NormalizingIncompleteDatabases

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1 NormalizingIncompleteDatabases Abstract 600MountainAvenue,MurrayHill,NJ07974USA AT&TBellLaboratories LeonidLibkin Databasesareoftenincompletebecauseofthepresence ofdisjunctiveinformation,duetoconicts,partialknowledgeandotherreasons.queriesagainstsuchdatabaseswithnullvalues[akg91,il84],isdisjunctiveinforticsofsuchdatabasesandprovenormalizationtheorems forset-andbag-basedcomplexobjects.thesetheorems provideuswithprogrammingprimitivesthatoneneeds inordertoobtainthelistofallpossibilitiesencodedbya complexobjectwithdisjunctions. theoryofansweringqueriesagainstincompletedatabases withdisjunctiveinformation,anduseittodesignpracticalalgorithmsforqueryevaluation.wedenetheseman- Themaingoalofthispaperistodevelopageneralwithdisjunctionsareknownintheliterature.The dierentdatabasesaremerged. mationthatoccursprimarilyintheareasofdesign ideaofusingand-ortreestodevelopanewobject Itmayalsoariseduetoconictsthatoccurwhen andplanning,aswasnoticedin[inv91a,inv91b]. waspresentedin[lw93a];however,ithadexponential oftenaskquestionsaboutvariouspossibilitiesencodedby thestoreddata,ratherthanthestoreddataitself.normalization,whichisamechanismforaskingsuchqueries, spacecomplexity. databaseone-by-one,ratherthanallatonce.ithaslinear spacecomplexityandallowsustospeedupmanyclasses rithmthatproducesobjectsrepresentedbyanincomplete ofqueries. andmorespaceecient.partialnormalizationallowsus todisregardsomeofthedisjunctionsiftheydonotaect agivenquery.wealsodesignanewnormalizationalgo- Westudytwowaysofmakingqueryevaluationfastercently,afunctionalquerylanguagefordatabaseswith disjunctionswasdesigned[lw93a]andimplemented Anumberofapproachestoqueryingdatabases aboutthedatastoredinadatabase,whereasconceptualqueriesaskquestionsaboutthedataencodedbencebetweenthestructuralandconceptualqueries, considerthefollowingexampleofanincompletedesignborrowedfrom[gl94],seegure1. [GL94].Inthesepaperstwokindsofquerieshave orienteddatamodelwithanadhocqueryfacility plexityinthismodelwasanalyzedin[imv89].re- wasexploitedin[inv91a,inv91b].thequerycom- beendistinguished:structuralqueriesaskquestions mentedinexistingdbpl.wepresentexperimentalresults thatdemonstratesubstantialimprovementoverstandard algorithms,bothinspaceandtime. 1Introduction Informationstoredindatabasesisusuallyincomplete.Oneofthetypicalsourcesofpartiality,along xyzv BB A1.1A1A HHHA2 A1.2A2.1 BB DESIGN Algorithmspresentedinthispaperhavebeenimple-theinformationinadatabase.Toillustratethedier- BBBqwklm A2.2 BBB B1 B p rsta2.3 B2 219Inthisgureverticalandhorizontallinesrepresent Figure1:Incompletedesign u

2 subpartsthatmustbeincludedinthedesign,while theslopinglinesrepresentpossiblechoices.for example,thewholedesignconsistsoftwoparts:a andb.anaiseitherana1orana2,andab consistsofab1andab2,whereab1iseithera worak.structuralqueriesaskaboutthestructure ofagivenobject.forexample,\whatistheleast expensivechoiceforb2"and\howmanysubparts doesa2have"areexamplesofstructuralqueries. Conceptualqueriesaskquestionsaboutpossible completeddesigns.forexample,\howmanycompleteddesignsarethere"and\isthereacompleted designthatcostsunder$100andhasreliabilityat least95%"areexamplesofconceptualqueries. Todistinguishordinarysetsfromcollectionsof disjunctivepossibilities,wecallthelatteror-sets,see [INV91a,LW93a,Rou91].Weusehitodenoteorsets.Intheexampleingure1,thewholedesigncan berepresentedasasetfa;bg,whileaisanor-set ha1;a2iandb2isanor-sethw;ki.notethator-sets havetwodistinctrepresentations.withrespectto structuralqueries,or-setsbehavelikesets,butwith respecttoconceptualqueries,anor-setdenotesone ofitselements.forexample,h1;2iisstructurallya two-elementset,butconceptuallyitisanintegerthat equalseither1or2. Amechanismforansweringconceptualqueries againstcomplexobjectswithor-sets,callednormalization,waspresentedin[lw93a].roughlyspeaking, itprovidesuswithasmallnumberofprogramming primitivesthat,whenrepeatedlyappliedtoanobject o,createanor-setthatlistsallpossibilitiesencoded byo(likecompleteddesigns).thisor-setiscalled thenormalformofo.thenconceptualqueriesare simplystructuralqueriesonnormalforms. Normalization,aspresentedin[LW93a],provides thesolidtheoreticalfoundationfordevelopinglanguagesinwhichconceptualqueriescanbeformulated.italsohasledtodevelopmentofaprototype [GL94].However,thereareseveraltheoreticalproblemsthatmustbeaddressedinordertodeveloppracticalmethodsforansweringconceptualqueries. Onlysetshavebeenconsideredin[INV91a, INV91b,LW93a,Rou91],butmanypractical languagesarebasedonbags(multisets).inthe pastfewyearsseveralapproachestodesignofbag languageshavebeenproposed.moreover,most approachesagreeonwhatconstitutesthebasicset ofbagoperations[alb91,gm93,lw93b,lw94]. Thus,webelievethenormalizationmechanism mustbeextendedtobags. Normalizationmaycauseexponentialblowupin thesizeofobjects.forobjectsofsizen,the sizeoftheirnormalformsisbounded(roughly) byn1:45n[lw93a].therefore,weneed