southeast asian bulletin of mathematics

... the beauty of mathematics, and of God, the sum of all wonders.The Beauty of MathematicsWonderful WorldBut:A-T-T-I-T-U-D-E1+20+20+9+20+21+4+5 = 100%THEN, look how far the love of God will ... that:While Hard Work and Knowledge will get you close, and Attitude willGet you there, It’s the Love of God that will put you over the top! Here’s a little mathematical formula that might helpAnswer...

... education on the benefits of going organic and for the growing market of organic retail sales. Acknowledgements The authors gratefully acknowledge funding from Department of Food Science and ... Department of Food Science and Technology, Faculty of Agriculture Technology (Bogor Agricultural University), Region Government of Bogor, Central Government of Bogor, and BRI (Bank Rakyat Indonesia). ... multi-theoretical framework of consumer decision making. British Food Journal. 104: 624-642. As. J. Food Ag-Ind. 2009, Special Issue, S363-S367 Asian Journal of Food and Agro-Industry...

... rolein the mathematics of today. Consider the conjectures of Langlands, including that of Shimura-Taniyama, leading to Wiles’s proof of Fermat’s last theorem.Drawing on the original work of mathematicians ... exposition of Sources in the Development of Mathematics The discovery of infinite products by Wallis and infinite series by Newton marked thebeginning of the modern mathematical era. The use of series ... practical support of my efforts to becomea mathematician. I dedicate this book to their memory.2.2 Johann Faulhaber and Sums of Powers 19as professor of mathematics at Basel, in spite of a salary...

... head of the Laboratory for Modeling in Solid Mechanics.Professor Manzhirov is also head of a branch of the Department of Applied Mathematics at Bauman Moscow State Technical University, professor ... professor of mathematics at MoscowState University of Engineering and Computer Science, vice-chairman of Mathematics and Mechanics Expert Council of the Higher Certification Committee of the RussianFederation, ... scientistin the fi elds of mechanics and applied mathematics, integralequations, and their applications.After graduating with honors from the Department of Mechanics and Mathematics of Rostov State...

... M. J. MOSSINGHOFFWe remark that a pure hill-climbing method would omit the resetting of bk−1to 0 at the end of Step 3 and would terminate as soon as none of theadjustments of Steps 2 or 3 ... of Mahler’s measure,” where this research began.LEHMER’S PROBLEM FOR POLYNOMIALS WITH ODD COEFFICIENTS365The root α1 of A1(x) in the proof of Theorem 6.2 is the smallest knownmeasure of ... that between two consecutive zeros of an(t)in(0, 1/2)there exists exactly one zero of pn(t), with two exceptions corresponding tothe absence of a zero of pn(t)att =1/3 and the extra zero...

... eigenvalues of A then f(λ1), . . . , f(λn) are eigenvalues of f(A),where f(·) is a polynomial.• the rank of a symmetric matrix is the number of non-zero eigenvalues it contains.• the rank of any ... the rank of any matrix A is equal to the number of non-zero eigenvalues of AA.• if we define the trace of a square matrix of order n as the sum of the n elements onits principal diagonal tr(A) ... function, p is the price of output, l,k are the amount of labor and capital employed by the firm (in units of output), w is the real wage and ris the real rental price of capital. The firm takes...

... dayCambridge hosts a Lucasian Professor of Mathematics, but Oxford’s equivalent is the SavilianProfessor of Geometry. And unless it should be thought that the association of mathematics with prediction ... ArnoldTHE HISTORY OF ASTRONOMYMichael HoskinTHE HISTORY OF LIFEMichael BentonTHE HISTORY OF MATHEMATICS Jacqueline StedallTHE HISTORY OF MEDICINE William BynumTHE HISTORY OF TIMELeofranc Holford-StrevensHIV/AIDS ... a little to redress the masculine bias of most depictions of the history of mathematics; it can, however, pay more than lip service to the mathematics of continentsother than Europe; and it...

... anelement of the set of positive integers because it will occur on the list eventually. Using the language of sets,we say that 0 is an element of the non-negative integers but 0 is not an element of ... property of addition2. commutative property of multiplication3. distributive property4. associative property of addition5. commutative property of addition6. associative property of multiplication7. ... places.Commutative Property of Addition:a + b = b + aCommutative Property of Multiplication:a · b = b · aWe know 5 + 3 = 3 + 5 because of the commutative property of addition. Similarly, 5...

... a way of defining logical implicationthat does not rely on any notion of truth, but only on manipulatingsequences of formulas, namely formal proofs or deductions. (Of course,any way of defining ... be the set of sentences of L=including• every sentence τ of Th(C), i.e. such that C |= τ,and•¬cr= csfor every pair of real numbers r and s such that r = s.Every finite subset of Σ is satisfiable. ... Definitions 6.4 and 6.5; the proof is similar in form tothe proof of Proposition 2.9.6.14. Use Definitions 6.4 and 6.5; the proof is similar in form tothe proof for Problem 2.10.vi PREFACEnot...

