Weighted Automata and Weighted Logics over Tree-like Structures

In theoretical computer science the connection between automata and logic is of fundamental importance. This connection was first considered by Büchi and Elgot in the 1960s, when they showed that the languages accepted by finite automata are precisely those languages that can be defined in monadic second-order logic (MSO). In this thesis we consider extensions of Büchi’s and Elgot’s theorem into two directions. First, we consider classes of objects which are more general than words and carry a tree-like structure. Second, we consider quantitative aspects and investigate weighted automata operating on these structures. The study of weighted automata goes back to the work of Schützenberger. He equipped the transitions of an automaton additionally with a weight and studied the behavior of such a device which is now a formal power series, i.e. a mapping assigning to a word an element of a semiring. A semiring is the algebraic structure that carries the weights. For example, the natural numbers form a semiring, but also the probabilistic semiring given by the interval [0, 1] together with the max-operator and the usual multiplication is a semiring. This thesis investigates different weighted automaton models over tree-like structures such as texts, nested words and hedges, which have already been considered in the literature. We characterize all automaton models logically. This is achieved by considering suitable adaptions of weighted logics. The formalism of weighted logics was introduced by Droste and Gastin [1] in 2005 and provides an extension of classical MSO which is now enriched with values from a semiring in order to add quantitative expressiveness. Since already for words the full weighted MSO is expressively stronger than weighted automata, we restrict the consideration to a syntactically defined fragment called sRMSO which was proposed in [2]. Now, rather than proving for each class of structures a characterization on its own, for instance by an induction over the structure of formulae, we use a translation technique and reduce the results to ones that have already been shown. This admits the advantage that it gives insight into the similarities of the different structures. More importantly, by using the translation technique we get decidability results for the emptiness and equivalence problems from corresponding results for trees. For the case of hedges and nested words the automaton models are straightforward generalizations of the unweighted models which had already been presented in the literature. For texts, however, no automaton model had been considered so far and the model of weighted