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Automata theory, one of the main branches of theoretical computer science, established its roots in the middle of the 20th century. One of its most fundamental concepts is that of a finite automaton, a basic yet powerful model of computation. In essence, finite automata provide a method to finitely represent possibly infinite sets of strings. Such a set of strings is also called a language, and the languages which can be described by finite automata are known as regular languages. Owing to their versatility, regular languages have received a great deal of attention over the years. Other formalisms were shown to be expressively equivalent to finite automata, most notably regular grammars, regular expressions, and monadic second order (MSO) logic. To increase expressiveness, the fundamental idea underlying finite automata and regular languages was also extended to describe not only languages of strings, or words, but also of infinite words by Büchi and Muller, finite trees by Doner and Thatcher and Wright, infinite trees by Rabin, nested words by Alur and Madhusudan, and pictures by Blum and Hewitt, just to name a few examples. In a parallel line of development, Schützenberger introduced weighted automata which allow the description of quantitative properties of regular languages. In subsequent works, many of these descriptive formalisms and extensions were combined and their relationships investigated. For example, weighted regular expressions and weighted logics have been developed as well as regular expressions for trees and pictures, regular grammars for trees, pictures, and nested words, and logical characterizations for regular languages of trees, pictures, and nested words. In this work, we focus on two of these extensions and their relationship, namely weighted automata and weighted logics. Just as the classical Büchi-Elgot-Trakhtenbrot Theorem established the coincidence of regular languages with languages definable in monadic second order logic, weighted automata have been shown to be expressively equivalent to a specific fragment of a weighted monadic second order logic by Droste and Gastin. We explore several aspects of weighted automata and of this weighted logic. More precisely, the thesis considers the following topics. In the first part, we extend the classical Feferman-Vaught Theorem to the weighted setting. The Feferman-Vaught Theorem is one of the fundamental theorems in model theory. The theorem describes how the computation of the truth value of a first order sentence in a generalized product of relational structures can be reduced to the computation of truth values of first order sentences in the contributing structures and the evaluation of an MSO sentence in the index structure. The theorem itself has a long-standing history. It builds upon work of Mostowski, and was shown in subsequent works to hold true for MSO logic. Here, we show that under appropriate assumptions, the Feferman-Vaught Theorem also holds true for a weighted MSO logic with arbitrary commutative semirings as weight structure. In the second part, we lift four decidability results from max-plus word automata to max-plus tree automata. Max-plus word and tree automata are weighted automata over the max-plus semiring and assign real numbers to words or trees, respectively. We show that, like for max-plus word automata, the equivalence, unambiguity, and sequentiality problems are decidable for finitely ambiguous max-plus tree automata, and that the finite sequentiality problem is decidable for unambiguous max-plus tree automata. In the last part, we develop a logic which is expressively equivalent to quantitative monitor automata. Introduced very recently by Chatterjee, Henzinger, and Otop, quantitative monitor automata are an automaton model operating on infinite words. Quantitative monitor automata possess several interesting features. They are expressively equivalent to a subclass of nested weighted automata, an automaton model which for many valuation functions has decidable emptiness and universality problems. Also, quantitative monitor automata are more expressive than weighted Büchi-automata and their extension with valuation functions. We introduce a new logic which we call monitor logic and show that it is expressively equivalent to quantitative monitor automata.

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Normalformen drücken semantische Eigenschaften einer Logik durch syntaktische Restriktionen aus. Sie ermöglichen es Algorithmen, Grenzen der Ausdrucksstärke einer Logik auszunutzen. Ein Beispiel ist die Lokalität der Logik erster Stufe (FO), die impliziert, dass Graph-Eigenschaften wie Erreichbarkeit oder Zusammenhang nicht FO-definierbar sind. Gaifman-Normalformen drücken die Bedeutung einer FO-Formel als Boolesche Kombination lokaler Eigenschaften aus. Sie haben eine wichtige Rolle in Model-Checking Algorithmen für Klassen dünn besetzter Graphen, deren Laufzeit durch die Größe der auszuwertenden Formel parametrisiert ist. Es ist jedoch bekannt, dass Gaifman-Normalformen im Allgemeinen nur mit nicht-elementarem Aufwand konstruiert werden können. Dies führt zu einer enormen Parameterabhängigkeit der genannten Algorithmen. Ähnliche nicht-elementare untere Schranken sind auch für Feferman-Vaught-Zerlegungen und für die Erhaltungssätze von Lyndon, Łoś und Tarski bekannt. Diese Arbeit untersucht die Komplexität der genannten Normalformen auf Klassen von Strukturen beschränkten Grades, für welche die nicht-elementaren unteren Schranken nicht gelten. Für diese Einschränkung werden Algorithmen mit elementarer Laufzeit für die Konstruktion von Gaifman-Normalformen, Feferman-Vaught-Zerlegungen, und für die Erhaltungssätze von Lyndon, Łoś und Tarski entwickelt, die in den ersten beiden Fällen worst-case optimal sind. Wichtig hierfür sind Hanf-Normalformen. Es wird gezeigt, dass eine Erweiterung von FO durch unäre Zählquantoren genau dann Hanf-Normalformen erlaubt, wenn alle Zählquantoren ultimativ periodisch sind, und wie Hanf-Normalformen in diesen Fällen in elementarer und worst-case optimaler Zeit konstruiert werden können. Dies führt zu Model-Checking Algorithmen für solche Erweiterungen von FO sowie zu Verallgemeinerungen der Algorithmen für Feferman-Vaught-Zerlegungen und die Erhaltungssätze von Lyndon, Łoś und Tarski. / Normal forms express semantic properties of logics by means of syntactical restrictions. They allow algorithms to benefit from restrictions of the expressive power of a logic. An example is the locality of first-order logic (FO), which implies that properties like reachability or connectivity cannot be defined in FO. Gaifman's local normal form expresses the satisfaction conditions of an FO-formula by a Boolean combination of local statements. Gaifman normal form serves as a first step in fixed-parameter model-checking algorithms, parameterised by the size of the formula, on sparse graph classes. However, it is known that in general, there are non-elementary lower bounds for the costs involved in transforming a formula into Gaifman normal form. This leads to an enormous parameter-dependency of the aforementioned algorithms. Similar non-elementary lower bounds also hold for Feferman-Vaught decompositions and for the preservation theorems by Lyndon, Łoś, and Tarski. This thesis investigates the complexity of these normal forms when restricting attention to classes of structures of bounded degree, for which the non-elementary lower bounds are known to fail. Under this restriction, the thesis provides algorithms with elementary and even worst-case optimal running time for the construction of Gaifman normal form and Feferman-Vaught decompositions. For the preservation theorems, algorithmic versions with elementary running time and non-matching lower bounds are provided. Crucial for these results is the notion of Hanf normal form. It is shown that an extension of FO by unary counting quantifiers allows Hanf normal forms if, and only if, all quantifiers are ultimately periodic, and furthermore, how Hanf normal form can be computed in elementary and worst-case optimal time in these cases. This leads to model-checking algorithms for such extensions of FO and also allows generalisations of the constructions for Feferman-Vaught decompositions and preservation theorems.