... ~whoseaddressIsAttributes(specifictelephonentm~wers):~tele~~Isdistancefrom~in(specifickinds of food)whosekind of foodIs(specificreviews)whosereviewisaddress(specificqualities of foods)whosequality of foodistelephone(specificprices)whosepriceIskind of foød(specificcreditcardss)whosecreditcardsarereview(specificnumber>whosedistancefromUtinmilesisquality of foodwithaminimumslicesize of price-withgridoncreditcardsCo~parisons°;withhorizontalgildbetween-withvertica!ridgreaterthanwith(n>divisionslessthangreaterthanorequaltolessthanorequaltoequaltoE’=v~tEI’iCi—r~r~i~rrJ:,RestartRefreshRuboutExitSystemSaveInputRetrieveInputDeleteInputsPlayInputShowInputShowParseTreeExecuteSaveOutputFindrestaurantswhosedistancefromutinmilesislessthanH.5Mor~AboveNAME:Sarilliq’JElLOCATION:233014.NorthLoopTELEPHONE:459—4121LIISTANCE_FROM_UT:0.r.iND OF Foot’:ME~ICANF:EVJE1.J:...piari.~fa’.joriteE.‘:‘rth~r~ier~u...frieridlySerViCE’.——Te~:asIlorithly~—LOC::2500.V—LOC:2500.OLIALIT, OF FUOLI:GOODPRI ... and“where.”TheDo-specialistreplacesthepredicateDO(fromtheverb“do”)withamorespecificverbchosenfromthoseacquiredforadomain.Although“do”doesnotappearasthemainverbveryofteninthedatabasequerytask,thetranslatorsdeduceitsimpliedpresenceinsomequeries—forinstanceinsuchcomparativequestionsas“WhatcountriescovermoreareathanPeruLdoes~?”.Thecomparativespecialistexaminesthetwoarguments of acomparisontodeterminewhetherthecomparisontobemadeisbetweentwoattributevalues(e.g.,Jack’sheightandsevenfeet)orbetweenanentityandsomevalue(e.g.,Jackandsevenfeet).Inthelattercase,TEAMtriestoidentifytheappropriateattribute of theentity(e.g.,Jack’sheight).2.3.4DatabaseSchemaThetranslationfromlogicalformtoSODAqueryrequiresknowingtheexactstructure of thetargetdatabaseandthemannerinwhichthepredicatesappearinginthelogicalformareassociatedwiththerelationsinthedatabase.Thisinformationisprovidedbythedatabaseschema,whichincludesthefollowinginformation8:•Definition of sortsinterms of databaserelations(subject)orfields(andfieldvalueforsortsderivedfromfeaturefields).8Theschematranslatoralsousescertaininformationintheconceptualschema,includingtaxonomicinformationinthesorthierarchyanddelineationinformationassociatedwithnonsortpredicates.—18—Figure5:AcquiringtheVirtualRelationsPKCONTandHEMICwindowforquestionsandanswers.WhentheDBEusesthemousetoselectone of theitemsfromthethreemenus,aset of questionsappearsinthequestion-answeringareaatthebottom of thedisplay,towhichhecanthenrespond.One of thegeneralprinciples of acquisitionisevidentfromthisdisplay,namely,thattheacquisitioniscenteredupontherelationsandfieldsinthedatabase,becausethisistheinformationmostfamiliartotheDBE.Theanswerstoeachquestioncanaffectthelexicon,theconceptualschema,andthedatabaseschema.TheDBEneednotbeaware of exactlywhyTEAMposesthequestionsitdoes—allhehastodoisanswerthemcorrectly.Eventheentriesdisplayedinthewordmenuowetheirpresencetoquestionsaboutthedatabase.TheDBEvolunteersentriestothismenuonlyinthecase of verbacquisition,tosupplyanadjectivecorrespondingtosomenounalreadyinTEAM’slexicon,ortoenterasynonymforsomelexicon-residentword.TheDBEisassumednottohaveanyknowledge of formallinguisticsor of natural-languageprocessingmethods.Heisassumed,however,toknowsomegeneralfactsaboutEnglish—forexample,whatpropernouns,verbs,plurals,andtenseare,butnothingmoredetailedthanthat.Ifmoresophisticatedlinguisticinformationisrequired,asinthecase of verbacquisition,TEAMproceedsbyaskingquestionsaboutsamplesentences,allowingtheDBEtorelyonhisintuitionasanativespeaker,andextractingtheinformationitneedsfromhisresponses.Virtualrelationsarespecifiediconically.Theleftside of Figure5showstheacquisition of avirtualrelationthatidentifiesthecontinent(PKCONT-CONTINENT,derivedfromWORLDC-CONTINENT) of apeak(PKCONT-NAME,fromPEAK-NAME)byperformingadatabasejoinonthePEAK-COUNTRYandWORLDC-CONTINENTfields.Similarly,therightside of Figure5showstheacquisition of thevirtualrelationthatencodesthehemisphere(HEMIC-HEMI) of acountry(HEMIC.NAME)byjoiningontheWORLDC-CONTINENTandCONT-NAMEfields.Ifhewishes,theDBEcanchangepreviousanswers.Incrementalupdatesarepossiblebecausemost of themethodsforupdatingthevariousTEAMstructures(lexicon,schemata)weredevisedtoundotheeffects of previousanswersbeforetheeffects of newanswerscouldbeasserted.HelpinformationisalwaysavailabletoassisttheDBEwhenheisunsurehowtoansweraquestion.Selectingthequestiontextwiththemouseproducesamoreelaboratedescription of theinformationTEAMistryingtoelicit,usuallyaccompaniedbypertinentexamples.Finally,theacquisitioncomponentkeepstrack of whatinformationremainstobesuppliedbeforeTEAMhastheminimumitneedstohandlequeries.TheDBEdoesnothavetodeterminehimselfhowmuchinformationissufficient;allhehastodoistoperceivethatnoacquisitionwindowindicatesremainingunansweredquestions. Of course,theDBEcanalwaysprovideinformationbeyondtheminimum—forexample,bysupplyingadditionalverbs,derivedadjectives,orsynonyms.—20—

