Table Of Contentmore information – www.cambridge.org/9780521898331
GRAPHSTRUCTUREAND
MONADICSECOND-ORDERLOGIC
ALanguage-TheoreticApproach
The study of graph structure has advanced significantly in recent years: finite graphs
cannowbedescribedalgebraically, enablingthemtobeconstructedoutofmorebasic
elements.Onecanobtainalgebraiccharacterizationsoftree-widthandclique-width,two
graph complexity measures that are important for the construction of polynomial-time
graphalgorithms.Separatelythepropertiesofgraphscanbestudiedinalogicallanguage
calledmonadicsecond-orderlogic.Inthisbook,thesetwofeaturesofgraphstructureare
broughttogetherforthefirsttimeinapresentationthatunifiesandsynthesizesresearch
overthelast25years.Theauthorsnotonlyprovideathoroughdescriptionofthetheory,
butalsodetailitsapplications,ontheonehandtotheconstructionofgraphalgorithms,and
ontheothertotheextensionofformallanguagetheorytofinitegraphs.Thisextension
combines algebraic notions (equational and recognizable sets) and logical ones (graph
transformationsspecifiedbylogicalformulas).Applicationsofthesetoolstolanguages
ofwordsandtermsarealsopresented.
Consequentlythebookwillbeofinteresttograduatestudentsandresearchersingraph
theory,finitemodeltheory,formallanguagetheoryandcomplexitytheory.
EncyclopediaofMathematicsandItsApplications
Thisseriesisdevotedtosignificanttopicsorthemesthathavewideapplicationinmath-
ematics or mathematical science and for which a detailed development of the abstract
theoryislessimportantthanathoroughandconcreteexplorationoftheimplicationsand
applications.
BooksintheEncyclopediaofMathematicsandItsApplicationscovertheirsubjects
comprehensively.Lessimportantresultsmaybesummarizedasexercisesattheendsof
chapters.Fortechnicalities,readerscanbereferredtothebibliography,whichisexpected
to be comprehensive. As a result, volumes are encyclopedic references or manageable
guidestomajorsubjects.
encyclopedia of mathematics and its applications
Allthetitleslistedbelowcanbeobtainedfromgoodbooksellersorfrom
CambridgeUniversityPress.
Foracompleteserieslistingvisitwww.cambridge.org/mathematics.
94 S.R.FinchMathematicalConstants
95 Y.JabriTheMountainPassTheorem
96 G.GasperandM.RahmanBasicHypergeometricSeries,2ndedn
97 M.C.PedicchioandW.Tholen(eds.)CategoricalFoundations
98 M.E.H.IsmailClassicalandQuantumOrthogonalPolynomialsinOneVariable
99 T.MoraSolvingPolynomialEquationSystemsII
100 E.OlivieriandM.EuláliaVaresLargeDeviationsandMetastability
101 A.Kushner,V.LychaginandV.RubtsovContactGeometryandNonlinearDifferentialEquations
102 L.W.BeinekeandR.J.Wilson(eds.)withP.J.CameronTopicsinAlgebraicGraphTheory
103 O.J.StaffansWell-PosedLinearSystems
104 J.M.Lewis,S.LakshmivarahanandS.K.DhallDynamicDataAssimilation
105 M.LothaireAppliedCombinatoricsonWords
106 A.MarkoeAnalyticTomography
107 P.A.MartinMultipleScattering
108 R.A.BrualdiCombinatorialMatrixClasses
109 J.M.BorweinandJ.D.VanderwerffConvexFunctions
110 M.-J.LaiandL.L.SchumakerSplineFunctionsonTriangulations
111 R.T.CurtisSymmetricGenerationofGroups
112 H.Salzmannetal.TheClassicalFields
