Boolean Models and Methods in Mathematics, Computer Science, and Engineering PDF
Preview Boolean Models and Methods in Mathematics, Computer Science, and Engineering
BooleanModelsandMethodsinMathematics, ComputerScience,andEngineering Thiscollectionofpaperspresentsaseriesofin-depthexaminationsofavarietyofad- vancedtopicsrelatedtoBooleanfunctionsandexpressions.Thechaptersarewrittenby someofthemostprominentexpertsintheirrespectivefieldsandcovertopicsranging fromalgebraandpropositionallogictolearningtheory,cryptography,computational complexity,electricalengineering,andreliabilitytheory.Beyondthediversityofthe questionsraisedandinvestigatedindifferentchapters,aremarkablefeatureofthecol- lectionisthecommonthreadcreatedbythefundamentallanguage,concepts,models, andtoolsprovidedbyBooleantheory.Manyreaderswillbesurprisedtodiscoverthe countlesslinksbetweenseeminglyremotetopicsdiscussedinvariouschaptersofthe book.Thistextwillhelpthemdrawonsuchconnectionstofurthertheirunderstanding oftheirownscientificdisciplineandtoexplorenewavenuesforresearch. Dr.YvesCramaisProfessorofOperationsResearchandProductionManagementand former Dean of the HEC Management School of the University of Lie`ge, Belgium. He is widely recognized as a prominent expert in the field of Boolean functions, combinatorialoptimization,andoperationsresearch,andhehascoauthoredmorethan 70papersand3booksonthesesubjects.Dr.Cramaisamemberoftheeditorialboard ofDiscreteOptimization,JournalofScheduling,and4OR–TheQuarterlyJournalof theBelgian,FrenchandItalianOperationsResearchSocieties. The late Peter L. Hammer (1936–2006) was a Professor of Operations Research, Mathematics,ComputerScience,ManagementScience,andInformationSystemsat RutgersUniversityandtheDirectoroftheRutgersUniversityCenterforOperations Research(RUTCOR).Hewasthefounderandeditor-in-chiefofthejournalsAnnalsof OperationsResearch,DiscreteMathematics,DiscreteAppliedMathematics,Discrete Optimization, and Electronic Notes in Discrete Mathematics. Dr. Hammer was the initiator of numerous pioneering investigations of the use of Boolean functions in 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. 100 E.OlivieriandM.Eula´liaVares LargeDeviationsandMetastability 101 A.Kushner,V.LychaginandV.Rubtsov ContactGeometryandNonlinearDifferential Equations 102 L.W.BeinekeandR.J.Wilson(eds.)withP.J.Cameron TopicsinAlgebraicGraphTheory 103 O.J.Staffans Well-PosedLinearSystems 104 J.M.Lewis,S.LakshmivarahanandS.K.Dhall DynamicDataAssimilation 105 M.Lothaire AppliedCombinatoricsonWords 106 A.Markoe AnalyticTomography 107 P.A.Martin MultipleScattering 108 R.A.Brualdi CombinatorialMatrixClasses 109 J.M.BorweinandJ.D.Vanderwerff ConvexFunctions 110 M.-J.LaiandL.L.Schumaker SplineFunctionsonTriangulations 111 R.T.Curtis SymmetricGenerationofGroups 112 H.Salzmannetal. TheClassicalFields 113 S.PeszatandJ.Zabczyk StochasticPartialDifferentialEquationswithLe´vyNoise 114 J.Beck CombinatorialGames 115 L.BarreiraandY.Pesin NonuniformHyperbolicity 116 D.Z.ArovandH.Dym J-ContractiveMatrixValuedFunctionsandRelatedTopics 117 R.Glowinski,J.-L.LionsandJ.He ExactandApproximateControllabilityforDistributed ParameterSystems 118 A.A.BorovkovandK.A.Borovkov AsymptoticAnalysisofRandomWalks 119 M.DezaandM.DutourSikiric´ GeometryofChemicalGraphs 120 T.Nishiura AbsoluteMeasurableSpaces 121 M.PrestPurity SpectraandLocalisation 122 S.Khrushchev OrthogonalPolynomialsandContinuedFractions 123 H.NagamochiandT.Ibaraki AlgorithmicAspectsofGraphConnectivity 124 F.W.KingHilbert TransformsI 125 F.W.KingHilbert TransformsII 126 O.CalinandD.