betternormalizationtools.onepossibilityisto normalizepartially.ifsomeofthedisjunctions donotaecttheconceptualquerythatisasked, thereisnoneedtounfoldthosedisjunctions.the problemofpartialnormalizationhasnotbeen addressedintheliterature. Normalization,aspresentedin[LW93a],requires thatthewholenormalformbecreatedbeforeany conceptualqueriescouldbeasked.therefore,it hasexponentialspacecomplexity.alternatively, onemaywanttoproducenormalformelements (e.g.completeddesigns)one-by-one,ratherthan allatonce,thusmakingthespaceusagelinear. Themaingoalofthepaperistoaddressthese shortcomingsofthenormalizationprocess.asthe outcome,weshallhavemuchbettertoolsforquerying databaseswithdisjunctiveinformationandmuch betterunderstandingoftheirstructure.themain contributionsofthispaperarelistedbelow. 1.Werigorouslydenenormalforms(orconceptual semantics)ofobjectswithor-setsandprove normalizationtheoremsgivingusasmallnumber ofoperationsthatconstructnormalforms.wedo thisforbothsetandbagsemantics. 2.Weproveapartialnormalizationresultthattells uswhenthenormalizationprocessneednotbe completedinordertoansweraconceptualquery. Wegivearestrictionontypesofobjectsforwhich thiscanbedone. 3.Wedesignalinearspacealgorithmthatproduces allelementsinthenormalform,andsuggestanew programmingprimitivebasedonit.thisprimitiveallowsustoexpressanumberofimportant queries(includingaclassofexistentialconceptual queries)inauniformfashion. 4.Weconsiderinteractionofdisjunctiveinformation withtraditionalformsofpartialinformation, representedviaordersonobjects,andproveboth normalizationandpartialnormalizationtheorems inthissetting. 5.Weimplementthenewspace-ecientalgorithm inthesystemforqueryingdatabaseswithdisjunctions[gl94].wecompareitwiththestandardalgorithmanddemonstratesubstantialimprovement.weshowhowthenewprogramming 220

3 primitivecanbeusedtogetherwithsomeheuristicstoanswerconceptualqueriesapproximately, whennormalizationprocessisveryexpensive. Organization.Wedenestructuralsemantics andnormalformsinsection2.normalization theoremsforsetsandbagsandpartialnormalization theoremareprovedinsection3.thespaceecientnormalizationalgorithmandaprogramming primitivebasedonitarepresentedinsection4. Normalizationinthepresenceofpartialinformation isstudiedinsection5.experimentalresultsare presentedinsection6. Remark.Ourapproachtodisjunctiveinformationas aformofpartialinformationshouldnotbeconfused withtheworkondisjunctivedeductivedatabases [LMR92].Fordierencesbetweentheseapproaches, see[inv91a,inv91b]. 2Semanticsandnormalforms Aswementionedbefore,objectswithor-setscan betreatedatthestructuralandconceptuallevels. Consequently,therearetwodierentsemanticsfor or-objects.oneofthemtreatsor-setsascollections, whiletheothertakesintoaccountthatanor-set denotesoneofitselements. Tostatethisprecisely,werstdenetypesof objects.therearetwotypesystemsofinterest:one dealingwithsetsandtheotherwithmultisets(bags): (ST) t:=bjttjftgjhti (BT) s:=bjssjfjsjgjhsi Herebrangesoveracollectionofbasetypessuchas integers,booleansetc.tt0istheproducttype;its elementsarepairs(x;y)wherexhastypetandyhas typet0.valuesofthesettypeftgarenitesetsof elementsoftypet.valuesoffjtjgandhtiarenite bagsandor-setsofvaluesoftypetrespectively.if Pfin(X)standsforthenitepowersetofXandPb(X) forthefamilyofnitebagsoverx,then,assuming thatadomaindbofeachbasetypeisgiven,wedene thestructuralsemanticsoftypesasfollows: [[b]]s=db [[tt0]]s=[[t]]s[[t0]]s [[ftg]]s=[[hti]]s=pfin([[t]]s)[[fjtjg]]s=pb([[t]]s) Anobjectwhosetypeisinthetypesystem(ST)is calledaset-basedcomplexobject.anobjectwhose typeisin(bt)iscalledabag-basedcomplexobject. Anyobjectcontainingor-setsisalsocalledanorobject. Weneedtwotranslationsbetween(ST)and(BT) andbetweenset-basedandbag-basedobjects.first, foranytypetin(st),wedenetbagin(bt)by replacingallsetbracketsbybagbrackets.type ssetisdenedassinwhichallbagbrackets arereplacedbysetbrackets.foranyobjectx ofan(st)typet,denexbagoftypetbagby replacingeachsetinxbyabagwiththesame elementsandallmultiplicitiesequal1.forexample, (f1;2g;f3;4g)bag=(fj1;2jg;fj3;4jg).conversely,for Yofa(BT)types,YSetoftypesSetisdenedby replacingeachbaginywiththesetcontainingall elementsofthatbag(i.e.duplicatesareeliminated). Forexample,fjfj1;1;2jg;fj1;2;2jgjgSet=ff1;2gg. Itshouldbenotedthat(tBag)Set=tforany (ST)typet,and(tSet)Bag=tforany(BT)type t.however,while(xbag)set=xforanysetbasedobjectx,itisnotnecessarilythecasethat (YSet)Bag=Yforabag-basedobjectY. Beforewedenetheconceptualsemantics,which willbecallednormalform,weneedthenotionofthe skeletonofatype.theskeletonsk(t)ofatypetis denedtobethetypeformedbyremovingallor-set bracketsfromt.thatis,sk(b)=b,sk(tt0)= sk(t)sk(t0),sk(ftg)=fsk(t)g,sk(fjtjg)=fjsk(t)jg andsk(hti)=sk(t). Next,wedeneabinaryrelationxlyamong objectswhosemeaningintuitivelyis\xisinthe conceptualrepresentationofy".(forexample,dl DESIGNidisacompleteddesign.) Foranyx;x0ofabasetype,x0lxix=x0. (x0;y0)l(x;y)ix0lxandy0ly. fjx01;:::;x0njglfjx1;:::;xnjgithereexistsa permutationonf1;:::;ngsuchthatx0ilx(i) foralli=1;:::;n. fx01;:::;x0nglfx1;:::;xkgithereexistsa partitionx1;:::;xnoffx1;:::;xkgsuchthatfor anyi=1;:::;nandforanyx2xi:x0ilx. xlhx1;:::;xkiixlxiforsomexi.(recall thatanor-setdenotesoneofitselements.) Notethatinthesetclauseitisnotenoughtoask forapermutationofelementsfx1;:::;xngthatwould satisfyx0ilx(i)becausesomeofthosex0imaythen bethesameandfx01;:::;x0ngwouldnotbeaset. Hence,weneedpartitions. Denition.ForanyobjectX,itsnormalform nf(x)isdenedastheor-sethx1;:::;xniofall objectsxisuchthatxilx.notethatthenormal formisalwaysnite. 221

