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 Büchi–Elgot–Trakhtenbrot 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 Kleene–Schützenberger 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 Büchi–Elgot–Trakhtenbrot 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.

weighted automata
formal power series
weighted logic
MSO
monadic second-order logic
Schützenberger
Droste and Gastin
Büchi, Elgot, Trakhtenbrot

• Front Matter
• Introduction
• Languages, Automata and Monadic Second-Order Logic
• Weighted Automata
• The Kleene–Schützenberger Theorem
• Weighted Monadic Second-Order Logic and Weighted Automata
• Summary and Further Research
• Back Matter

Konstanz, Germany
Laura Wirth

Book Title
Weighted Automata, Formal Power Series and Weighted Logic

Authors
Laura Wirth

Series Title
BestMasters

DOI
https://doi.org/10.1007/978-3-658-39323-6

Softcover ISBN
978-3-658-39322-9
Published: 14 October 2022

eBook ISBN
978-3-658-39323-6
Published: 13 October 2022

Series ISSN
2625-3577

Series E-ISSN
2625-3615

Edition Number
1

Number of Pages
XI, 190

Number of Illustrations
63 b/w illustrations

Topics
Computational Mathematics and Numerical Analysis, Computational Science and Engineering