ResearchsupportedbyONRContractN00014-91-J-4052,ARPAOrder8225.

FunctionalProgrammingFormalisms GerdHillebrand,ParisKanellakis, DepartmentofComputerScience Providence,RhodeIsland02912 andsridharramaswamy foroodbmethods BrownUniversity CS-94-28 May1994

BrownUniversity,Providence,RI02912,U.S.A. foroodbmethods GerdHillebrand,ParisKanellakis,andSridharRamaswamy FunctionalProgrammingFormalisms minglanguagesare(1)applicativeprogramschemasand(2)typedlambda (OODBs)andinparticulartothedescriptionofmethods. calculi.werelatetheseprogrammingformalismstoobject-orienteddatabases Abstract.Twowell-studiedfunctionalformalismsinthetheoryofprogram- databasesisptime. classes,methods,inheritance,nameoverloading,andlatebinding.from[4],we databasequerylanguagesbyshowingthatitsexpressivepoweroverniteordered applicativeprogramschemaswithadditionalkeyobject-orientedfeaturessuchas presentitssyntaxandsemanticsandsurveythestate-of-the-artofconsistency studyingdatabaseschemaevolution.wethenrelatemswithmoreconventional checkingorsignatureinferenceforthislanguage,aproblemwhichcanbeusedin Despiteitssimplicityandapplicability,MSdoesnotdirectlymodelthetuple, Thelanguageofmethodschemas(MS)isaprogrammingformalismbasedon set,andlistcomplexstructuresthatarequitecommonindatabases.also,it calculus/algebra,datalog(withorwithoutnegation),andthecomplexobject doesnottreatfunctionsasobjects,i.e.,methodsaredierentfromobjects.it ispossibletoachievethesetwocapabilitiesusingthetypedlambdacalculus subsumesmostconventionaldatabasequerylanguagesincludingtherelational withequality(tlc=)asadatabasequerylanguage,evenwithoutanyobjectorientedfeatures.from[25],weillustratehowthispurefunctionallanguage calculus/algebra(withorwithoutpowerset). withtheexpressivetlc=andfacilitatesoperationsonsetsofobjects. OODBsmustbeafunctionallanguagethatcombinestheobject-orientedMS 1Introduction Object-orienteddatabases(OODBs)representanewgenerationofdatabase technology,whichhasdevelopedthroughacombinationofadvancedprogramminglanguagefeaturesandsuccessfulrelational/networkdatabasetechniques, see[9,30,44].bothprogramminglanguagetheoryanddatabasetheoryhave Inconclusion,wearguethattheappropriateprogrammingformalismfor ResearchsupportedbyONRContractN00014-91-J-4052,ARPAOrder8225.

2beenproposedasstartingpointsforOODBformalfoundations.Theyareboth legitimatestartingpoints,butoneshouldbeawarethatanaivesynthesisofsuch questionhowtomatchdistinctprogrammingparadigmsinonelanguage. foroodbs"fromafunctionalperspective.weviewtheorganizationofdata distincttheoriesisnotnecessarilyharmonious.forexample,itislargelyanopen Inthispaperweexploretheproblem\whatistheappropriateformalmodel GerdHillebrand,ParisKanellakis,andSridharRamaswamy intoobjectswithmethodsasthecorenewfeatureofoodbs.thisisafunctioncentric(ormethod-centric)viewofdataasopposedtothestructure-centricview ofdatainrelationaldatabases(e.g.,in[28,41]). Becauseofthefunction-centriccharacterofOODBswebelievethatthisisoneof advocatedinthepast,e.g.,[15,38],butmuchremainstobedoneinthisarea. themostpromisingapproachestotheproblemoftheirformalization.notethat someoftheactualquerylanguagesinoodbprototypesarefunctional[9]and thatrecentresearchoncomplex-objectsandtypeshasemphasizedfunctional models[13,14]. Interestingfunctionalapproachestodatabasequerylanguageshavebeen anoodbformalismcalledmethodschemas(ms)in[4].weshowthatmethod schemas(overordereddatabases)expressexactlytheptimequeries.(2)we culi[19,11,12].werelatethesetwoprogrammingformalismstooodbsand inparticulartothedescriptionofoodbmethods. guagesare(1)applicativeprogramschemas[21,23]and(2)typedlambdacal- presentthetypedlambdacalculusasaformalismforexpressingawiderangeof relationalandcomplex-objectdatabasequeries[25].inthiscase,theelementary Inordertoaccomplishthis:(1)Wemodifyapplicativeprogramschemasinto Twowell-studiedfunctionalformalismsinthetheoryofprogramminglan- timequeries(whichproperlycontaintheptimequeries)areexpressible,becauseofalimitedabilitytocreateandmanipulateliststructures.so,interms themoreconvenient)framework. ofexpressivepower,thetypedlambdacalculusisthemoregeneral(evenifnot andsemanticsofmethodschemasandmotivatethemwithanoodbexample.methodschemaconsistencyisaformof\typeinference,"whichinvolves nameoverloading,andlatebinding.insection2.1,weexplainthesyntax programschemaswithadditionalfeaturessuchasclasses,methods,inheritance, generality,inpurefunctionalform.morespecically,weproceedasfollows. oftheexpressivepowerofmethodschemaswedemonstratehowmanynotions fromrelationalandobject-orienteddatabasescanbeformulated,withnolossof MSisasimpleprogrammingformalismforOODBs,basedonapplicative Byouroverviewoftheapproachesin[4,25]andournewcharacterization state-of-the-artonconsistencychecking.weusevariousalgorithmictechniques andtechniquesfromthetheoryofprogramschemas,e.g.,[8,33]. schemaevolutionproblem[4,10,26,39,43,45].insection2.2,wesurveythe checkingformethodschemasisanalgorithmicformalizationofthedatabase consistencyatrun-timeinsomeinterpretation.amoreprecisetermforthis problemwouldbesignatureinference.theincrementalversionofconsistency decidingatcompile-timewhetheragivenmethodschemacanleadtoanin- InSection3,weexaminetheexpressivepowerofMSbyrelatingittothemore