4 Lemma1IfXisoftypet,thenanyxlXisof typesk(t).inparticular,foranyor-objectxoftype t,itsnormalformnf(x)isoftypehsk(t)i. 2 Inotherwords,thenormalformofanobjectlists allpossibilitiesthatareencodedbythedisjunctions presentinthatobject.eachnormalformentryisa regularcomplexobject,i.e.doesnothaveanyor-sets. 3Normalizationtheorems Thegeneralideaofthenormalizationtheoremsis togivealistofoperationsthatcanberepeatedly appliedtoanobjectuntilthenormalformis produced.suchalistwasrstpresentedin[lw93a]; herewegofurtherinseveralaspects.first,weclearly distinguishbetweensetandbagsemantics.second, weproveapartialnormalizationresultthatcanbe viewedasnormalizationatintermediatetypes.that is,whilethestandardnormalizationtheoremsnda uniquerepresentationofanobjectoftypetattype hsk(t)i,thepartialnormalizationresultndssucha representationattypeswheresis\between"tand hsk(t)i.toguaranteeuniqueness,somerestrictions ontypesmustbeimposed. Weneedalanguagetoexpresstheoperationsused fornormalizingobjects.weadopttheframeworkof [LW93a]whichinturnisbasedon[BBW92]andnds itsoriginsin[ab88,bbn91].theoperatorstogether withtheirmostgeneraltypesaregiveningure2. Recallbrieythesemanticsofthegeneralandset operators.fgiscompositionoffunctions;(f;g)is pairformation.1and2aretherstandthesecond projections.!alwaysreturnstheuniqueelementofa specialbasetypeunit.eqisequalitytest;idisthe identityandcondisconditional.forsetoperations: Kfgisthefunctionthatrepresentstheconstantfg; formssingletons:(x)=fxg;[takesunionof twosets;attenssetsofsets:(ff1;2g;f2;3gg)= f1;2;3g;map(f)appliesftoallelementsofaset; and2ispair-with:2(1;f2;3g)=f(1;2);(1;3)g. Operatorsonor-setsareexactlythesameasoperatorsonsetsexceptthattheprexorisadded.Operatorsonbagsaresimilartothoseonsets,butadditiveunionthataddsupmultiplicitiesisused.Also, atteningforbagsisadditive:b(fjb1;:::;bnjg)= B1]:::]Bn. Finally,andbprovideinteractionbetweensets andor-setsandbetweenbagsandor-sets.assume thatx=fx1;:::;xngandy=fjy1;:::;ynjgwhere Xi=hxi1;:::;xiniiandYi=hyi1;:::;yinii.LetFbe thefamilyof\choice"functionsfromf1;:::;ngton Generaloperators g:u!sf:s!t fg:u!t f:u!sg:u!t (f;g):u!st 1:st!s2:st!t!:t!uniteq:tt!boolid:t!t c:boolf:s!tg:s!t cond(c;f;g):s!t Operatorsonsets Kfg:unit!ftg2:sftg!fstg [:ftgftg!ftg:t!ftg f:s!t mapf:fsg!ftg:fftgg!ftg Operatorsonbags Kfjjg:unit!fjtjgb2:sfjtjg!fjstjg ]:fjtjgfjtjg!fjtjgb:t!fjtjg f:s!t bmapf:fjsjg!fjtjgb:fjfjtjgjg!fjtjg Operatorsonor-sets Khi:unit!htior2:shti!hsti or[:htihti!htior:t!hti f:s!t ormapf:hsi!htior:hhtii!hti Interaction :fhtig!hftgi b:fjhtijg!hfjtjgi Figure2:Operatorsofor-NRLandbor-NRL suchthat1f(i)niforalli.then (X)=hfxif(i)ji=1;:::;ngjf2Fi b(y)=hfjyif(i)ji=1;:::;njgjf2fi Themaindierencebetweenthesetwodenitionsis thatduplicatesareremovedfromsetsbutnotfrom bags.forexample,(fh1;3i;h2;3ig)evaluatesto hf1;2g;f1;3g;f2;3g;f3gi,butb(fjh1;3i;h2;3ijg)is equaltohfj1;2jg;fj1;3jg;fj2;3jg;fj3;3jgi. Denition(seealso[LW93a]).Thelanguage or-nrlovertypesystem(st)includesallgeneral operators,setoperators,or-setoperatorsand.the languagebor-nrlovertypesystem(bt)includes allgeneraloperators,bagoperators,or-setoperators andb. 222

5 3.1Normalizingtypes Denethefollowingrewriterulesontypes: shti!hstihsit!hstihhtii!hti fhtig!hftgi fjhsijg!hfjsjgi Denetherewritesystem(STR)on(ST)typesas thethreerulesintherstlineandfhtig!hftgi. Therewritesystem(BTR)on(BT)typesisdened asthetopthreerulesandfjhsijg!hfjsjgi.weuse thenotations?!?!tifsrewritestotinzeroormore steps.recall[dj90]thatanormalformofarewrite systemisatermthatcannotbefurtherrewritten. Proposition2(see[LW93a])Both(STR)and (BTR)areterminatingChurch-Rosserrewritesystems.Consequently,eachtypehasauniquenormal formthatcanbecalculatedashsk(t)iforanytypet thatinvolvesor-sets Normalizingcomplexobjects Itwassuggestedin[LW93a]toassignfunctionsin thelanguagetotherewriterulessothatforevery rewritingfromstottherewouldbeanassociated denablefunctionoftypes!t.thegoalofthis assignmentistoobtainafunctionoftypes!hsk(s)i thatproducesthenormalformsforobjectsoftypes. Insubsection3.3weexplainhowtodothisfor bags.subsection3.4dealswithsets.werecall theresultof[lw93a]andexplainhownormalization processforsetsinteractswithduplicateelimination. Insubsection3.5weconsiderthecasewhenthetarget typeisnotsk(s)butanintermediatetypetsuchthat s?!?!t?!?!hsk(t)i.wendtypestforwhichany objectoftypeswouldhaveauniquerepresentation attypet;theprocessofndingsucharepresentation iscalledpartialnormalization. 