... apart of itintoanequivalentquerytree.Thesetransformationsincludereplacement of operators(e.g.Cartesianproductandselectionbyajoin),insertion of newoperators(e.g.anadditionalprojecttoeliminatefieldsasearlyaspossible),andrearrangement of operatorstoachievelowerprocessingcost.1ThisworkwasdoneattheComputerSciencesDepartment,University of Wisconsin—Madison.37Forthispurpose,wewillpresentthetechniquesusedforthisphase of queryprocessinginProjectMULTOS,fortheimplementation of adataserverinwhichdataobjectsareconstitutedbymultimediadocumentswithcomplexinternalstructures.2.TheMULTOSSystemTheMULTOSmultimediadocumentserverhasbeendesignedandimplementedwithinprojectMULTOS(BERT85](BERT86I,whichispart of theEuropeanStrategicProgrammeforResearchinInformationTechnology(ESPRIT).Theinternalstructure of thedocumentserverconsists of acertainnumber of components:thetypehandlermaintainstypedefinitionsandmanagesalltheoperationsontypes;thestoragesubsystemprovidesaccessmethodsfordocumentretrievalandallowsthestorage of largedatavalues,theoperationandstructuretranslatormapsdocumentleveloperationsontothedatastructures of thestoragesubsystem.Thequeryprocessorisresponsiblefortheexecution of queries.Itchecksthesyntacticcorrectness of thequeryandperformsquerydecompositionandqueryoptimization.Theresult of queryexecutionistheset of identifiers of alldocumentssatisfyingthequery.ThequeryprocessorisalsothemodulewhichperformsType-Levelprocessing of thequeries.2.1.TheDocumentModelAmultimediadocumentisacollection of componentswhichcontaindifferenttypes of multimediainformation,andmaybefurtherstructuredinterms of othercomponents(suchasthebody of apaperthatiscomposed of sectionsandparagraphsandcontainsimagesandattributesembeddedintext).Forthesereasons,wecanconsidermultimediadocumentsascomplexdataobjects.Thesecomplexstructures,whichcanvarygreatlyfromonedocumentinstancetoanother,cannotbeadequatelydescribedwiththestructuringmechanisms of traditionaldatamodels.Thus,animportantissueconcernstheadoption of asuitableconceptualmodel.ThedatamodeladoptedinMULTOSisdefinedinIMULT86],andisbasedontheideasexpressedinRABI85]andIBARB85I.DocumentPlaceDateReçeiver+SendelLetter-~-Nam~____NamerAd~1S~StreetCityCountryStreetCityCountryFigure1:Example of DocumentType:Generic_LetterDocument---PlaceDate~~e~er+Send~tter_BodyfTiietc.etc.~~~~--Company_LogoProduct_DescriptionSignatureProduct_PresentationProduct_CostFigure2:Example of DocumentType:Business_Letter2~CUSTOMERORDERCity_INDCity_IND/CPCCIDJIocity=Panscrcity=Pat~/—ocity=Pa~i~~/—\CIDNCID1Cname_~CIDNCIDcnameNcname\/IcnameNcnameC_PC0_PCY0_PC\/ircrlame,age,pnamecxlOlD~l0lDiTcname,age,pnamelTcname,age,pnameQueryTreePT1PT2Figure1:AlternativeProcessingTreesforANon—RecursiveQueryIfnoteveryjoincanbeprocessedusingajoinindex,thenjoinswithjoinindicesmaybecombinedwithmoretraditionaljoinalgorithms.LetusconsiderthequerywhosequerytreeisgiveninFigure2.Supposethatonlyonejoinindexexistsforthatquery.Twocasescanoccur:thereisajoinindexonORDERandPART,orajoinindexonCUSTOMERandORDER.Inthefirstcase,atraditionaljoinprecedesthejoinusingjoinindex.ThetraditionaljoinproducesrelationA,whichisthenusedbothinasemi—joinwiththejoinindex(toselecttherelevantsubset of thejoinindex)andinthefinaljoinusingthejoinindex.Inthesecondcase,thejoinusingthejoinindexprecedesthetraditionaljoin.Figure2showsthePT’scorrespondingtoeachcase.CUSTOMERORDERCPC0PCC_PCJl(CID,OlD)0_PCN/PARTN\/PPCNJl(OID,PID)NP_PC~QueryTreeJoinIndexonORDERandPARTJoinIndexonCUSTOMERandORDERFigure2:ProcessingTreesforDifferentJoinIndices4.Optimization of RecursiveQueriesRecursivequeriescanbemappedintoloops of relationalalgebraoperationsBancilhon861,wheretheoperations of iterationiuseasinputtheresultsproducedbyiteration(i—i).InJagadish87],itisshownthatthemostimportantclass of recursivequeries,calledlinearqueries,canbemappedintoprogramsconsisting of relationalalgebraoperationsandtransitiveclosure.Thus,thetransitiveclosureoperator,extensivelyusedforfix—pointcomputations,is of majorimportanceandrequiresefficientimplementation.InValduriez86a],wehaveillustratedthevalue of joinindicesforoptimizingrecursivequeries,andparticularlytransitiveclosure.14Ajoinindexcapturesthesemanticlinksthatexistbetweentuples.Ifweviewthejoin of