113 S.PeszatandJ.ZabczykStochasticPartialDifferentialEquationswithLévyNoise
114 J.BeckCombinatorialGames
115 L.BarreiraandY.PesinNonuniformHyperbolicity
116 D.Z.ArovandH.DymJ-ContractiveMatrixValuedFunctionsandRelatedTopics
117 R.Glowinski,J.-L.LionsandJ.HeExactandApproximateControllabilityforDistributedParameterSystems
118 A.A.BorovkovandK.A.BorovkovAsymptoticAnalysisofRandomWalks
119 M.DezaandM.DutourSikiric´GeometryofChemicalGraphs
120 T.NishiuraAbsoluteMeasurableSpaces
121 M.PrestPurity,SpectraandLocalisation
122 S.KhrushchevOrthogonalPolynomialsandContinuedFractions
123 H.NagamochiandT.IbarakiAlgorithmicAspectsofGraphConnectivity
124 F.W.KingHilbertTransformsI
125 F.W.KingHilbertTransformsII
126 O.CalinandD.-C.ChangSub-RiemannianGeometry
127 M.Grabischetal.AggregationFunctions
128 L.W.BeinekeandR.J.Wilson(eds.)withJ.L.GrossandT.W.TuckerTopicsinTopologicalGraphTheory
129 J.Berstel,D.PerrinandC.ReutenauerCodesandAutomata
130 T.G.FaticoniModulesoverEndomorphismRings
131 H.MorimotoStochasticControlandMathematicalModeling
132 G.SchmidtRelationalMathematics
133 P.KornerupandD.W.MatulaFinitePrecisionNumberSystemsandArithmetic
134 Y. Crama and P. L. Hammer (eds.) Boolean Models and Methods in Mathematics, Computer Science,
andEngineering
135 V.BerthéandM.Rigo(eds.)Combinatorics,AutomataandNumberTheory
136 A. Kristály, V. D. Ra˘dulescu and C.Varga Variational Principles in Mathematical Physics, Geometry,
andEconomics
137 J.BerstelandC.ReutenauerNoncommutativeRationalSerieswithApplications
138 B.CourcelleandJ.EngelfrietGraphStructureandMonadicSecond-OrderLogic
139 M.FiedlerMatricesandGraphsinGeometry
140 N.VakilRealAnalysisthroughModernInfinitesimals
141 R.B.ParisHadamardExpansionsandHyperasymptoticEvaluation
142 Y.CramaandP.L.HammerBooleanFunctions
143 A.Arapostathis,V.S.BorkarandM.K.GhoshErgodicControlofDiffusionProcesses
144 N.Caspard,B.LeclercandB.MonjardetFiniteOrderedSets
145 D.Z.ArovandH.DymBitangentialDirectandInverseProblemsforSystemsofIntegralandDifferential
Equations
146 G.DassiosEllipsoidalHarmonics
encyclopedia of mathematics and its applications
Graph Structure and
Monadic Second-Order Logic
ALanguage-TheoreticApproach
BRUNO COURCELLE
UniversitédeBordeaux
JOOST ENGELFRIET
UniversiteitLeiden
CAMBRIDGEUNIVERSITYPRESS
Cambridge,NewYork,Melbourne,Madrid,CapeTown,
Singapore,SãoPaulo,Delhi,MexicoCity
CambridgeUniversityPress
TheEdinburghBuilding,CambridgeCB28RU,UK
PublishedintheUnitedStatesofAmericabyCambridgeUniversityPress,NewYork
www.cambridge.org
Informationonthistitle:www.cambridge.org/9780521898331
©B.CourcelleandJ.Engelfriet2012
Foreword©M.Nivat2012
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished2012
PrintedintheUnitedKingdomattheUniversityPress,Cambridge
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary
LibraryofCongressCataloguinginPublicationdata
Courcelle,B.
Graphstructureandmonadicsecond-orderlogic:alanguage-theoreticapproach/BrunoCourcelle,
JoostEngelfriet.
p. cm.–(Encyclopediaofmathematicsanditsapplications;138)
Includesbibliographicalreferencesandindex.