3.3Normalizingbag-basedcomplexobjects Weassociatethefollowingfunctionswiththerewrite rules: or2:shti!hsti or1:hsit!hsti or:hhtii!hti b:fjhsijg!hfjsjgi: Hereor1=ormap((2;1))or2(2;1)ispairwithovertherstargument. Now,following[LW93a],wedenethefunction appb(r):s!twhererisarewritestrategythat rewritesstot.firstassumethattisatypeandpa positioninthederivationtreefortsuchthatapplying arewriterulewithassociatedfunctionftotatp yieldstypes.wedeneafunctionappb(t;p;f):t! sshowingtheactionofrewriterulesonobjectsby inductiononthestructureoft: ifpistherootofthederivationoft,then appb(t;p;f)=f; ift=t1t2andpisint1,thenappb(t;p;f)= (appb(t1;p;f)1;2); ift=t1t2andpisint2,thenappb(t;p;f)= (1;appb(t2;p;f)2); Ifpisint0,thenappb(fjt0jg;p;f)= bmap(appb(t0;p;f)); Ifpisint0,thenappb(ht0i;p;f)= ormap(appb(t0;p;f)). Forarewritestrategyr:=tf1?!t1f2?!:::fn?! tn=t0suchthattherewriterulewithassociated functionfiisappliedatpositionpi,weextend appbtoappb(t;t0;r):t!t0byappb(t;t0;r)= appb(tn?1;pn;fn):::appb(t1;p2;f2)appb(t;p1;f1). Theorem3(Normalizationforbags)Forany bag-basedor-objectxoftypetandanyrewritestrategyr:t?!?!hsk(t)i,thefollowingholds: appb(t;hsk(t)i;r)(x)=nf(x) 3.4Normalizingset-basedcomplexobjects Thenormalizationtheoremforset-basedobjectswas provedin[lw93a],thoughdetailswerenotexplained there.herewegiveitsstatementthatfollows immediatelyfromtheorem3. Letrbearewritingt1!:::!tnwherealltis aretypesfrom(st).byrbagwemeantherewriting tbag 1!:::!tBag nof(bt)types.notethatif t1?!?!tnisin(str),thentbag 1?!?!tBag nisin(btr). Theorem4(Normalizationforsets)Forany set-basedor-objectxandanyrewritestrategy r:t?!?!hsk(t)i,thefollowingholds: (appb(tbag;hsk(tbag)i;rbag)(xbag))set=nf(x) Inotherwords,turnxintoabag-object,andapply rbagbyusingappbtoobtainsomeobjecty.then nf(x)=yset. 223

6 Notethatthestatementoftheorem4isdifferentfrom(andinfactstrongerthan)thenormalizationtheoremin[LW93a],whichstatedthat (appb(tbag;hsk(tbag)i;rbag)(xbag))setdoesnotdependonthechoiceofr,anddenednormalforms astheresultofapplicationofanysuchrewritingr. Thequestionarisesifitispossibletoconstructthe normalformwithoutusingthebagsemantics.the answertothisquestionisnegative.toseethis,dene app(t;t0;r)forset-basedobjectsinthesamewaywe denedappb,butusingmapinsteadofbmaptomap oversets,andusinginsteadofb. Proposition5Thereexistset-basedobjectsxof typetsuchthatfornorewritingr:t?!?!hsk(t)i isapp(t;hsk(t)i;r)(x)thenormalformofx.2 Themainreasonthatitisimpossibletoexpress normalizationbymeansofappinor-nrlisthatduplicateeliminationdoesnotcommutewithnormalization.thatis,nf(xset)isgenerallydierentfrom nf(x)set,whilenf(ybag)set=nf(y).wemustadmit herethatproposition5contradictsaclaimmadein [LW93a]thatnormalizationdoesnotaddexpressivenesstoor-NRL.Itdoesnotenhancebor-NRL,but doesaddexpressivepowertoor-nrl. 3.5Partialnormalization Supposethataconceptualqueryasksaquestion aboutpossibilitiesthatareencodedonlybysome ofthedisjunctions,andthatitdoesnottakeinto accountotherdisjunctionspresentinagivenobject. Dowehavetocompletethenormalizationprocessto answersuchaquery?ifaqueryqcanbeansweredby havinganobjectoftypes,andwehaveanobjectxof typetsuchthatt?!?!s,canwendarepresentation ofxattypestoanswerq? Inthissectionweexplainwhensuchapartial normalizationcanbeperformed.firstnoticethat itisnotalwayspossible.takex=hhh1;2i;h2;3iii oftypehhhintiii.thenor(x)=hh1;2i;h2;3iiand ormap(or)(x)=hh1;2;3ii{thesearetwodierent objectsofthesametypehhintii. Theorem9belowsaysthatessentiallyweonlyhave toexcludesituationslikethis.weconsiderbagshere; theresultforsetscanbereadilyobtained,justas theorem4wasobtainedfromtheorem3. First,weneedacriterionthatwouldcheckifatype scanberewrittentot.(wedidnothavethisproblem before,asitwaseasytocheckift=hsk(s)i.)let tsmeanthatsisobtainedfromtbyremoving someoftheor-setbrackets,i.e.shasfewer disjunctions.nowwedeneanewrelationcon typesusingtherulesbelow. tct tct0scs0 tt0css0 tcs fjtjgcfjsjg tt0t0cs tchsi Proposition6Theaboverulesaresoundandcompletefor?!?!.Thatis,s?!?!tisCt. 2 ThelastruleforCintroducesanewvariablet0 insteadofsuggestingaproofsearchstrategy.one mightthinkthatthisleadsto(atleast)exponential timealgorithmsforverifyingsct.(thissomewhat resemblesthesituationwiththecutruleinsequent calculus.althoughitcanbeeliminated,thecost isahyperexponentialblow-upintheprooflength, cf.[gir87].)fortunately,thisphenomenonisnot observedforourrewritesystem. Proposition7Thereexistsalineartimecomplexity algorithmthat,giventwotypessandt,returnstrue ifs?!?!tandfalseotherwise. 2 Nowwesaythatatypetisa-typeifit doesnothaveasubtypeoftheformhhvii.we nextdenetheconceptofa-rewritingbetweentypes.intuitively,-rewritingsresolveallambiguities arisingfromsubtypesofformhhvii.formally,let sandtbetwodistinct-typessuchthats?!?!t. Letrbearewritingbetweensandt:s=s0?! s1?!:::?!sn=t.foreachi=0;:::;n?1,let s1i;:::;smi ibeallthetypessuchthatsi?!sji(in onestep)andsji?!?!t.letpjibethepositioninsi atwhichrewriteruleisappliedtoobtainsjifromsi, j=1;:::;mi. Thentherewritingr:s?!?!tisa-rewriting (writtenasr:s?!?!t)ifeithern=1(onestep rewriting)orn>1anditsatisesthefollowingtwo propertiesforeveryi=0;:::;n?2: 1.Ifoneofsjisisa-type,thensi+1isa-type. 2.Ifallsjihavesubtypesofformhhvii,then(a) si+1=sjisuchthatthereisnopliclosertothe rootthanpji,and(b)si+2isobtainedfromsi+1 byapplyingtherulehhvii?!hvionthenewly createdsubtypehhvii. 224