twotuplesasanarcconnectingthosetupleidentifiers,ajoinindexcanrepresentdirectedgraphsinaverycompactway.Therefore,itwillbeveryusefultooptimizegraphoperationsliketransitiveclosure.LetusconsideragainthePARTrelation:PART(pname,weight,price,spname)wherespnameisthename of asubpart(orcomponentpart).AssumingthatPIDandSPIDstandforPARTtupleidentifiers,thenwecanhavetwojoinindices(eachclusteredonitsfirstattribute)Jl1(PID,SPID)J12(SPID,PID)J11associatesapart_idwithitssubpart_id’s,whileJ12associatesasubpart_idwithitsparentpart_id.Therefore,Jl1iswellsuitedfortraversalsinthepart—subpartdirection.J12allowsefficienttraversalsthatfollowthesubpart—partdirection.Assumingthatarecursivequeryismappedintoaconceptualquerytree of relationalalgebraoperatorsandtransitiveclosure,thequeryoptimizationalgorithmdiscussedinSection3stillapplies.However,theintroduction of transitiveclosureyieldsalargersolutionspace,sincetransitiveclosuremaybepermutedwithotherrelationaloperators(e.g.,selectandjoin).Thetransitiveclosureoperatorcanbeimplementedefficientlybyaloop of joins,unions,andpossiblydifference(forcyclicrelations).Superiorperformanceisconsistentlyattainedwhentransitiveclosureisappliedusingjoinindexratherthantheprimarycopy of therelationValduriez86a].Forinstance,letusconsidertherecursivequeryonthePARTrelation“listthecomponentpartsandtheirpricesforpartA”.Figure3illustratesthecorrespondingquerytreeandapossibleprocessingtree,inwhichtransitiveclosure(notedIC)isappliedtothejoinindex.Theselection“pname=A”precedesthetransitiveclosuresothatonlythosepartsthatare(transitively)components of partAareproduced.InthePT,theresult of thetransitiveclosureonjoinindexJIisaset of pairs(PID of A,PID of asubpart of A).Therefore,anadditionaljoinwithrelationPARTisnecessarytocompletethequery.Thus,themostcomplexpart of thequeryisdoneonsmalldatastructures(selectionindex,joinindex).Thevalue of performingthetransitiveclosureusingjoinindexistoavoidrepeatedaccesstorelationPART,whichispotentiallymuchlargerthanthejoinindex.pname_INDPARTJl1(PID,SPID)PARTapname=Aapname=A/PARTSPID1~1PID‘ITpname,price‘~pname,priceQueryTreePTwithJoinIndexFigure3:Processing of aRecursiveQuerywithJoinIndex5.ConclusionJoinindicesaredatastructuresespeciallydesignedtospeedupjoinoperations.Theincorporation of joinindicesinastoragemodelprovidesthequeryoptimizerwithalargersolutionspaceandhencemoreopportunitiesforoptimization.Wehaveillustratedtheuse of joinindicestooptimizenon—recur-15abstraction,andmaybeconvenientlydescribedlevelbylevel.(Forexample,whatarethelegalsequences of joins;whatsubgraphcanbetheimplementation of ajoin,whatinterestingsortordersshouldbeconsidered?)Thesearchedstrategyspaceconsists of thosestrategiesthathavebeenconsidered(explicitlyorimplicitly)inthesearch.Typically,werestrictattentiontodetail-levelstrategies,sincethisiswherestrategyeliminationoccurs.(Rulesforeliminatingsuboptimalstrategiesarediscussedbelow).Anoptimizerfullysearchesitspotentialstrategyspaceifitperformssufficientstrategygenerationandcostevaluationguaranteethatithasfoundthelowestcostexpressioninthespace.(Incontrast,directedsearchestrytoquicklyfindagoodsolution).Ifallelaborations of someabstract-levelsubgraphhavebeensearched,wesaythat ... apart of itintoanequivalentquerytree.Thesetransformationsincludereplacement of operators(e.g.Cartesianproductandselectionbyajoin),insertion of newoperators(e.g.anadditionalprojecttoeliminatefieldsasearlyaspossible),andrearrangement of operatorstoachievelowerprocessingcost.1ThisworkwasdoneattheComputerSciencesDepartment,University of Wisconsin—Madison.37Forthispurpose,wewillpresentthetechniquesusedforthisphase of queryprocessinginProjectMULTOS,fortheimplementation of adataserverinwhichdataobjectsareconstitutedbymultimediadocumentswithcomplexinternalstructures.2.TheMULTOSSystemTheMULTOSmultimediadocumentserverhasbeendesignedandimplementedwithinprojectMULTOS(BERT85](BERT86I,whichispart of theEuropeanStrategicProgrammeforResearchinInformationTechnology(ESPRIT).Theinternalstructure of thedocumentserverconsists of acertainnumber of components:thetypehandlermaintainstypedefinitionsandmanagesalltheoperationsontypes;thestoragesubsystemprovidesaccessmethodsfordocumentretrievalandallowsthestorage of largedatavalues,theoperationandstructuretranslatormapsdocumentleveloperationsontothedatastructures of thestoragesubsystem.Thequeryprocessorisresponsiblefortheexecution of queries.Itchecksthesyntacticcorrectness of thequeryandperformsquerydecompositionandqueryoptimization.Theresult of queryexecutionistheset of identifiers of alldocumentssatisfyingthequery.ThequeryprocessorisalsothemodulewhichperformsType-Levelprocessing of thequeries.2.1.TheDocumentModelAmultimediadocumentisacollection