functionalcharacterizationintermsofequationsispresentedin[24].notealso ofptime(analogoustothelogicalcharacterizationsof[27,42]).adierent that,datalogwithnegationhasbeenusedin[5]toprovidesemanticstoanumber ofvariationsofmethodschemas.insection3.1weoutlinetheinput-output 31,6].Forniteorderedinterpretationsthisisafunctionalcharacterization conventionalformalismsofrelationalcalculus[20]anddatalogwithnegation[18, FunctionalProgrammingFormalismsforOODBMethods 3 conventions,insection3.2wearguethateveryptimequeryisexpressible, Datalog,complexstructures(suchassets,tuplesorlists)arenotdirectlypartof andinsection3.3thatonlyptimequeriesareexpressible. theframeworkbuthavetobeencoded(intorelationsfordatalog,intotheclass functionalframeworkbutdonottreatfunctionsasobjects,i.e.,methodsare dierentfromobjects.toachievethisuniformityonehastoexaminehigherorderframeworksrelatedtothelambdacalculus.anotherproblemisthat,asin somedisadvantages,however.methodschemasareanexampleofapractical itincorporatesmanyadditionaldesirableobject-orientedfeatures.thereare theveryrobustconceptofptimedatabasequeriesthatisattractivebecause OurexpressivepowerresultindicatesthatMSisasyntacticvariationon semantics,andthetypesofthesimplytypedlambdacalculuswithequality notionoftypeshereisstructuralandquitedierentfromthenotionofclasses. (TLC=).Wereferto[25]forhowtoremoveequalityfromthiscalculus.Forthe hierarchyandmethodsforms).intypedlambdacalculiwecanuseasimple typedisciplinetoencodecomplexstructuresinafarmoredirectfashion.the problemoftypeinferencewereferto[29]. manydatabasequerylanguagesincludingtherelationalcalculus/algebra[20], lus/algebra(withorwithoutpowerset)[1,3].weconsiderembeddingssuch Datalog(withorwithoutnegation)[18,31,6],andthecomplexobjectcalcu- thatforeachqueryexpressedthereisalambdatermwhichwhenappliedtothe InSection4,wepresentsomebasicbackgroundonthesyntax,theoperational inputdatabase. database;mostimportantly,ifthequeryexpressedisinptimethenthenormal formcanbecomputedinanumberofreductionstepspolynomialinthesizeof inputdatabase(whichisalist-of-tupleslambdaterm)normalizestotheoutput Itispossibleto\naturallyembed"inthetypedlambdacalculuswithequality basedontheanalysisofreductionsequencesof[40,34]. tuplesandlists,aswellastheimportantconceptoflistiteration.insection5.2 wepresent,asanextendedexample,theembeddingoftherelationalalgebra. Ouranalysisoftheexpressivepowerofthesimplytypedlambdacalculusis languagetypesystems,i.e.,onlysimpletypeswithoutanypolymorphism.itis thatthesefeaturesallowtheembeddingofhigher-orderqueries,e.g.,thesecondorderqueriesof[22,17].notethatweonlyusetheverycoreofprogramming InSection5.1wediscusstheinput-outputconventionsforrepresentingsets, of[25]inthetypedlambdacalculus. schemasandtheirrecursivedenitioncanallbesimulatedusingthexpoints interestingthatthe(otherwiseveryconvenient)useofmethodnamesinmethod Wereferto[25]forthemoreadvancedxpointandpowersetfeatures.Note Theembeddingswepresentformtherststepinunderstandingtheconnec-