-C.Chang Sub-RiemannianGeometry 127 M.Grabischetal. AggregationFunctions 128 L.W.BeinekeandR.J.Wilson(eds.)withJ.L.GrossandT.W.Tucker TopicsinTopological GraphTheory 129 J.Berstel,D.PerrinandC.Reutenauer CodesandAutomata 130 T.G.Faticoni ModulesoverEndomorphismRings 131 H.Morimoto StochasticControlandMathematicalModeling 132 G.Schmidt RelationalMathematics 133 P.KornerupandD.W.Matula FinitePrecisionNumbersSystemsandArithmetic encyclopedia of mathematics and its applications Boolean Models and Methods in Mathematics, Computer Science, and Engineering Editedby YVES CRAMA Universite´deLie`ge PETER L. HAMMER cambridgeuniversitypress Cambridge,NewYork,Melbourne,Madrid,CapeTown,Singapore, Sa˜oPaulo,Delhi,Dubai,Tokyo,MexicoCity CambridgeUniversityPress 32AvenueoftheAmericas,NewYork,NY10013-2473,USA www.cambridge.org Informationonthistitle:www.cambridge.org/9780521847520 (cid:2)c YvesCramaandPeterL.Hammer2010 Thispublicationisincopyright.Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithoutthewritten permissionofCambridgeUniversityPress. Firstpublished2010 PrintedintheUnitedStatesofAmerica AcatalogrecordforthispublicationisavailablefromtheBritishLibrary. LibraryofCongressCataloginginPublicationdata Booleanmodelsandmethodsinmathematics,computerscience,andengineering/ editedbyYvesCrama,PeterL.Hammer. p. cm.–(Encyclopediaofmathematicsanditsapplications;134) Includesbibliographicalreferencesandindex. ISBN978-0-521-84752-0 1.Algebra,Boolean. 2.Probabilities. I.Crama,Yves,1958– II.Hammer,P.L.,1936– III.Title. IV.Series. QA10.3.B658 2010 511.3(cid:2)24–dc22 2010017816 ISBN978-0-521-84752-0Hardback CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyofURLsfor externalorthird-partyInternetWebsitesreferredtointhispublicationanddoesnot guaranteethatanycontentonsuchWebsitesis,orwillremain,accurateorappropriate. Contents Preface pagevii Introduction ix Acknowledgments xiii Contributors xv AcronymsandAbbreviations xvii PartI AlgebraicStructures 1 CompositionsandClonesofBooleanFunctions 3 ReinhardPo¨schelandIvoRosenberg 2 DecompositionofBooleanFunctions 39 JanC.Bioch PartII Logic 3 ProofTheory 79 AlasdairUrquhart 4 ProbabilisticAnalysisofSatisfiabilityAlgorithms 99 JohnFranco 5 OptimizationMethodsinLogic 160 JohnHooker PartIII LearningTheoryandCryptography 6 ProbabilisticLearningandBooleanFunctions 197 MartinAnthony 7 LearningBooleanFunctionswithQueries 221 RobertH.Sloan,Bala´zsSzo¨re´nyi,andGyo¨rgyTura´n v vi Contents 8 BooleanFunctionsforCryptographyand Error-CorrectingCodes 257 ClaudeCarlet 9 VectorialBooleanFunctionsforCryptography 398 ClaudeCarlet PartIV GraphRepresentationsandEfficient ComputationModels 10 BinaryDecisionDiagrams 473 BeateBollig,MartinSauerhoff,DetlefSieling,andIngoWegener 11 CircuitComplexity 506 MatthiasKrauseandIngoWegener 12 FourierTransformsandThresholdCircuitComplexity 531 JehoshuaBruck 13 NeuralNetworksandBooleanFunctions 554 MartinAnthony 14 DecisionListsandRelatedClassesofBooleanFunctions 577 MartinAnthony PartV ApplicationsinEngineering 15 HardwareEquivalenceandPropertyVerification 599 J.-H.RolandJiangandTizianoVilla 16 SynthesisofMultilevelBooleanNetworks 675 TizianoVilla,RobertK.Brayton,andAlbertoL. Sangiovanni-Vincentelli 17 BooleanAspectsofNetworkReliability 723 CharlesJ.Colbourn Preface Boolean models and methods play afundamental roleintheanalysis of abroad diversityofsituationsencounteredinvariousbranchesofscience. The objective of this collection of papers is to highlight the role of Boolean theoryinanumberofsuchareas,rangingfromalgebraandpropositionallogicto learning theory, cryptography, computational complexity, electrical