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BooleanFunctions
Writtenbyprominentexpertsinthefield,thismonographprovidesthefirstcompre-
hensiveandunifiedpresentationofthestructural,algorithmic,andappliedaspectsof
thetheoryofBooleanfunctions.
ThebookfocusesonalgebraicrepresentationsofBooleanfunctions,especiallydis-
junctiveandconjunctivenormalformrepresentations.Itpresentswithinthisframework
thefundamentalelementsofthetheory(Booleanequationsandsatisfiabilityproblems,
primeimplicantsandassociatedshortrepresentations,dualization),anin-depthstudy
ofspecialclassesofBooleanfunctions(quadratic,Horn,shellable,regular,threshold,
read-oncefunctionsandtheircharacterizationbyfunctionalequations),andtwofruit-
fulgeneralizationsoftheconceptofBooleanfunctions(partiallydefinedfunctionsand
pseudo-Booleanfunctions).Severaltopicsarepresentedhereinbookformforthefirst
time.
Becauseoftheuniquedepthandbreadthoftheunifiedtreatmentthatitprovidesand
itsemphasisonalgorithmsandapplications,thismonographwillhavespecialappeal
for researchers and graduate students in discrete mathematics, operations research,
computerscience,engineering,andeconomics.
Dr.YvesCramaisProfessorofOperationsResearchandProductionManagementand
the former Director General of the HEC Management School of the University of
Liège,Belgium.HeiswidelyrecognizedasaprominentexpertinthefieldofBoolean
functions,combinatorialoptimization,andoperationsresearch,andhehascoauthored
morethanseventypapersandthreebooksonthesesubjects.Dr.Cramaisamemberof
theeditorialboardofDiscreteAppliedMathematics,DiscreteOptimization,Journal
ofScheduling,and4OR–TheQuarterlyJournaloftheBelgian,FrenchandItalian
OperationsResearchSocieties.
The late Peter L. Hammer (1936–2006) was a Professor of Operations Research,
Mathematics,ComputerScience,ManagementScience,andInformationSystemsat
RutgersUniversityandtheDirectoroftheRutgersUniversityCenterforOperations
Research(RUTCOR).Hewasthefounderandeditor-in-chiefofthejournalsAnnals
of Operations Research, Discrete Mathematics, DiscreteApplied Mathematics, Dis-
creteOptimization,andElectronicNotesinDiscreteMathematics.Dr.Hammerwas
theinitiatorofnumerouspioneeringinvestigationsoftheuseofBooleanfunctionsin
operationsresearchandrelatedareas,ofthetheoryofpseudo-Booleanfunctions,and
ofthelogicalanalysisofdata.Hepublishedmorethan240papersand19bookson
thesetopics.
encyclopedia of mathematics and its applications
founding editor g.-c. rota
EditorialBoard
R.Doran,P.Flajolet,M.Ismail,T.-Y.Lam,E.Lutwak
Thetitlesbelow,andearliervolumesintheseries,areavailablefrombooksellersorfrom
CambridgeUniversityPressatwww.cambridge.org.
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.Lions,andJ.HeExactandApproximateControllabilityforDistributed
ParameterSystems
118 A.A.BorovkovandK.A.BorovkovAsymptoticAnalysisofRandomWalks
119 M.DezaandM.DutourSikiriéGeometryofChemicalGraphs
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.TuckerTopicsin
TopologicalGraphTheory
129 J.Berstel,D.Perrin,andC.ReutenauerCodesandAutomata
130 T.G.FaticoniModulesoverEndomorphismRings
131 H.MorimotoStochasticControlandMathematicalModeling
132 G.SchmidtRelationalMathematics
133 P.KornerupandD.W.MatulaFinitePrecisionNumbersSystemsandArithmetic
134 Y.CramaandP.L.HammerBooleanModelsandMethodsinMathematics,Computer
Science,andEngineering
135 V.BerthéandM.RigoCombinatorics,AutomataandNumberTheory
136 A.Kristály,V.D.Raˇdulescu,andC.VargaVariationalPrinciplesinMathematicalPhysics,
Geometry,andEconomics
137 J.BerstelandC.ReutenauerNoncommutativeRationalSerieswithApplications
138 B.CourcelleGraphStructureandMonadicSecond-OrderLogic
139 M.FiedlerMatricesandGraphsinGeometry
140 N.VakilRealAnalysisthroughModernInfinitesimals
141 R.B.ParisHadamardExpansionsandHyperasymptoticEvaluation
Boolean Functions
Theory, Algorithms, and Applications
YVES CRAMA
UniversityofLiège,Belgium
PETER L. HAMMER
cambridge university press
Cambridge,NewYork,Melbourne,Madrid,CapeTown,
Singapore,SãoPaulo,Delhi,Tokyo,MexicoCity
CambridgeUniversityPress
32AvenueoftheAmericas,NewYork,NY10013-2473,USA
www.cambridge.org
Informationonthistitle:www.cambridge.org/9780521847513
©YvesCramaandPeterL.Hammer2011
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished2011
PrintedintheUnitedStatesofAmerica
AcatalogrecordforthispublicationisavailablefromtheBritishLibrary.