ISBN978-0-521-89833-1(hardback)
1. Logic,Symbolicandmathematical–Graphicmethods. I. Engelfriet,Joost. II. Title.
QA9.C748 2012
511.3–dc23 2012008159
ISBN978-0-521-89833-1Hardback
Additionalresourcesforthispublicationatwww.labri.fr/perso/courcell/TheBook.html
CambridgeUniversityPresshasnoresponsibilityforthepersistenceor
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ToDominiqueandLouco
Contents
ForewordbyMauriceNivat page xi
Introduction 1
1 Overview 16
1.1 Context-freegrammars 17
1.2 Inductivesetsofpropertiesandrecognizability 28
1.3 Monadicsecond-orderlogic 40
1.4 Twographalgebras 46
1.5 Fixed-parametertractability 53
1.6 Decidabilityofmonadicsecond-orderlogic 56
1.7 Graphtransductions 58
1.8 Monadicsecond-orderlogicwithedgesetquantifications 68
1.9 Relationalstructures 74
1.10 References 78
2 Graphalgebrasandwidthsofgraphs 80
2.1 Algebrasandterms 81
2.2 Graphs 87
2.3 TheHRalgebraofgraphswithsources 99
2.4 Tree-decompositions 121
2.5 TheVRalgebraofsimplegraphswithports 144
2.6 Many-sortedgraphalgebras 176
2.7 References 185
3 Equationalandrecognizablesetsinmany-sortedalgebras 188
3.1 Theequationalsetsofanalgebra 189
3.2 Transformationsofequationsystems 206
3.3 Intermezzoonautomata 221
3.4 Therecognizablesetsofanalgebra 227
3.5 References 259
viii Contents
4 Equationalandrecognizablesetsofgraphs 260
4.1 HR-equationalsetsofgraphs 261
4.2 HR-recognizablesetsofgraphs 281
4.3 VR-equationalsetsofsimplegraphs 292
4.4 VR-recognizablesetsofsimplegraphs 305
4.5 HR-andVR-equationalandrecognizablesets 312
4.6 References 313
5 Monadicsecond-orderlogic 315
5.1 Relationalstructuresandlogicallanguages 315
5.2 Graphpropertiesexpressibleinmonadic
second-orderlogic 331
5.3 Monadicsecond-orderlogicandrecognizability 358
5.4 Decidablemonadicsecond-ordertheories 408
5.5 Logicalcharacterizationofrecognizability 409
5.6 Equivalencesoflogicalformulas 416
5.7 References 425
6 Algorithmicapplications 427
6.1 Fixed-parametertractablealgorithmsformodel-checking 428
6.2 Decompositionandparsingalgorithms 433
6.3 Monadicsecond-orderformulascompiledintofiniteautomata 439
6.4 Othermonadicsecond-orderproblemssolvedwithautomata 493
6.5 References 503
7 Monadicsecond-ordertransductions 505
7.1 Definitionsandbasicproperties 506
7.2 TheEquationalityTheoremfortheVRalgebra 534
7.3 Graphtransductionsusingincidencegraphs 555
7.4 TheEquationalityTheoremfortheHRalgebra 559
7.5 Decidabilityofmonadicsecond-ordersatisfiabilityproblems 566
7.6 Questionsaboutlogicalcharacterizationsofrecognizability 574
7.7 References 576
8 Transductionsoftermsandwords 578
8.1 Terminology 580
8.2 Tree-walkingtransducers 585
8.3 Thebasiccharacterization 591
8.4 Fromjumpingtowalking 593
8.5 Fromglobaltolocaltests 595
8.6 Multibottom-uptree-to-wordtransducers 604
8.7 Attributegrammarsandmacrotreetransducers 610
8.8 Nondeterminism 613
8.9 VR-equationalsetsoftermsandwords 614
8.10 References 618
