讲稿数据库七讲lecture073

Functional-DependencyFunctional-DependencyWenowconsidertheformaltheorythattellsuswhichfunctionaldependenciesareimpliedlogicallybyagivensetoffunctionalWethendevelopalgorithmstogeneratelosslessdecompositionsintoBCNFand3NFandhighernormalform.ClosureofaSetClosureofaSetofFunctionalGivenasetFsetoffunctionaldependencies,therearecertainotherfunctionaldependenciesthatarelogicallyimpliedbyF.IfABBForthenweinferthatAThesetofallfunctionaldependencieslogicallyimpliedbyFistheclosureofF,wedenotetheclosureofFbyF+.ClosureofaSetofFunctionalWeClosureofaSetofFunctionalWecanfindF+,theclosureofF,byArmstrong’s,then(reflexivity自反律(augmentation增广律if,thenif,then,(transitivity传递律Theserulessound(donotgenerateanyincorrectfunctionaldependencies)complete(generateallfunctionaldependenciesthathold).RR=(A,B,C,G,H,I)F={ABACCGHCGIBsomemembersofAHbytransitivityfromABandBAGbyaugmentingACwithG,togetAGandthentransitivitywithCGCGbyaugmentingCGItoinferCGandaugmentingofCGHtoinferCGIandthenProcedureforComputingProcedureforComputingTocomputetheclosureofasetoffunctionaldependenciesF+=foreachfunctionaldependencyfinapplyreflexivityandaugmentationrulesonaddtheresultingfunctionaldependenciesto+foreachpairoffunctionaldependenciesf1andf2inFiff1andf2canbecombinedusingthenaddtheresultingfunctionaldependencytoFuntilF+doesnotchangeanyWeshallseeanalternativeprocedureforthisClosureofFunctionalDependenciesAdditionalClosureofFunctionalDependenciesAdditionalIfIfholdsandthenholds,thenholdsandIfholdsandholds,thenTheaboverulescanbeinferredfromArmstrong’s已知关系模式RU={A已知关系模式RU={A，B，C，D,E,F={AB→C,C→A,BC→D,D→EG,BE→C,CG→BD,判断BD→AC是否为F逻辑蕴解：由D→EG知…又知BE→C，C→A所以BE→A,BE→AC②由①、②知，BD→AC，所以BD→AC被F所涵88.4.2Closureof8.4.2ClosureofAttributeGivenasetofattributesdefinetheclosureofunderF(denotedby+)asthesetofattributesthatarefunctionallydeterminedbyunderFAlgorithmtocompute+,theclosureofresult:=while(changestoresult)doforeachinFdoFifresultthenresult:=resultExampleofAttributeExampleofAttributeSetR=(A,B,C,G,H,F={AACCGHCGIBH}1.result=(ACandA2.result=(CGHandCGAGBC)(CGIandCGAGBCH)3.result=4.result=已知关系模式中：已知关系模式中：求（AB）F+。example：设R={A,example：设R={A,B,C},F={A→B,B→C}+X分别为A,B,C时求。解：当X=A时XF=ABC当X=B时，XF+=BC当X=C时，XF+=Cexample：example：已知关系模式中R={A，B，C，D,E,F={AB→C,C→A,BC→D,D→EG,BE→C,CG→BD,求结论WhatWhatwecandowiththeattributeUsesUsesofAttributeSetIsAGacandidateIsAGasuperDoesAGR?==Is(AG)+IsanysubsetofAGaDoesAR?==Is(A)+DoesGR?==Is(G)+TestingforTotestifisasuperkey,wecompute+,andcheck+containsallattributesof已知关系模式中:F={AC→B，BC→D，A→BE，E→GC}求AB,BC,AC是否为关系R的侯选码UsesUsesofAttributeThereareseveralusesoftheattributeclosureTestingfunctionalTocheckifafunctionaldependencyholds(or,inotherwords,isinF+),justcheckif+.Thatis,wecompute+byusingattributeclosure,andthencheckifitcontains.Isasimpleandcheaptest,andveryexample：已知关系模式中example：已知关系模式中R={A，B，C，D,E,F={AB→C,C→D,BC→D,D→EG,BE→C,CG→BD,求(AC)+，判断AC→BD是否属于结论集，因此AC→BD属于BD为(AC)+的子example：example：已知关系模式中R={A，B，C，E,H,P,G},F={AC→PE,PG→A,B→CE,A→P,GA