of componentswhichcontaindifferenttypes of multimediainformation,andmaybefurtherstructuredinterms of othercomponents(suchasthebody of apaperthatiscomposed of sectionsandparagraphsandcontainsimagesandattributesembeddedintext).Forthesereasons,wecanconsidermultimediadocumentsascomplexdataobjects.Thesecomplexstructures,whichcanvarygreatlyfromonedocumentinstancetoanother,cannotbeadequatelydescribedwiththestructuringmechanisms of traditionaldatamodels.Thus,animportantissueconcernstheadoption of asuitableconceptualmodel.ThedatamodeladoptedinMULTOSisdefinedinIMULT86],andisbasedontheideasexpressedinRABI85]andIBARB85I.DocumentPlaceDateReçeiver+SendelLetter-~-Nam~____NamerAd~1S~StreetCityCountryStreetCityCountryFigure1:Example of DocumentType:Generic_LetterDocument---PlaceDate~~e~er+Send~tter_BodyfTiietc.etc.~~~~--Company_LogoProduct_DescriptionSignatureProduct_PresentationProduct_CostFigure2:Example of DocumentType:Business_Letter2~CUSTOMERORDERCity_INDCity_IND/CPCCIDJIocity=Panscrcity=Pat~/—ocity=Pa~i~~/—\CIDNCID1Cname_~CIDNCIDcnameNcname\/IcnameNcnameC_PC0_PCY0_PC\/ircrlame,age,pnamecxlOlD~l0lDiTcname,age,pnamelTcname,age,pnameQueryTreePT1PT2Figure1:AlternativeProcessingTreesforANon—RecursiveQueryIfnoteveryjoincanbeprocessedusingajoinindex,thenjoinswithjoinindicesmaybecombinedwithmoretraditionaljoinalgorithms.LetusconsiderthequerywhosequerytreeisgiveninFigure2.Supposethatonlyonejoinindexexistsforthatquery.Twocasescanoccur:thereisajoinindexonORDERandPART,orajoinindexonCUSTOMERandORDER.Inthefirstcase,atraditionaljoinprecedesthejoinusingjoinindex.ThetraditionaljoinproducesrelationA,whichisthenusedbothinasemi—joinwiththejoinindex(toselecttherelevantsubset of thejoinindex)andinthefinaljoinusingthejoinindex.Inthesecondcase,thejoinusingthejoinindexprecedesthetraditionaljoin.Figure2showsthePT’scorrespondingtoeachcase.CUSTOMERORDERCPC0PCC_PCJl(CID,OlD)0_PCN/PARTN\/PPCNJl(OID,PID)NP_PC~QueryTreeJoinIndexonORDERandPARTJoinIndexonCUSTOMERandORDERFigure2:ProcessingTreesforDifferentJoinIndices4.Optimization of RecursiveQueriesRecursivequeriescanbemappedintoloops of relationalalgebraoperationsBancilhon861,wheretheoperations of iterationiuseasinputtheresultsproducedbyiteration(i—i).InJagadish87],itisshownthatthemostimportantclass of recursivequeries,calledlinearqueries,canbemappedintoprogramsconsisting of relationalalgebraoperationsandtransitiveclosure.Thus,thetransitiveclosureoperator,extensivelyusedforfix—pointcomputations,is of majorimportanceandrequiresefficientimplementation.InValduriez86a],wehaveillustratedthevalue of joinindicesforoptimizingrecursivequeries,andparticularlytransitiveclosure.14Ajoinindexcapturesthesemanticlinksthatexistbetweentuples.Ifweviewthejoin of twotuplesasanarcconnectingthosetupleidentifiers,ajoinindexcanrepresentdirectedgraphsinaverycompactway.Therefore,itwillbeveryusefultooptimizegraphoperationsliketransitiveclosure.LetusconsideragainthePARTrelation:PART(pname,weight,price,spname)wherespnameisthename of asubpart(orcomponentpart).AssumingthatPIDandSPIDstandforPARTtupleidentifiers,thenwecanhavetwojoinindices(eachclusteredonitsfirstattribute)Jl1(PID,SPID)J12(SPID,PID)J11associatesapart_idwithitssubpart_id’s,whileJ12associatesasubpart_idwithitsparentpart_id.Therefore,Jl1iswellsuitedfortraversalsinthepart—subpartdirection.J12allowsefficienttraversalsthatfollowthesubpart—partdirection.Assumingthatarecursivequeryismappedintoaconceptualquerytree of relationalalgebraoperatorsandtransitiveclosure,thequeryoptimizationalgorithmdiscussedinSection3stillapplies.However,theintroduction of transitiveclosureyieldsalargersolutionspace,sincetransitiveclosuremaybepermutedwithotherrelationaloperators(e.g.,selectandjoin).Thetransitiveclosureoperatorcanbeimplementedefficientlybyaloop of joins,unions,andpossiblydifference(forcyclicrelations).Superiorperformanceisconsistentlyattainedwhentransitiveclosureisappliedusingjoinindexratherthantheprimarycopy of therelationValduriez86a].Forinstance,letusconsidertherecursivequeryonthePARTrelation“listthecomponentpartsandtheirpricesforpartA”.Figure3illustratesthecorrespondingquerytreeandapossibleprocessingtree,inwhichtransitiveclosure(notedIC)isappliedtothejoinindex.Theselection“pname=A”precedesthetransitiveclosuresothatonlythosepartsthatare(transitively)components of partAareproduced.InthePT,theresult of thetransitiveclosureonjoinindexJIisaset of pairs(PID of A,PID of asubpart of A).Therefore,anadditionaljoinwithrelationPARTisnecessarytocompletethequery.Thus,themostcomplexpart of thequeryisdoneonsmalldatastructures(selectionindex,joinindex).Thevalue of performingthetransitiveclosureusingjoinindexistoavoidrepeatedaccesstorelationPART,whichispotentiallymuchlargerthanthejoinindex.pname_INDPARTJl1(PID,SPID)PARTapname=Aapname=A/PARTSPID1~1PID‘ITpname,price‘~pname,priceQueryTreePTwithJoinIndexFigure3:Processing of aRecursiveQuerywithJoinIndex5.ConclusionJoinindicesaredatastructuresespeciallydesignedtospeedupjoinoperations.Theincorporation of joinindicesinastoragemodelprovidesthequeryoptimizerwithalargersolutionspaceandhencemoreopportunitiesforoptimization.Wehaveillustratedtheuse of joinindicestooptimizenon—recur-15abstraction,andmaybeconvenientlydescribedlevelbylevel.