Table Of ContentBestMasters
Mit “BestMasters” zeichnet Springer die besten Masterarbeiten aus, die an renommi-
erten Hochschulen in Deutschland, Österreich und der Schweiz entstanden sind. Die
mit Höchstnote ausgezeichneten Arbeiten wurden durch Gutachterinnen und Gutachter
zur Veröffentlichung empfohlen und behandeln aktuelle Themen aus unterschiedlichen
Fachgebieten der Naturwissenschaften, Psychologie, Technik und Wirtschaftswis-
senschaften. Die Reihe wendet sich an Personen aus Praxis und Wissenschaft gleicherma-
ßen und soll insbesondere auch dem wissenschaftlichen Nachwuchs Orientierung geben.
Springer awards “BestMasters” to the best master’s theses which have been com-
pleted at renowned Universities in Germany, Austria, and Switzerland. The studies
received highest marks and were recommended for publication by supervisors.
They address current issues from various fields of research in natural sciences,
psychology, technology, and economics. The series addresses practitioners as well
as scientists and, in particular, offers guidance for early stage researchers.
Laura Wirth
Weighted Automata,
Formal Power Series and
Weighted Logic
Laura Wirth
University of Konstanz
Konstanz, Germany
ISSN 2625-3577 ISSN 2625-3615 (electronic)
BestMasters
ISBN 978-3-658-39322-9 ISBN 978-3-658-39323-6 (eBook)
https://doi.org/10.1007/978-3-658-39323-6
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Fachme-
dien Wiesbaden GmbH, part of Springer Nature 2022
This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher,
whether the whole or part of the material is concerned, specifically the rights of translation,
reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any
other physical way, and transmission or information storage and retrieval, electronic adaptation,
computer software, or by similar or dissimilar methodology now known or hereafter developed.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this
publication does not imply, even in the absence of a specific statement, that such names are
exempt from the relevant protective laws and regulations and therefore free for general use.
The publisher, the authors, and the editors are safe to assume that the advice and information in
this book are believed to be true and accurate at the date of publication. Neither the publisher
nor the authors or the editors give a warranty, expressed or implied, with respect to the material
contained herein or for any errors or omissions that may have been made. The publisher remains
neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Responsible Editor: Marija Kojic
This Springer Spektrum imprint is published by the registered company Springer Fachmedien
Wiesbaden GmbH, part of Springer Nature.
The registered company address is: Abraham-Lincoln-Str. 46, 65189 Wiesbaden, Germany
Tomymom,foralwaysbeingbymyside,
andinlovingmemoryofOpaHelmut,thebiggestfanofmystudies.
Acknowledgements
IwouldliketoexpressmydeepestgratitudetomysupervisorProf.Dr.SalmaKuhl-
mann for her patient guidance throughout my studies. Special thanks shall also go
to my second supervisor Prof. Dr. Sven Kosub, whom I could always ask to clarify
computer-scienti(cid:28)caspectsofanykind. Ihadtheadvantageofbeingabletoaskboth
of them for advice and answers to resolve di(cid:30)culties at all times. Theirremarks and
commentshavebeenavaluablehelpandpowerfulencouragement.
Furthermore,IwouldliketothankProf.Dr.ManfredDrosteforourexchangeregarding
Section 2.5. I could ask him to obtain references, and he gave me a hint that (cid:28)nally
resulted in a proof of Theorem 2.5.19, which generalizes the classical result of B(cid:252)chi,
ElgotandTrakhtenbrot.
I owe a particular debt to Dr. Lothar Sebastian Krapp, who calmly supported me in
overcoming all problems that arose during this work. His deep mathematical insight
has given rise to many invaluable remarks and comments, which have essentially im-
provedthepresentationinmanyrespects. Inparticular,Iwishtothankhimfortaking
thetroubletoattentivelyproofreadseveraliterationsofthiswork. Beyondthat,Iam
luckyenoughtohavehimasafriend.
Most warmly, I thank my family (cid:21) my sister Anna-Lena, my parents Bernd and
Manuela, as well as Oma Anne and Oma Ria (cid:21) for their less mathematical but more
loving and emotional support, while I was studying and writing obscure theses that
noneofthemislikelytoeverread.