engineering, andreliabilitytheory. Thechaptersarewrittenbysomeofthemostprominentexpertsintheirfields and are intended for advanced undergraduate or graduate students, as well as for researchers or engineers. Each chapter provides an introduction to the main questions investigated in a particular field of science, as well as an in-depth discussionofselectedissuesandasurveyofnumerousimportantorrepresentative results.Assuch,thecollectioncanbeusedinavarietyofways:somereadersmay simply skim some of the chapters in order to get the flavor of unfamiliar areas, whereasothersmayrelyonthemasauthoritativereferencesorasextensivesurveys offundamentalresults. Beyondthediversityofthequestionsraisedandinvestigatedindifferentchap- ters,aremarkablefeatureofthecollectionisthepresenceofan“Ariane’sthread” createdbythecommonlanguage,concepts,models,andtoolsofBooleantheory. Manyreaderswillcertainlybesurprisedtodiscovercountlesslinksbetweenseem- inglyremotetopicsdiscussedinvariouschaptersofthebook.Itishopedthatthey will be able to draw on such connections to further their understanding of their ownscientificdisciplinesandtoexplorenewavenuesforresearch. Thecollectionintendstobeausefulcompanionandcomplementtothemono- graph by Yves Crama and Peter L. Hammer, Boolean Functions: Theory, Algo- rithms, and Applications. Cambridge University Press, Cambridge, U.K., 2010, which provides the basic concepts and theoretical background for much of the materialhandledhere. vii Introduction The first part of the book, “Algebraic Structures,” deals with compositions and decompositionsofBooleanfunctions. Aset F ofBoolean functions iscalledcomplete ifevery Boolean function is a composition of functions from F; it is aclone if it is composition-closed and containsallprojections.In1921,E.L.Postfoundacompleteness criterion,that is, a necessary and sufficient condition for a set F of Boolean functions to be complete.Twentyyearslater,hegaveafulldescriptionofthelatticeofBoolean clones.Chapter1,byReinhardPo¨schelandIvoRosenberg,providesanaccessible andself-containeddiscussionof“CompositionsandClonesofBooleanFunctions” andoftheclassicalresultsofPost. Functional decomposition of Boolean functions was introduced in switching theory in the late 1950s. In Chapter 2, “Decomposition of Boolean Functions,” JanC.Biochproposesaunifiedtreatmentofthistopic.Thechaptercontainsboth apresentationofthemainstructuralpropertiesofmodulardecompositionsanda discussionofthealgorithmicaspectsofdecomposition. PartIIofthecollectioncoverstopicsinlogic,whereBooleanmodelsfindtheir historicalroots. In Chapter 3, “Proof Theory,” Alasdair Urquhart briefly describes the more important proof systems for propositional logic, including a discussion of equa- tionalcalculus,ofaxiomaticproofsystems,andofsequentcalculusandresolution proofs.Theauthorcomparestherelativecomputationalefficiencyofthesediffer- ent systems and concludes with a presentation of Haken’s classical result that resolutionproofshaveexponentiallengthforcertainfamiliesofformulas. The issue of the complexity of proof systems is further investigated by John FrancoinChapter4,“ProbabilisticAnalysisofSatisfiabilityAlgorithms.”Central questions addressed in this chapter are: How efficient is a particular algorithm whenappliedtoarandomsatisfiabilityinstance?Andwhatdistinguishes“hard” from “easy” instances? Franco provides a thorough analysis of these questions, startingwithapresentationofthebasicprobabilistictoolsandmodelsandcovering advancedresultsbasedonabroadrangeofapproaches. ix
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