(Forexample,whatarethelegalsequences of joins;whatsubgraphcanbetheimplementation of ajoin,whatinterestingsortordersshouldbeconsidered?)Thesearchedstrategyspaceconsists of thosestrategiesthathavebeenconsidered(explicitlyorimplicitly)inthesearch.Typically,werestrictattentiontodetail-levelstrategies,sincethisiswherestrategyeliminationoccurs.(Rulesforeliminatingsuboptimalstrategiesarediscussedbelow).Anoptimizerfullysearchesitspotentialstrategyspaceifitperformssufficientstrategygenerationandcostevaluationguaranteethatithasfoundthelowestcostexpressioninthespace.(Incontrast,directedsearchestrytoquicklyfindagoodsolution).Ifallelaborations of someabstract-levelsubgraphhavebeensearched,wesaythat ... abasefortheoptimizerbuildsaveryclearandconciseframeworkformodularization of theDBI’soptimizercode,andfacilitatesincrementaldevelopmentandtesting.Therulesaretranslatedbytheoptimizergeneratorintoexecutablesourcecode,whichiscompiledandlinkedwiththesupportfunctionsandotherdatabasesoftware.Interpretativetechniqueswereruledoutbyourresearchbecausetheresultingoptimizers,includingoneprototypeimplementationundertakeninProlog,arelimitedtotheinterpreter’ssearchstrategy,andboundtobeslow.Oneinterestingdesignissuethatremainsistoprovidegeneralsupportforpredicatesassomeform of predicateislikelytoappearinalldatamodels.WritingtheDBIcodeforpredicates,andoperatorargumentsingeneral,wasthehardestpart of developingouroptimizerprototypes.Inthecurrentdesign,theDBImustdesignhisorherowndatastructuresandprovidealltheoperationsonthemforbothruleconditionsandargumenttransferfunctions.Itmaybedifficulttoinventagenerallysatisfyingdefinitionandsupportforpredicates,butitwouldbeasignificantimprovement of theoptimizergenerator.Thefactthatpredicatesareaspecialcase of argumentsposesanadditionalchallenge,sincetheoveralldesign of theargumentdatastructuremuststillremainwiththeDBI.Moregenerally,werealizethattheoptimizergeneratorworkslargelyonthesyntacticlevel of thealgebra.Thesemantics of thedatamodelarelefttotheDBI’scode.Thishastheadvantage of allowingtheDBImaximalfreedomwiththetype of datamodelimplemented,butithasthedisadvantage of leavingasignificantamount of codingtotheDBI.Wethereforewouldliketoincorporatesomesemanticknowledge of thedatamodelintothedescriptionfile.This,however,isalong-termgoaltowhichwehavenotyetgivenmuchattention.6.AcknowledgementsTheauthorappreciatesthegeneroussupportandadvicebyDavidDeWitt,MikeCarey,andthestudentmembers of theEXODUSproject.ReferencesBarrl98l.A.BarrandE.A.Feigenbaum,TheHandbook of ArtificialIntelligence,WilliamKaufman,Inc.,LosAltos,CA.(1981).Blasgenl977.M.BlasgenandK.Eswaran,“StorageandAccessinRelationalDatabases,”IBMSystemsJournal16(4)(1977).Carey1986.MJ.Carey,DJ.DeWitt,D.Frank,G.Graefe,J.E.Richardson,EJ.Shekita,andM.Muralikrishna,“TheArchitecture of theEXODUSExtensibleDBMS:APreliminaryReport,”Proceedings of theInt’lWorkshoponObject-OrientedDatabaseSystems,pp.52-65(September1986).Careyl986a.MJ.Carey,DJ.DeWitt,J.E.Richardson,andE.J.Shekita,“ObjectandFileManagementintheEXODUSExtensibleDatabaseSystem,”Proceedings of theConferenceonVeryLargeDataBases,pp.91-100(August1986).DeWittl984.DJ.DeWitt,R.Katz,F.Olken,L.Shapiro,M.Stonebraker,andD.Wood,“ImplementationTechniquesforMainMemoryDatabaseSystems,”Proceedings of theACMSIGMODConference,pp.1-8(June1984).Graefel987.G.GraefeandDJ.DeWitt,“TheEXODUSOptimizerGenerator,”Proceedings of theACMSIGMODConference,pp.160-171(May1987).42operations.Aswewillsee,thisiscriticaltoaplug-compatiblesoftwaretechnology.Second,thereareotheralgorithmsbesidesthethreelistedabove.Toaddanewinvertedfileretrievalalgorithm,onesimplyaddsanothercase.Thesameappliesforotherabstractoperations(e.g.,abstractrecordinsertion,deletion,etc.).Theytooareimplementedascasestatements,andnewalgorithmsareincludedasadditionalcases.Introducinganotheralgorithmtoprocessanabstractfileorabstractlinkoperationiscalledalgorithmextensibility.Third,anelement of queryoptimizationcanbeseeninthisexample.WhenaRET(AF,Q)istobeprocessed,thecheapestavailablealgorithmisselected.Thisisaccomplishedbyevaluatingcostfunctionsforeachalgorithmandchoosingthecheapestalgorithm.Thecodetemplateformakingthisoptimizationdecisionhasthefollowingorganization:$RET(AF,Q)/*takenfromINDEXmodule*/retum(minimum _of ($RET(F,Q),/*Mgicostfunction~/$USE_1_INDEX(AF,Q,$RET(I~,Q~)~$ACC(F,p)),1*A1g2costfunction*1$USE_n_IND1CES(AF,Q,$RET(I~,Q~).