Finally,IwishtothankmyfellowstudentsCarlEggen,MoritzLink,PatrickMichalski,
including Philipp Huber, with whom I could share my enthusiasm for the subject of
thiswork.
vii
Abstract
A basic concept from Theoretical Computer Science for the speci(cid:28)cation of formal
languagesare(cid:28)niteautomata. Byequippingthestatesandtransitionsofthese(cid:28)nite
automatawithweights,oneobtainsthequantitativemodelofweightedautomata. The
includedweightsmaymodele.g.theamountofresourcesneededfortheexecutionof
atransition,theinvolvedcosts,orthereliabilityofitssuccessfulexecution. Toobtain
a uniform model, the underlying weight structure is usually modeled by an abstract
semiring. Thebehaviorofaweightedautomatonisthenrepresentedbyaformalpower
series. Aformalpowerseriesisde(cid:28)nedasamapassigningtoeachwordoveragiven
alphabet an element of the semiring, i.e. some weight associated with the respective
word.
In this work, we put emphasis on the expressive power of weighted automata. More
precisely, the main objective is to represent the behaviors of weighted automata by
expressively equivalent formalisms. These formalisms include rational operations on
formalpowerseries,linearrepresentationsbymeansofmatrices,andweightedmonadic
second-orderlogic.
To this end, we (cid:28)rst exhibit the classical language-theoretic results of Kleene, B(cid:252)chi,
ElgotandTrakhtenbrot,whichconcentrateontheexpressivepowerof(cid:28)niteautomata.
We further derive a generalized version of the B(cid:252)chi(cid:21)Elgot(cid:21)Trakhtenbrot Theorem
addressing formulas, which may have free variables, whereas the original statement
concernsonlysentences. Thenweusethelanguage-theoreticapproachesandmethods
asstartingpointforourinvestigationswithregardtoformalpowerseries. Weestablish
Sch(cid:252)tzenberger’sextensionofKleene’sTheorem,referredtoasKleene(cid:21)Sch(cid:252)tzenberger
Theorem. Moreover, we introduce a weighted version of monadic second-order logic,
which is due to Droste and Gastin, and analyze its expressive power. By means of
thisweightedlogic,wederiveanextensionoftheB(cid:252)chi(cid:21)Elgot(cid:21)TrakhtenbrotTheorem.
Thus, we point out relations among the di(cid:27)erent speci(cid:28)cation approaches for formal
powerseries. Further,werelatethenotionsandresultsconcerningformalpowerseries
totheirrespectivecounterpartsinLanguageTheory.
Overall, our investigations shed light on the interplay between languages, classical as
wellasweightedautomata,formalpowerseriesandmonadicsecond-orderlogic. Hence,
the topic of this work lies at the interface between Theoretical Computer Science,
AlgebraandLogicor,moregenerally,ModelTheory.
ix
Contents
1. Introduction 1
2. Languages, Automata and Monadic Second-Order Logic 7
2.1. WordsandFormalLanguages . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2. FiniteAutomata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3. Kleene’sTheorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4. MonadicSecond-OrderLogicforWords . . . . . . . . . . . . . . . . . . 26
2.5. TheB(cid:252)chi(cid:21)Elgot(cid:21)TrakhtenbrotTheorem. . . . . . . . . . . . . . . . . . 33
3. Weighted Automata 55
3.1. SemiringsandFormalPowerSeries . . . . . . . . . . . . . . . . . . . . . 56
3.2. WeightedAutomataandTheirBehavior . . . . . . . . . . . . . . . . . . 62
3.3. LinearRepresentations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4. The Kleene(cid:21)Sch(cid:252)tzenberger Theorem 85
4.1. OperationsonFormalPowerSeries . . . . . . . . . . . . . . . . . . . . . 85
4.2. RationalFormalPowerSeries . . . . . . . . . . . . . . . . . . . . . . . . 95
4.3. TheKleene(cid:21)Sch(cid:252)tzenbergerTheorem. . . . . . . . . . . . . . . . . . . . 99
ClosurePropertiesofRecognizableFormalPowerSeries . . . . . . . . . 99
WeightedAutomataandLinearSystemsofEquations . . . . . . . . . . 108