7 Thisdenitionresolvesambiguitiesarisingfrom subtypesofformhhvii.therstpropertysays thattheyneednotbeintroducedunlessabsolutely necessary,andthesecondpropertydictatesthatonce wecannotavoidintroducingasubtypehhvii,itmust bedoneasclosetotherootaspossible,andthen gottenridofatthenextstepoftherewriting.togive anexample,hfhtigis!hfhtigsi!hhftgisi! hhftgsii!hftgsiisa-rewriting,buttheone thatachievesthesameresultbydoinghfhtigis! hhftgiisrstisnotbecauseintroductionofthe doubleor-setsubtypecanbeavoided. Proposition8Letsandtbe-typesands?!?!t. Thenthereexistsa-rewritingr:s?!?!t. 2 Usingthisproposition,wecanformulatethepartial normalizationtheorem. Theorem9(PartialNormalization)Letsandt be-typessuchthats?! rewritingsr1;r2:s?!?!t.thenforanytwo-?!tandforanyobjectxof types,thefollowingholds: appb(s;t;r1)(x)=appb(s;t;r2)(x) Thistheoremtellsusthatanyobjectofa-type shasanunambiguousrepresentationofa-typetif sct.thisrepresentationisobtainedbyapplying any-rewritestrategythatrewritesstot. Onemaywonderifrestrictingrewritingstorewritingsonlyisreallynecessary,andifso,are boththeconditionson-rewritingsnecessary.the followingpropositionshowsthatitis. Proposition10Itispossibletond-typessand t,anobjectxoftypesandtworewritingsr1andr2 fromstotwhichviolateeithertherstorthesecond propertyof-rewritingssuchthatappb(s;t;r1)(x)6= appb(s;t;r2)(x). 2 4Normalizationalgorithmsand primitives Thereis,ofcourse,atrivialnormalizationalgorithm basedonthegeneralnormalizationtheorems.we presentitbelowforbag-basedcomplexobjects. IfXisnotanor-object,thennf(X)=hXi. IfXis(x;y)oftypest,thennf(X)= orcartprod(nf(x);nf(y))ifbothsandtinvolve or-sets,nf(x)=or1(nf(x);y)ifonlysinvolves or-setsandnf(x)=or2(x;nf(y))ifonlyt involvesor-sets. IfX=fjx1;:::;xnjg,thennf(X)= b(fjnf(x1);:::;nf(xn)jg). Thisalgorithmdoescalculatethenormalform,as followsfromtheorem3.itcanbereadilyadaptedto theset-basedcomplexobjects. Theproblemwiththisalgorithmisitsexponential spacecomplexity,asshownin[lw93a].itcreatesthe wholenormalformbeforeanyconceptualqueriescan beasked.webelieveitwouldbemorereasonableto designanewevaluationstrategy,thatproducesthe elementsinthenormalformone-by-one.thenthe spaceusagewouldbelinearand,inaddition,some conceptualqueriescanbeevaluatedmuchfaster. Forexample,foranexistentialqueryoveranormal form,satisabilitycannowbeveriedforeachnewly producedentry.iftheconditionissatised,the evaluationstopswithoutproducingallelementsin thenormalform.thatis,ifxisoftypetand pisoftypesk(t)!bool,andwewanttond outifthereisanelementofnf(x)thatsatisesp (e.g.isthereacheapreliabledesign?),thenwe shouldbeabletostopwhensuchanelementis found.thequery9pwhichwillbeshownlaterin thissectiondoespreciselythat.notethatusing thestraightforwardnormalizationalgorithm,even evaluationof9(x:true)requiresexponentialspace asthenormalformmustbeproducedrst! Theevaluationstrategythatwearegoingto presentisessentiallythedepthrstsearchonthe and-ortreeunderlyingacomplexobject.this strategywillworkforbothset-andbag-based complexobjects,assetsandbagswillbetranslated intoliststogiveanorderofevaluation.usingthis evaluationstrategy,weshallalsosuggestnew,more exible,normalizationprimitives. Wecreateaspecialdatastructure,calledannotated complexobjects,torepresentand-ortrees.basically, anannotationgivesachoiceofanelementforeachorsetandalsocontainslocalconditionstellingwhether allpossibilitiesencodedbyanobjectareexhausted. Foreachobjecttypet,wehaveanewannotatedtype A(t)andtheinitialtranslationt!A(t).Fromeach annotatedobject,wecangetanentryinthenormal form.attheheartofthealgorithmliesaprocedure thattakesanannotatedobjectandproducesthe \next"one.thisenablesustolistallnormalform entriessequentially. Wetranslatesetsandbagsintolists,assuming someordering.nomatterwhichorderingischosen, thealgorithmwillproduceallnormalformentries. However,theorderinwhichtheyareproduceddoes 225

