups, which allows us to improve the upper bound on the maximal orbit-increasing suffix length. Then, we investigate the situation for reversible (and complete) automata and obtain that every word is expandable with respect to these automata. Finally, we give a lower bound example for the length of an orbit-increasing suffix

Dynamics of wind turbine operational states

Modern wind turbines gather an abundance of data with their Supervisory Control And Data Acquisition (SCADA) system. We study the short-term mutual dependencies of a variety of observables (e.g. wind speed, generated power and current, rotation frequency) by evaluating Pearson correlation matrices on a moving time window. The analysis of short-term correlations is made possible by high frequency SCADA-data. The resulting time series of correlation matrices exhibits non-stationarity in the mutual dependencies of different measurements at a single turbine. Using cluster analysis on these matrices, multiple stable operational states are found. They show distinct correlation structures, which represent different turbine control settings. The current system state is linked to external factors interacting with the control system of the wind turbine. For example at sufficiently high wind speeds, the state represents the behavior for rated power production. Moreover, we combine the clustering with stochastic process analysis to study the dynamics of those states in more detail. Calculating the distances between correlation matrices we obtain a time series that describes the behavior of the complex system in a collective way. Assuming this time series to be a stochastic process governed by a Langevin equation, we estimate the drift and diffusion terms to understand the underlying dynamics. The drift term, which describes the deterministic behavior of the system, is used to obtain a potential. Dips in the potential are identified with the cluster states. We study the dynamics of operational states and their transitions by analyzing the development of the potential over time and wind speed. Thereby, we further characterize the different states and discuss consequences for the analysis of high frequency wind turbine data