$ACC(F,p))1*A1g3costfunction*1)Twopoints.First,thecostfunctions$USE_1_INDEXand$USE_n_INDICEShavealgorithmstodeterminewhichindicestousetoprocess~inordertoachievetheoptimalperformance.(Astherearemanysuchalgorithms,thealgorithmthatisactuallyusedcouldbeaparametertothe$RET(AF,Q)costfunction.Doingsowouldprovideasimplemeansbywhichoptimizingalgorithmscouldbechanged).Theinformationaboutwhatindicestouseisretainedaspart of theoptimizationprocess,andislaterusedwhentheselectedalgorithmsareexecuted.Second,notethatactualexpressionsforthecostfunctionsforindexfileretrieval$RET(I~,Q~)anddatafileaccessing$ACC(F,p)arenotspecified;onlywhentheimplementation of thesefilesisgiveninamoduleexpressioncanactualcostexpressionsbeassignedtothesefunctions.Insummary,thebuildingblocks of DBMSsareparameterizedandnonparameterizedfiletypes.Thesoftwarebuildingblocksarethealgorithmsthatmapoperationsonabstractfilestooperationsonconcretefiles.Thecostmodelbuildingblocksarecostfunctionsthatareassociatedwiththeseoperationmappings.Composition of buildingblocksisconsideredinthenextsection.4.SyntheticDBMSsandSyntheticPerformanceModelsAsmentionedintheprevioussection,asimpleinvertedDBMS,similartoINGRES,isdescribedbythemoduleexpressionNDEXHEAP,BPLUS].Inthisparticularexample,datafileoperationsreferencedintheINDEXalgorithmsaremappedtoheapfileoperations,andindexfileoperationsaremappedtoB+treeoperations.LetCbeaconceptualfile,Dbeitscorrespondingdatafile,I~beanindexfile,andQbeaselectionpredicate.Thealgorithmstoretrieveconceptualrecordsareacomposition of INDEX,BPLUS,andHEAPretrievalandaccessalgorithms:RET(C,Q)(case(use_$RET(C,Oj_to_choose_cheapest) of Mgi:RET_HEAP(D,Q);Alg2:USE_1_INDEX(C,Q,RET_BPLUS(IJ,Q~).ACC_IiEAP(D,p));A1g3:USE_n_INDICES(C,Q,RET_BPLUS(I~,Q~).ACC_HEAP(D,p));TheabovealgorithmswerederivedbyreplacingRET(D,Q)withRET_HEAP(D,Q),RET(I~,Q~)withRET_BPLUS(IJ,Q~).andACC(D,p)withACC_HEAP(D,p).34MESH,MESHisacomplexnetwork of pointers.Ateachstepinthesearch,thetransformationperformedistheonewhichcarriesthemostpromisethatitwilleventually,viasubsequenttransformations,leadtotheoptimalqueryevaluationplan.Thecrucialelementinthissearchstrategyisthepromisecalculation,calledthepromiseevaluationfunction.Itmustincludethecurrentqueryandplan,otherqueriesandplanswhichhavewerefoundearlierinthesearchprocess,andinformationaboutthetransformationruleinvolved.Themostnaturalmeasureforpromiseisthecostimprovement of theaccessplans.3.Modularization of DBICodeInanextensibledatabasesystem,therearealwayssomepartsintheoptimizer(andinothercomponentsaswell)thatcannotbeexpressedinarestricted,e.g.rule-basedlanguage.ThesepartsarebestwrittenintheDBI’simplementationlanguage.AsoftwaretoolisusedtocombinetherulesandtheDBI’ssourcecode.Foreasyextensibility,itisveryimportanttoassisttheDBIindividingthecodeintomeaningful,independentmodules.Notonlyisamodularoptimizereasiertoimplement,weenvisionthisasanaidformaintainingadatabasemanagementsystemthatevolvesovertime.Someoptimizercomponentscanonlybedefinedafterthedatamodelhasbeendefined(data-model-dependentcomponents),andhencemustbeprovidedbytheDBI.Inthissection,wewillbrieflyreviewthesecomponents,andhowtheyarebrokenintomodules.Wegenerallyassociatetheseprocedureswithone of theconceptsthatwehaveintroducedearlier,namelyoperators,methods,andrules.3.1.Data-Model-DependentDataStructuresTherearetwotypes of data-model-dependentdatastructuresthatareimportantintheoptimizationprocess.First,thereareargumentsforoperatorsandmethods.Second,inalmostallcasesitisdesirabletomaintainsomedictionaryinformationforintermediateresultsinaquerytree.Wetermsuchdictionaryinformationproperties of theintermediateresults.Sincedefiningthesedatastructuresispart of customizinganextensibledatabasesystem,theoptimizationcomponent of suchasystemmusttreatthesestructuresas“blackboxes”.InEXODUS,wedefineanduseaproceduralinterfacetomaintainandqueryproperties.Furthermore,wedistinguishbetweenoperatorandmethodarguments,andbetweenoperator-dependentandmethod-dependentproperties.Asanexamplefromarelationalsystem,cardinalityandtuplewidthareoperator-dependentproperties,whereassortorderisamethod-dependentproperty.3.2.RulesandConditionsIntheEXODUSoptimizationconcept,theset of operators,theset of