8 dependonthetranslation,andcanbeusedfor additionaloptimizations. Inwhatfollows,wepresentthealgorithmforsetbasedcomplexobjects.Thealgorithmforbag-based complexobjectscanbeobtainedbyrepeatingit verbatimandreplacing\set"by\bag".wedenote thetypeoflistsoftypetby[t]. Denition(Annotatedcomplexobjects).Type K(kind)hasfourpossiblevalues:B(base),P (product),s(set),ando(or-set).foreachtypet, weproduceanannotatedtypea(t)asfollows: A(b)=Kbifbisabasetype. A(st)=Kbool(A(s)A(t)): A(ftg)=Kbool[A(t)]: A(hti)=Kbool[(A(t)bool)]: Thebooleanvalueinthesetranslationissetto trueiftherearestillentriesencodedbytheobject thathavenotbeenlookedat.foror-sets,the booleancomponentinsidelistsisusedforindicating theelementthatiscurrentlyusedasthechoicegiven bythator-set.inallalgorithmsonlyoneentryin suchalistwillhavethetruebooleancomponent. Nowwedenethreefunctions:initial:t!A(t) producestheinitialannotationofanobject;pick: A(t)!sk(t)producesanelementofthenormalform givenbyanannotation;end:a(t)!boolreturns trueiallpossibilitiesencodedbyitsargumenthave beenexhausted. Thedenitionsofinitialandpickaregivenin gure3.byvoidwemeanaspecialobjectused toindicatetheendoftheprocessofgoingoverthe normalform.p1{p5giveasimpliedversionofpick inwhichvoidisnotpropagatedtothetoplevel.such propagationisdonetodetectinconsistenciesencoded byemptyor-sets. Thefunctionendalwaysreturnstrueon(B;x). Onanyotherannotatedobjectx=(k;c;v),endx= :c.wealsodeneafunctionreset:a(t)!a(t)that disregardstheannotationofanobjectandrestores theinitialone.thedenitionalmostverbatim repeatsinitialandisomittedhere. Arecursivealgorithmfornextisgiveningure 4.Weusethe[]bracketsforlists.Foranylist X=[x1;:::;xn],Xoistandsfor[x1;:::;xi?1]and X1idenotes[xi+1;:::;xn](theymaybeempty).We Thatis,a::xputsaasthenewheadbeforethelist Nowwecanproducethefollowingalgorithmthat listselementsofthenormalformofanor-objecto. Calculatingnorm(cond,init,update,out)(o) acc:=init; ao:=initialo; last:=endao; while:(cond(pickao)_last) doacc:=update(pickao,acc); ao:=nextao; last:=endao end; returnout((pickao,last),acc) Figure5:Algorithmfornorm ao:=initialo; repeatprint(pickao); ao:=nextao untilend(ao) Theorem11Foranyor-objecto,thealgorithm aboveprintsallelementsofnf(o)andnothingelse. Moreover,ithaslinearspacecomplexity. 2 Althoughnoduplicateeliminationisdoneinthis algorithm,itdoesnotproduceunnecessarycopies. Corollary12Letobeanor-objectsuchthatall or-setsinitarepairwisedisjoint.thentheabove algorithmprintseachentryinnf(o)exactlyonce.2 Thecorrectnessresultsuggestsaddingnew,more exiblenormalizationprimitivestoor-nrl.we proposethefollowingonecallednorm. cond:sk(t)!bool update:sk(t)u!u out:(sk(t)bool)u!s init:u norm(cond,init,update,out):t!s Its\semantics"isgivenbythealgorithmingure 5.Intuitively,theoutputvalueisaccumulatedin acc,condisusedtobreaktheloopifthecondition issatised,lastindicatesifallpossibilitieshavebeen lookedat,andoutformstheoutput. Now,anumberoffunctionscanbedenedusing norm.hereweconsiderjusttwo.intherst denition,pisoftypesk(t)!bool. 9pnorm(p;false;x:y:false;1) normalizenorm(x:false;hi;x:y:or(x)or[y;2) 226

9 I1initialx=(B;x)ifxisofbasetype. I2initial(x;y)=(P;true;(initialx;initialy)). I3initialfx1;:::;xng=(S;true;[initialx1;:::;initialxn]). I4initialhx1;:::;xni=(O;true;[(initialx1;true);(initialx2;false);:::;(initialxn;false)]). I5initialhi=(O;false;[]). P1pick(B;x)=x. P2pick(P;c;(x;y))=ifcthen(pickx;picky)elsevoid. P3pick(S;c;[x1;:::;xn])=ifcthenfpickx1;:::;pickxngelsevoid. P4pick(O;c;[x1;:::;xn])=ifcthenpick1(xi)elsevoidwhere2(xi)=true. P5pick(O;c;[])=void. Figure3:Denitionsofinitial(I1{I5)andpick(P1{P5) next(p;c;(x;y))=(p;true;(x;nexty)) :end(nexty) next(b;x)=(b;x) Base next(p;c;(x;y))=(p;true;(nextx;resety)) end(nexty):end(nextx) Pair Set next(p;c;(x;y))=(p;false;(x;y)) end(nexty)end(nextx) next(s;c;[])=(s;false;[]) end(nextx1)next(s;true;[x2;:::;xn])=(s;c0;x0) next(s;c;x)=(s;c0;resetx1::x0) next(s;c;x)=(s;true;nextx1::[x2;:::;xn]) Or-set :end(nextx1) next(o;c;[])=(o;false;[]) 2(xi)X1i6=[]end(next1(xi)) 2(xi):end(next1(xi)) 2(xi)X1i=[]end(next1(xi)) next(o;c;x)=(o;false;x) Figure4:Algorithmfornext 227

10 Corollary13Foranyor-objecto,1)9p(o)=(x;c) wherexisanormalformentrysatisfyingpifc= falseandtherearenonormalformentriessatisfying pifc=true,and2)normalize(o)isitsnormalform. 2Notethat9pisveryusefulinevaluationofexistential queries.ifanentrythatsatisespisfound,9pstops andreturnsthatentrywithoutproducingallother normalformentries.incontrasttothestandard algorithmthatrequiresexponentialspacetoevaluate suchqueriesevenifpisx:true,9(x:true)needs lineartimeandspacetobeevaluated. Asanotherapplicationofthenewevaluation strategy,itispossibletorunnormalizationfora giventime,andgetthebestentryinthenormalform obtainedinthattime.thisisoftenhelpfulifan approximatesolutionissatisfactory. Space-ecientevaluationofrecursive queriesusingnormalization.nowweshowa somewhatsurprisingapplicationofournormalizationalgorithm{itdealswithalgorithmicexpressive powerofquerylanguages.recallthattheabiteboul- BeerialgebraA&B[AB88]isthenestedrelationalalgebra(generalandsetoperatorsingure2)plusthe powersetoperator.whilethenestedrelationalalgebracannotexpressrecursivequeriessuchastransitiveclosure(tc)[lw94],a&bcanexpresstcby rstproducingallpossiblerelationsonagivensetof nodesandthenselectingthosethatcontainagiven oneandaretransitive.ofcoursethiswayofcomputingtcusesexponentialspace.aremarkableresultof[sp94]saysthatnomatterhowwewritean A&B-expressiontocomputetc,itwilluseexponential space.however,itisbasedonacontrivedrestriction thata\natural"evaluationstrategyisused.ifthis restrictionisdropped,thenitispossibletodevisean