→B,GC→A,PAB→G,ABCP→H},证明BG→HE为F逻辑蕴含证UsesUsesofAttributeComputingclosureof8.4.3SetsoffunctionaldependenciesmayhaveredundantdependenciesthatcanbeinferredfromtheothersFor8.4.3SetsoffunctionaldependenciesmayhaveredundantdependenciesthatcanbeinferredfromtheothersForexample:ACisredundantC,A{ABPartsofafunctionaldependencymaybe{ABAE.g.:onsimplifiedcan{ABAIntuitively,Intuitively,acanonicalcoverofFisa“minimal”setoffunctionaldependenciesequivalenttoF,havingnoredundantdependenciesorredundantpartsofdependenciesExtraneous(无关的Extraneous(无关的ConsiderasetFoffunctionaldependenciesandthefunctionaldependencyinF.AttributeAisextraneousinifA{})andFlogicallyimplies(FA)}.AttributeAisextraneousinifAandthesetoffunctionaldependencies(F–{}){(–A)}logicallyimpliesExample:GivenExample:GivenF={AC,ABCBisextraneousinABbecause{AC,ABC}logicallyimpliesC(I.e.theresultofdroppingBfromABAttributeAisextraneousinifAandFlogicallyimplies(F–{})A)–Example:GivenExample:GivenF={AC,ABCisextraneousinABsinceABCcanbeinferredevenafterdeletingCfromABCDAttributeAisextraneousinifAandthesetoffunctionaldependencies(F–{}){(–A)}logicallyimpliesTestingifanTestingifanAttributeisConsiderasetFoffunctionaldependenciesandthefunctionaldependencyinF.TotestifattributeAisextraneousincompute({}–A)+usingthedependenciesin checkthat({}–A)+contains;ifitdoes,AisextraneousinTotestifattributeAisextraneousin1.compute+ usingonlythedependenciesinF’=(F–{}){(–A)}, checkthat containsA;ifitdoes,AisextraneousExample:GivenExample:GivenF={AC,ABCBisextraneousinABC,TotestifattributeAisextraneousincompute({}–A)+usingthedependenciesin checkthat({}–A)+contains;ifitdoes,AisextraneousinExample:GivenFExample:GivenF={AC,ABCisextraneousinABCD,TotestifattributeAisextraneousin1.compute+ usingonlythedependenciesinF’=(F–{}){(–A)}, checkthat containsA;ifitdoes,AisextraneousAAcanonicalcoverforFisasetofFcsuchFlogicallyimpliesalldependenciesinFc,FclogicallyimpliesalldependenciesinF,NofunctionaldependencyinFccontainsanextraneousattribute,andEachleftsideoffunctionaldependencyinFcisCanonicalCanonicalTocomputeacanonicalcoverforF:Usetheunionruletoreplaceanydependenciesin11and12with11Findafunctionaldependencywithanextraneousattributeeitherinorin/*Note:testforextraneousattributesdoneusingFc,notF*/Ifanextraneousattributeisfound,deleteitfromuntilFdoesnotNote:Unionrulemaybecomeapplicableaftersomeextraneousattributeshavebeendeleted,soithastobeR=(A,B,C)FR=(A,B,C)F={ABCBCAB,AB1)CombineABCandABintoASetisnow{ABC,BC,AB2)ifweapplytheextraneitytesttoABCAisextraneousinABCheckiftheresultofdeletingAfromABisimpliedbytheotherdependenciesYes:infact,BCisalreadypresent!3)Setisnow{ABC,B4)Cis4)CisextraneousinACheckifACislogicallyimpliedbyAandtheotherYes:usingtransitivityonA–CanuseattributeclosureofAinmorecomplexcasesandBAB5)ThecanonicalcoverCanonicalCanonicalWemaygetdifferentFcwiththeR=(R=(A,B,C)F={ABCBAC,CR=(A,B,FR=(A,B,F={ABCBAC,C1)firstapplytheextraneitytesttoAWefindbothBandCareextraneousunder2)ifwedeleteB,andFc={AC,BW