Handbook of Automata Theory Volume I Theoretical Foundations

898 Pages·2021·9.734 MB·English

by  Jean-Éric Pin

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H andbook of
Automata Theory
Volume I
Theoretical Foundations
Edited by Jean-Éric Pin
Editor:
Jean-Éric Pin
Institut de Recherche en Informatique Fondamentale (IRIF)
Université de Paris and CNRS
Bâtiment Sophie Germain, Case courier 7014
8 Place Aurélie Nemours
75205 Paris Cedex 13
E-mail: [email protected]
Volume I:
2020 Mathematics Subject Classification: 68Q45; 03B50, 03D05, 08A70, 08B20, 15A80, 16Y60,
20E18, 20M05, 20M07, 20M35, 28A05, 68Q32, 68Q42, 68Q70, 68R10, 68T05
Keywords: finite automata, Hopcroft’s algorithm, regular languages, regular expressions, finite
transducers, weighted automata, automata on infinite words, automata on trees, picture l anguages,
algorithmic learning, descriptional complexity, Boolean circuits, synchronizing automata, road
colouring problem, varieties of languages, profinite topology, descriptive set theory, equational
theories, language equations, forest algebras
ISBN Vol. I 978-3-98547-002-0
ISBN Vol. II 978-3-98547-003-7
ISBN Set 978-3-98547-006-8 (set of both volumes)
Bibliographic information published by the Deutsche Nationalbibliothek
The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie;
detailed bibliographic data are available on the Internet at http://dnb.dnb.de.
Published by EMS Press, an imprint of the
European Mathematical Society – EMS – Publishing House GmbH
Institut für Mathematik
Technische Universität Berlin
Straße des 17. Juni 136
10623 Berlin, Germany
https://ems.press
2021 EMS Press
©
Cover drawing of Jacques de Vaucanson’s digesting duck (canard digérant) published in
Scientific American Vol. 80 (3), 1899. Fractal tree on the first page by Nicolas Janey.
Typeset using the authors’ LaTeX sources: Marco Zunino, Savona, Italy
Printing and binding: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany
Printed on acid free paper
♾
9 8 7 6 5 4 3 2 1
Dedicatedtothememoryof
ProfessorZoltánÉsik
(1951–2016)