evaluationstrategythatcomputestcinpolynomial space,asshownin[ah95]. Itwasprovedin[LW93a]thathasessentiallythe expressivepowerofthepowersetoperator.hence, wecanviewor-nrlasanextensionofa&bwithorsets.nowweexplainhowtousenormtocomputetc space-ecientlyinthislanguage.weusesomemetanotation,buteverythingcanbeexpressedinor-nrl. LetR:fbbgbeanonemptybinaryrelation. DeneNR=map(1)R[map(2)R(thesetof nodesofr)andn2rascartprod(nr;nr).nowlet PR=map(z:or[(or(fg);or(z)))(N2R) Thatis,foreachpairofnodes(x;y),thesetPR containsanelementhfg;f(x;y)gi.letrc:ft tgfttg!fttgcomputetherelational composition(itcanbedoneinanylanguagethat containsrelationalalgebraasasublanguage).let ebeoftypebb(i.e.anedge).dene ce=s:(rc(s;s)=s)&(rs)&(e62s) Finally,lettce=norm(ce;();x:();21)(PR). Proposition14tceevaluatestotrueifeisintc(R) anditevaluatestofalseotherwise.consequently, tc(r)canbecomputedinpolynomialspaceusing norm. 2 Thispropositioncanberegardedasacounterpartof theresultof[ah95]sayingthattccanbeevaluated ina&busingpolynomialspaceunderaspecial evaluationstrategy.hereweusedourspace-ecient strategyfornormalizationtoachievethesameresult. 5Objectswithpartialinformation andantichainsemantics Theantichainsemantics,denedin[Lib95,LW93a] andbasedontheideasfrom[bjo91,lib91],is usedforobjectswithpartialinformation.thekey ideaisthatthenotionofpartialitycanbeconveyed byorderings,withxymeaningthatyismore informativethanx. Thisorderingisusuallygivenforbasetypes.For example,anullvalueni(noinformation)isless informativethananyintegerorboolean.forpairs, (x;y)(x0;y0)ixx0andyy0.itwasexplained in[lw93a]thatthefollowingtwoorderings,wellknowninsemanticsofconcurrency[gun92],mustbe usedforsetsandor-setsrespectively: Xv[Y,8x2X9y2Y:xy Xv]Y,8y2Y9x2X:xy Usingtheseorderingssuggestsanewsemanticsin whichanobjectcandenoteanyotherobjectthat ismoreinformative.thisallowseliminationof redundanciesgivenbycomparableelements,because Xv[YimaxXv[YandXv]YiminXv]Y, wheremaxxandminxaresetsofmaximaland minimalelementsofx. InmaxXandminXelementsarepairwiseincomparable.Suchsetsarecalledantichains.Using Afin(A)forthefamilyofantichainsoveraposetA, wedenethefollowing(structural)antichain-based semantics.hereweconsideronlyset-basedobjects. [[b]]a=(db;b) [[ts]]a=[[t]]a[[s]]a [[ftg]]a=(afin([[t]]a);v[)[[hti]]a=(afin([[t]]a);v]) 228

11 Asfollowsfromtheclaimsabove,foreachobjectx oftypetthereexistsasemanticallyequivalentobject xin[[t]]adenedbythefollowingrules: x=xforxofabasetype. (x;y)=(x;y): fx1;:::;xng=maxfx1;:::;xng: hx1;:::;xni=minhx1;:::;xni: Consequently,foreachoperationf:s!tin or-nrl,wedeneanewoperationfathattakes x2[[s]]aandreturnsf(x)2[[t]]a.itisknown(see [Lib92,LW93a])thataisanisomorphismbetween [[fhtig]]aand[[hftgi]]a.usingtheseoperationsfa,itis possibletodeneappa(t;t0;r):t!t0thatappliesa rewritestrategyr:t?!?!t0,exactlyinthesameway aswedenedapp,butusingtheindexaeverywhere. Thefollowingtworesultsstatethenormalization theoremfortheantichainsemantics,andthepartial normalizationtheorem. Theorem15Letx2[[t]]abeanobjectoftypet suchthattinvolvesor-sets.then,foranyrewriting r:t?!?!hsk(t)i,thefollowingholds: appa(t;hsk(t)i;r)(x)=nf(x) Theorem16Letsandtbetwo-typessuchthat s?!?!t.thenforanytwo-rewritingsr1;r2: s?!?!tandanyx2[[s]]a, appa(s;t;r1)(x)=appa(s;t;r2)(x) 6Experimentalresults Thebasicnormalizationalgorithmandthenew spaceecientnormalizationalgorithmhavebeen implementedinthesystemor-sml1[gl94],which isadatabaseprogramminglanguagebuiltontopof StandardMLofNewJersey[HMT90]. Werananumberofexpermentstocomparethe speedofthebasicalgorithmwiththenewalgorithm describedinthispaper.asourtestobjects,wechose objectsthatareknowntocauseexponentialblow-up inthesizeofthenormalform[lw93a].inaddition, theseobjectsarenotwellsuitedfortheor-sml duplicateeliminationalgorithm[gl94],sowecould comparethespeedofthestandardalgorithmsforsets andbags. Inthetablebelow,therstcolumnshows(approximately)thenumberofentiresinthenormalform. Entriesthemselvesarerelativelysmall.Thesecond 1[GL94]describestheversionofOR-SMLinwhichthe primitivenormisnotavailable. columnshowsrunningtime2forthestandardalgorithmforsets;thatis,attheendduplicatesareeliminated.thethirdcolumnisrunningtimeforthestandardalgorithmforbags.thelastcolumnisrunning timeforthenewalgorithm.notethatwecompare timeratherthanspace.despiteitsspaceeciency, thennewalgorithmstillhastocomputeexponentially manyentries.thereareseveralreasonswhygures inthelastcolumnarebetter;amongthemiswinning intimeduetonotrunninggarbagecollections. #entriestime(1)time(2)time(3) >19,000>11min0.9sec 1.8sec >59,000>90min8.9sec 5.8sec >175,000>16hr31.1sec19.1sec >525,000>2days1min35sec59sec >1:5106notdoneoutof memory3min9sec >4:5106notdonesame9min56sec >14106notdonesame31min51sec Wehavealsoconsideredanapplicationofthe normalizationalgorithmwhereonehastoselecta normalformentrywhichisbestaccordingtosome criterionf.ifthenormalformislarge,itispossible torunthealgorithmforagiventime,returning thebestentrythatwasfoundsofar.inoneof ourexamples,withalmost3.5billionentriesinthe normalform(goingoverthemtakesabout5days), weobatinedthevalueoffwithin7%oftheoptimal byrunningthealgorithmforonly15seconds,andthe valuewithin4%oftheoptimalin30minutes. 