5. Weighted Monadic Second-Order Logic and Weighted Automata 119
5.1. SyntaxandSemantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.2. ResultsofDrosteandGastin . . . . . . . . . . . . . . . . . . . . . . . . 136
FromWeightedFormulastoWeightedAutomata . . . . . . . . . . . . . 139
FromWeightedAutomatatoWeightedFormulas . . . . . . . . . . . . . 146
LocallyFiniteSemirings . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
6. Summary and Further Research 157
6.1. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
6.2. FurtherResearch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
A. Appendix 167
A.1. ModelTheoryandMonadicSecond-OrderLogic . . . . . . . . . . . . . 167
A.2. MonadicSecond-OrderLogicforWords . . . . . . . . . . . . . . . . . . 176
References 181
Index 185
xi
1. Introduction
AfundamentalconcernofComputerScienceisthestudyandprocessingofinformation
anddata. Anyinformationrequirestobespeci(cid:28)ed,anddependingontheapplication
context, a suitable representation or speci(cid:28)cation is chosen in order to interpret the
information in a targeted manner. More precisely, the processing of data, and in
particular the amount of resources required for this, depends on the chosen method
or formal model for the representation of the conveyed information. Especially when
workingwithin(cid:28)niteobjectsorstructures,itisessentialtohavea(cid:28)nitespeci(cid:28)cationof
theconveyedinformation. Inthiswork,wefocusonthedescription-orientedresearch
inTheoreticalComputerScience,ratherthanonalgorithm-orientedresults. However,
thetworesearchareasreallyarehighly,eveninseparably,connected.
Theinformationconveyingstructuresthatareconsideredinthisworkareformallan-
guages and formal power series. Formal languages are sets of words over a given al-
phabet,andtheformalpowerseries,whichconcernus,aremapsassociatingelements
ofanabstractsemiringtowordsoveragivenalphabet. Thus,languagesmodelqual-
itativeinformationconcerningthemembershipofwords,whereasformalpowerseries
provide quantitative data about words. In fact, formal power series can be regarded
asquantitativeextensionsoflanguages.
In Theoretical Computer Science, the basic tool for the speci(cid:28)cation of languages
are (cid:28)nite automata. Historically, (cid:28)nite automata originate in the mid-1950s in the
work of Kleene [18]. In his fundamental result, which is commonly referred to as
Kleene’sTheorem, Kleenecharacterizedthelanguagesthatarerecognizableby(cid:28)nite
automata as rational languages1. Mainly motivated by decidability questions, the
expressiveequivalenceof(cid:28)niteautomataandmonadicsecond-orderlogicwasderived
independentlybyB(cid:252)chi[4],Elgot[15]andTrakhtenbrot2[43]intheearly1960s. Their
equivalence result, referred to as B(cid:252)chi(cid:21)Elgot(cid:21)Trakhtenbrot Theorem, establishes a
veryearlyconnectionbetweenthetheoryof(cid:28)niteautomataandMathematicalLogic.
Thetwoapproachesoftencomplementeachotherinasynergeticway,andtheirrelation
ishighlyrelevantformultipleapplicationdomains,e.g.inveri(cid:28)cationandknowledge
representation, for the design of combinatorial and sequential circuits, as well as in
natural language processing. In Theorem 2.5.19, we further present an extension of
theB(cid:252)chi(cid:21)Elgot(cid:21)TrakhtenbrotTheoremtomonadicsecond-orderformulas,whichmay
havefreevariables,whereastheoriginalresultaddressesonlysentences. Theproofof
1Alanguageiscalledrationalifitcanbeconstructedfrom(cid:28)nitelymany(cid:28)nitelanguagesbyapplying
therationaloperationsunion,concatenationandKleenestar.
2Duetodi(cid:27)erenttransliterationfromtheCyrillic,variousspellingsofthisnamearecommoninLatin
script.
© The Author(s), under exclusive license to
Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2022
L. Wirth, Weighted Automata, Formal Power Series and Weighted Logic,
BestMasters, https://doi.org/10.1007/978-3-658-39323-6_1