LibraryofCongressCataloginginPublicationData
Crama,Yves,1958–
Booleanfunctions/YvesCrama,PeterL.Hammer.
p. cm.–(Encyclopediaofmathematicsanditsapplications)
Includesbibliographicalreferencesandindex.
Contents:Theory,algorithms,andapplications
ISBN978-0-521-84751-3(hardback)
1. Algebraicfunctions. 2. Algebra,Boolean. I. Hammer,P.L.,1936–2006. II. Title.
QA341.C732011
511.3(cid:1)24–dc22 2011009690
ISBN978-0-521-84751-3Hardback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyofURLsfor
externalorthird-partyInternetWebsitesreferredtointhispublicationanddoesnotguaranteethat
anycontentonsuchWebsitesis,orwillremain,accurateorappropriate.
ToEdith,
bywayofapologyforcountlessdays
spentinfrontofthecomputer.
YC
Contents
Contributors pagexiii
Preface xv
Acknowledgments xix
Notations xxi
PartI Foundations
1 Fundamentalconceptsandapplications 3
1.1 Booleanfunctions:Definitionsandexamples 3
1.2 Booleanexpressions 8
1.3 Duality 13
1.4 Normalforms 14
1.5 TransforminganarbitraryexpressionintoaDNF 19
1.6 OrthogonalDNFsandnumberoftruepoints 22
1.7 Implicantsandprimeimplicants 24
1.8 Restrictionsoffunctions,essentialvariables 28
1.9 Geometricinterpretation 31
1.10 MonotoneBooleanfunctions 33
1.11 RecognitionoffunctionalandDNFproperties 40
1.12 OtherrepresentationsofBooleanfunctions 44
1.13 Applications 49
1.14 Exercises 65
2 Booleanequations 67
2.1 Definitionsandapplications 67
2.2 ThecomplexityofBooleanequations:Cook’stheorem 72
2.3 OntheroleofDNFequations 74
2.4 Whatdoesitmeanto“solveaBooleanequation”? 78
2.5 Branchingprocedures 80
2.6 Variableeliminationprocedures 87
vii
viii Contents
2.7 Theconsensusprocedure 92
2.8 Mathematicalprogrammingapproaches 95
2.9 Recenttrendsandalgorithmicperformance 103
2.10 MoreonthecomplexityofBooleanequations 104
2.11 Generalizationsofconsistencytesting 111
2.12 Exercises 121
3 PrimeimplicantsandminimalDNFs 123
PeterL.HammerandAlexanderKogan
3.1 Primeimplicants 123
3.2 Generationofallprimeimplicants 128
3.3 Logicminimization 141
3.4 Extremalandtypicalparametervalues 159
3.5 Exercises 165
4 Dualitytheory 167
YvesCramaandKazuhisaMakino
4.1 Basicpropertiesandapplications 167
4.2 Dualitypropertiesofpositivefunctions 176
4.3 Algorithmicaspects:Thegeneralcase 183
4.4 Algorithmicaspects:Positivefunctions 189
4.5 Exercises 198
PartII SpecialClasses
5 Quadraticfunctions 203
BrunoSimeone
5.1 Basicdefinitionsandproperties 203
5.2 WhyarequadraticBooleanfunctionsimportant? 205
5.3 Specialclassesofquadraticfunctions 207
5.4 QuadraticBooleanfunctionsandgraphs 209
5.5 Reducibilityofcombinatorialproblems
toquadraticequations 218
5.6 Efficientgraph-theoreticalgorithmsforquadraticequations 230
5.7 Quadraticequations:Specialtopics 243
5.8 Primeimplicantsandirredundantforms 250
5.9 Dualizationofquadraticfunctions(Contributedby
OyaEkinKaras¸an) 263
5.10 Exercises 266
6 Hornfunctions 269
EndreBoros
6.1 Basicdefinitionsandproperties 269
6.2 ApplicationsofHornfunctions 273
6.3 FalsepointsofHornfunctions 277
