Weighted Automata, Formal Power Series and Weighted Logic

The main objective of this work is to represent the behaviors of weighted automata by expressively equivalent formalisms: rational operations on formal power series, linear representations by means of matrices, and weighted monadic second-order logic.
First, we exhibit the classical results of Kleene, Büchi, Elgot and Trakhtenbrot, which concentrate on the expressive power of finite automata. We further derive a generalization of the BüchiElgotTrakhtenbrot Theorem addressing formulas, whereas the original statement concerns only sentences. Then we use the language-theoretic methods as starting point for our investigations regarding power series. We establish Schützenberger's extension of Kleene's Theorem, referred to as KleeneSchützenberger Theorem. Moreover, we introduce a weighted version of monadic second-order logic, which is due to Droste and Gastin. By means of this weighted logic, we derive an extension of the BüchiElgotTrakhtenbrot Theorem. Thus, we point out relations among the different specification approaches for formal power series. Further, we relate the notions and results concerning power series to their counterparts in Language Theory.
Overall, our investigations shed light on the interplay between languages, formal power series, automata and monadic second-order logic.

Autorentext

Laura Wirth completed her Master's thesis in Mathematics at the University of Konstanz in 2022. It was supervised by Prof. Salma Kuhlmann as well as Prof. Sven Kosub and received the highest grade with honors.

Inhalt
Introduction.- Languages, Automata and Monadic Second-Order Logic.- Weighted Automata.- The KleeneSchützenberger Theorem.- Weighted Monadic Second-Order Logic and Weighted Automata.- Summary and Further Research.