Preface
The Handbook of Automata Theory has its origins in the research programme Au-
toMathA (Automata:from Mathematics to Applications, 2005–2010), a multidisci-
plinary programme of the European Science Foundation at the crossroads of mathe-
matics,theoreticalcomputerscience,andapplications.Itisdesignedtoprovideabroad
audienceofresearchersandstudentsinmathematicsandcomputersciencewithacom-
prehensiveoverviewofresearchinautomatatheory.
Automata theory is one of the longest established areas in computer science. It
wasbornoversixtyyearsago,withtheseminalworkofKleene,whofirst formalised
the early attemptsof McCullochandPitts, andwas originallymotivatedbythe study
ofneuralnetworks. Formanyyears,itsmainapplicationshavebeencomputerdesign,
compilationofprogramminglanguages,andpatternmatching.Butoverthelasttwenty
years,applicationsofautomatatheoryhaveconsiderablydiversified,andnowinclude
verificationmethodstocopewithsuchemergingtechnicalneedsasnetworksecurity,
mobileintelligentdevices,andhighperformancecomputing.
Atthesame time,themathematicalfoundationsofautomatatheoryrely on more
and more advanced parts of mathematics. While only elementary graph theory and
combinatorics were required in the early sixties, new tools from non-commutative
algebra(semigroups,semiringsandformalpowerseries),logic,probabilitytheory,and
symbolic dynamics have been successively introduced, and the latest developments
borrowideasfromtopologyandgeometry.
Itwastimetogatherthesemathematicaladvancesandtheirnumerousapplications
in a reference book. The Handbook of Automata Theory is intended to serve this
purpose.Itcomprisesthirty-ninechapters,presentedintwovolumes:
VolumeI: Theoreticalfoundations
VolumeII: Automatainmathematicsandselectedapplications
Together, the two volumes cover most of the topics related to automata. Volume I
presents, in the first part, the basic models of the theory: finite automata working
on finite words, infinite words, finite trees and infinite trees, transducers, weighted
automata and max-plus automata, and two-dimensional models. In the second part,
complexity and algorithmic issues are discussed extensively, including connections
with circuit complexity and finite model theory. In the third part, the algebraic and
topologicalaspectsofautomatatheoryaretreated.
VolumeII first offers a wide range of connectionsbetween automataand mathe-
matics, includinggroup theory, numbertheory, symbolicdynamics, finitemodelthe-
ory, and fractal-type images. Secondly, selected applications are covered, including
viii Preface
message-passingsystems,symbolicmethods,synthesis,timedautomatonmodel,veri-
ficationofhigher-orderprograms,analysisofprobabilisticprocesses,naturallanguage
processing,formalverificationofprograms,andquantumcomputing.
Much of this material had never been published in a book before, making the
Handbookauniquereferenceinautomatatheory.
DuetothelengthoftheHandbook,thechaptersaredividedintotwovolumes.For
the convenienceof the reader, the front matterand the index appear in both volumes
(paginatedwithromannumerals).
Asthisprojectstartedovertenyearsago,somerecentdevelopmentsmaynothave
been addressed. Nevertheless, the reader will be able to find updates and possible
correctionson
https://ems.press/isbn/978-3-98547-006-8
Acknowledgements. IwouldliketothanktheEuropeanScienceFoundation,andin
particular the Standing Committee for Physical & Engineering Sciences (PESC), for
funding the research programme AutoMathA within their Research Networking Pro-
gramme(2005–2010).TheHandbookwouldnothavebeenpossiblewithouttheirmoral
andfinancialsupport.TheprogrammeAutoMathAbroughttogetheraresearchcommu-
nityofwidescope;itsjointworkandeffortshavebeenvitalforcomposingthepresent
handbook. The AutoMathA project was initially launched by Jorge Almeida (Lis-
boa),StefanoCrespiReghizzi(Milano)andmyself.Letmealsothanktheothermem-
bers of the AutoMathA Steering Committee: Jacques Duparc, Jozef Gruska, Juhani
Karhumäki,MikołajBojańczyk,SørenEilers, StuartW.Margolis,TatianaJajcayova,
VéroniqueBruyère,WernerKuich,WolfgangThomas,andZoltánÉsik.Sadlyandun-
expectedly,ZoltánÉsik passedawayduringthe finalstagesoftheHandbookproject.
WededicatetheHandbooktothememoryofthisgreatscientistandfriend.Theconstant
supportoftheAutoMathASteeringCommitteeduringthepreparationofthishandbook
was an invaluable help. Narad Rampersad’s assistance during the early stage of the
Handbookwasalsoparticularlyappreciated.
All the authors are particularly indebted to Jeffrey Shallit. As one of the few
nativeEnglish-speakingauthorsofthebook,Jeffreyhasacceptedthedauntingtaskof
reviewingallthechaptersintheirentirety.Henotonlydetectedaconsiderablenumber
of English mistakes, buthe also greatly improvedthe style and mathematicalcontent
ofthechapters.Iwouldthereforeliketoexpressmydeepestthankstohim.
TheadvisoryboardconsistingofSørenEilers(Copenhagen)andWolfgangThomas
(Aachen)wasinstrumentalindefiningtheearlyversionoftheHandbookproject.Iam
particularlyindebtedtoWolfgangThomasforhisadviceandconstantencouragement
andhelpduringthelonggestationperiodofthishandbook.
Ofcourse,thishandbookwouldnothavebeenpossiblewithouttheauthorsofthe
thirty-nine chapters. I would like to thank them all for their high quality scientific
contributionandtheirperseveranceduringthechapterreviewprocess.
Preface ix
For their patience and extreme care in the production of the Handbook, I would
liketothankthetypesetterMarcoZuninoandallthepeopleofEMSPressIhavebeen
working with: Apostolos Damialis, Sylvia Fellmann, Thomas Hintermann, Manfred
Karbe,VeraSpillner,andSimonWinter. SpecialthankstoNicolasJaneywhokindly
designedthefractaltreeonthefirstpage.
Jean-ÉricPin
Managingeditor
Paris,2021