7Conclusion Inthispaperwehavestudiedvarioustechniquesfor normalizingdatabaseswithdisjunctiveinformation representedbyor-sets.thisproblemisparticularly importantintheareasofapplicationsuchasdesign andplanning,aswellasmergingdatabases.queries againstsuchdatabasesoftenaskquestionsabout possibilitiesencodedbythedatabase,ratherthan theinformationthatisstoredthere.werigorously denedtheconceptofnormalizationforbothset andbagsemantics.weexplainedhownormalforms thatlistallpossibilitiesencodedbyanincomplete objectcanbecalculated.onlyalimitednumber ofoperationsareneededforcalculationofnormal forms,andthesequenceinwhichtheyareapplied isirrelevantforbothsetandbagsemantics. Sincenormalformscanbeofsizeexponentialinthe sizeoftheobjects,weneedbettertoolsforanswering conceptualqueries.wedemonstratedtwo.partial 2OnSGIChallengeXL{8R MHzprocessorswith 1GigabyteRAM. 229

12 normalizationallowsustoanswerquerieswithout normalizingcompletely.wehavealsodesignedanew space-ecientnormalizationalgorithm. Thereareimmediatepracticalbenetsoftheresultspresentedinthispaper.ThenewspaceecientalgorithmhasbeenimplementedinOR-SML{ asystemforqueryingdatabaseswithdisjunctions.in additiontobeingspaceecientandfasterthanthe standardalgorithm,itallowsmorecontroloverthe processofnormalization.thismakesthenormalizationtechniquesapplicableinpracticalproblems,such ascomputerautomateddesign. Acknowledgements:ThankstoRickHull, TomaszImielinskiandKumarVadapartyforrightfullydisputingtheclaimin[LW93a]thattheproduceallnormalizationisthewaytoanswerconceptual queries.iamverygratefultopeterbuneman,elsa Gunter,JonRiecke,ValTannenandLimsoonWong fortheircomments,helpandcriticism,andtoanthonykoskyforacarefulreadingofthemanuscript. References [AB88]S.Abiteboul,C.Beeri,Onthepowerof languagesforthemanipulationofcomplex objects,inproc.ofint.workshopon TheoryandApplicationsofNestedRelations andcomplexobjects,darmstadt,1988. [AH95]S.AbiteboulandG.Hillebrand.Spaceusage infunctionalquerylanguages.inlncs893: Proc.ICDT-95,pages437{454. [AKG91]S.Abiteboul,P.KanellakisandG.Grahne. Ontherepresentationandqueryingofsetsof possibleworlds.tcs78(1991),159{187. [Alb91]J.Albert.Algebraicpropertiesofbagdata types.invldb-91,pages211{219. [BBN91]V.Breazu-Tannen,P.Buneman,andS.Naqvi. Structuralrecursionasaquerylanguage.In Proc.ofDBPL-91,pages9{19. [BBW92]V.Breazu-Tannen,P.Buneman,andL.Wong. Naturallyembeddedquerylanguages.InLNCS 646:Proc.ICDT-92,pages140{154. [BDW91]P.Buneman,S.Davidson,A.Watters,A semanticsforcomplexobjectsandapproximate answers,jcss43(1991),170{218. [BJO91]P.Buneman,A.Jung,A.Ohori,Usingpowerdomainstogeneralizerelationaldatabases, TCS91(1991),23{55. [DJ90]N.DershowitzandJ.-P.Jouannand.Rewrite systems.in:handbookoftheoreticalcomputerscience,northholland,1990,pages243{ 320. [Gir87]J.-Y.Girard.\ProofsandTypes",Cambridge, [GM93]S.GrumbachandT.Milo.Towardstractable algebrasforbags.inpods-93,pages49{58. [Gun92]C.Gunter.\SemanticsofProgrammingLanguages".TheMITPress,1992. [GL94]E.GunterandL.Libkin.OR-SML:Afunctionaldatabaseprogramminglanguagefor disjunctiveinformationanditsapplications. LNCS856:Proc.DEXA-94,pages [HMT90]R.Harper,R.Milner,andM.Tofte.\The DenitionofStandardML",TheMITPress, [IL84]T.Imielinski,W.Lipski.Incompleteinformationinrelationaldatabases.J.ofACM 31(1984),761{791. [INV91a]T.Imielinski,S.Naqvi,andK.Vadaparty. Incompleteobjects adatamodelfordesign andplanningapplications.insigmod-91, pages288{297. [INV91b]T.Imielinski,S.Naqvi,andK.Vadaparty. Queryingdesignandplanningdatabases. InLNCS566:DOOD-91,pages524{545. Springer-Verlag. [IMV89]T.Imielinski,R.vanderMeydenandK.Vadaparty.Complexitytailoreddesign:Anew methodologyfordatabasedesign.toappear injcss.extendedabstractinpods-89. [Lib91]L.Libkin,Arelationalalgebraforcomplex objectsbasedonpartialinformation,inlncs 495:MFDBS-91,pages36{41. [Lib92]L.Libkin,Anelementaryproofthatupperand lowerpowerdomainconstructionscommute, BulletinoftheEATCS,48(1992),175{177. [Lib95]L.Libkin.Approximationindatabases.In LNCS893:Proc.ICDT-95,pages411{424. [LW93a]L.LibkinandL.Wong.Semanticrepresentationsandquerylanguagesforor-sets.In PODS-93,pages37{48. [LW93b]L.LibkinandL.Wong.Someproperties ofquerylanguagesforbags.indbpl-93, SpringerVerlag,1994,pages97{114. [LW94]L.LibkinandL.Wong.Newtechniquesfor studyingsetlanguages,baglanguagesand aggregatefunctions.inpods-94,pages155{ 166. [LMR92]L.Lobo,J.MinkerandA.Rajasekar.\FoundationsofDisjunctiveLogicProgramming".The MITPress,1992. [Rou91]B.Rounds,Situation-theoreticaspectsof databases,inproc.conf.onsituationtheory andapplications,cslivol.26,1991,pages [SP94]D.SuciuandJ.Paredaens.Anyalgorithm inthecomplexobjectalgebrawithpowerset needsexponentialspacetocomputetransitive closure.inpods-94,pages201{