methods,transformationrules,andimplementationrulesarethecentralcomponentsthattheDBIspecifiestoimplementanoptimizer.Therulesarenon-procedural;theyaregivenasequivalencelawsthatthegeneratortranslatesintocodetoperformtreetransfonnations.Each of theserulesshouldbeself-contained.Onlythenisitpossibletoexpandtherulesetsafelyasthedatamodelevolves.Therulesexpressequivalence of querytrees.Treeexpressions,i.e.algebraicexpressions,embodytheshape of atreeandtheoperatorsinit.Forsomerules,however,applicabilitydoesnotdependonthetree’sshapeandtheoperatorsalone.Forexample,sometransformationsmightonlybepossibleifanoperatorargumentsatisfiesacertaincondition.SinceoperatorargumentsshouldbedefinedbytheDBI,suchconditionscannotbeexpressedinadatamodelindependentform.WeallowtheDBItoaugmentruleswithsourcecodetoinspecttheoperatorarguments,thedatadictionary,etc.3.3.CostFunctionsAsmentionedearlier,processingcostoccursbyexecutingaparticularalgorithm.Thecostcalculationiscloselyrelatedtotheprocessingmethodbeingexecuted.Hence,weassociatecostfunctionswiththemethods,andcalculatethecost of aqueryexecutionplanasthesum of thecosts of themethodsinvolved.Theparameters of acostfunctionarethecharacteristics of thedatastreamsservingasinputsintothemethod,e.g.thenumber of dataobjectsineachinputdatastream,andthemethodargument,e.g.apredicate.3.4.PropertyFunctionsThecharacteristics of thedatastreamwhichareneededasparameterstothecostfunctionsaredatamodel-dependent.Thus,theymustbedefinedbytheDBI.Weattachcharacteristics,whichwecallproperties,withboththeoperatorsandthemethods.Operators(andtheirarguments)determinethelogicalproperties of anodeinaquerytree,e.g.cardinality.Aparticularalgorithmormethodchosendefinesphysicalproperties of an39Lohmanl985.0.Lohman,C.Mohan,L.Haas,D.Daniels,B.Lindsay,P.Selinger,andP.Wilms,“QueryProcessinginR*,”pp.31-47inQueryProcessinginDatabaseSystems,ed.J.W.Schmidt,Spnnger,Berlin(1985).Mackertl986.L.F.MackertandG.M.Lohman,“R*OptimizerValidationandPerformanceEvaluationforLocalQueries,”Proceedings of theACMSIGMODConference,pp.84-95(May1986).Mackertl986a.L.F.MackertandG.M.Lohman,“R*OptimizerValidationandPerformanceEvaluationforDistributedQueries,”Proceedings of theConferenceonVeryLargeDataBases,pp.149-159(August1986).Richardson1987.J.E.RichardsonandMJ.Carey,“ProgrammingConstructsforDatabaseSystemImplementationinEXODUS,”Proceedings of theACMSIGMODConference,pp.208-219(May1987).Selingerl979.P.GriffithsSelinger,M.M.Astrahan,D.D.Chamberlin,R.A.Lone,andT.G.Price,“AccessPathSelectioninaRelationalDatabaseManagementSystem,”Proceedings of theACMSIGMODConference,(June1979).4Joinindicescouldbeusedinmanydifferentstoragemodels.However,inordertosimplifyourdiscussionregardingqueryoptimization,wepresenttheintegration of joinindicesinasimplestoragemodelwithsingleattributeclusteringandselectionindices.Thenweillustratetheimpact of thestoragemodelwithjoinindicesontheoptimization of non—recursivequeries,assumedtobeSPJqueries.Inparticular,efficientaccessplans,wherethemostcomplex(andcostly)part of thequerycanbeperformedthroughindices,canbegeneratedbythequeryoptimizer.Finally,weillustratetheuse of joinindicesintheoptimization of recursivequeries,wherearecursivequeryismappedintoaprogram of relationalalgebraenrichedwithatransitiveclosureoperator.2.StorageModelwithJoinIndicesThestoragemodelprescribesthestoragestructuresandrelatedalgorithmsthataresupportedbythedatabasesystemtomaptheconceptualschemaintothephysicalschema.Inarelationalsystemimplementedonadisk—basedarchitecture,conceptualrelationscanbemappedintobaserelationsonthebasis of twofunctions,partitioningandreplicating.Allthetuples of abaserelationareclusteredbasedonthevalue of oneattribute.Weassumethateachconceptualtupleisassignedasurrogatefortupleidentity,calledaTID(tupleidentifier).ATIDisavalueuniqueforalltuples of arelation.Itiscreatedbythesystemwhenatupleisinstantiated.TID’spermitefficientupdatesandreorganizations of baserelations,sincereferencesdonotinvolvephysicalpointers.Thepartitioningfunctionmapsarelationintooneormorebaserelations,whereabaserelationcorrespondstoaTIDtogetherwithanattribute,severalattributes,oralltheconceptualrelation’sattributes.Therationaleforapartitioningfunctionistheoptimization of projection,bystoringtogetherattributeswithhighaffinity,i.e.,frequentlyaccessedtogether.ThereplicatingfunctionreplicatesoneormoreattributesassociatedwiththeTID of therelationintooneormorebaserelations.Theprimaryuse of replicatedattributesisforoptimizingselectionsbasedonthoseattributes.Anotheruseisforincreasedreliabilityprovidedbythoseadditionaldatacopies.inthispaper,weassumeasimplestoragemodel...