4PTIMEquerieswepresenthereamethodschemaframework(Section3)andthe characterizationsofptimewereferto[24,32]. databasetheoryoneciency,wecanrestrictourselvestoptimequeries.for tionsbetweenfunctionalprogramminganddatabases.giventheemphasisof basisforalambdacalculusframework(section5and[25]).forotherfunctional Inconclusion,inSection6wecommentonthepossiblecombinationofMS GerdHillebrand,ParisKanellakis,andSridharRamaswamy withtlc=aswellassomeopenresearchquestions. Weassumetheexistenceofthefollowingdisjointcountablesetsofsymbols: 2MethodSchemas(MS) 2.1SyntaxandOperationalSemantics ofclasses(c1;c2;:::)andofmethods(m1;m2;:::).foreachmethodm,the arityofmisanintegergreaterorequaltozero.asignatureisanexpression (butsincetheword\type"hasbeenmisusedinmanydierentcontextswe expressionis!c1.inthisformalismthesignaturesplaytheroleof\types" c1;:::;ck 1!ck,whereeachci,for1ik,isaclass.Fork=1,the prefertousethemoreprecisetermsignature). integerinf1;:::;kg;(2)eachvertexlabeledbyamethodofarityjhasjchildren fork1isatriple(m;(c1;:::;ck);t),wheremisamethodofarityk,for 1ik,eachciisaclass,andtisak-term(i.e.,k-termsareourprogramming (Fork=1,themethodmiszero-ary.)Acodeddenitionofmatc1;:::;ck, (dag)suchthat:(1)eachvertexisuniquelylabeledeitherbyamethodorbyan formalism). ck)),wheremisamethodofarityk 1andtheremainingpartisasignature. Ak-termforsomek1isaniterootedordereddirectedacyclicgraph Abasedenitionofmatc1;:::;ck 1,fork1isapair(m;(c1;:::;ck 1! nition,also2-terms,3-terms,etc.theideaisthatk-termsrepresentfunctions ofkarguments,whereacopyofargumentiisplacedateachinputlabeledi. goingfromlefttorightbecausethechildrenofeachvertexareordered. theyareconstantinputs. Verticescanalsohaveoutdegreezeroandbelabeledbyzero-arymethods,i.e., inputsandeachvertexlabeledwithanintegerisaninput;(4)therootiscalled theoutput;(5)thedagcomeswithauniquelydeneddepthrstsearchorder whichareorderedfromlefttoright;(3)allverticesofout-degreezeroarecalled Forexample,thetermstandt0inFigure1are1-terms.Theyare,byde- LetsbeasinglearcwithtaillabeledPriceandheadlabeled1,andtbeasin Figure1.Thesedenitionscanberepresentedsyntacticallyusinganintuitive Asanexample,considerthefollowingdenitions: lambdanotation insteadofthegraphicalk-termnotation asshownbelow. (Cost;(Part);t) (Cost;(Basepart);s) (Price;(Basepart!Int))

FunctionalProgrammingFormalismsforOODBMethods Assemblingcost \\\\\ ZZZZ SumCost 5 ZZZZ ZZZZ 1Subparts Head CostSum 1Cost Notethatweusethesymbolcolon(:)forrepresentingbasemethodsandthe Fig.1.Termtandt0 Tail symbolequal(=)forcodedmethods. Denition1.Amethodschemaisa4-tuple(C;;0;1)where: 1.CisanitesetofclassesandisaforestonC.Wesaythatcc0ifcisa Price@Basepart:Int Cost@Basepart=x:Price(x) 3.1isasetofcodeddenitionswithclassesfromC.LetM0bethesetof 2.0isasetofbasedenitionswithclassesfromC. descendantofc0inthisforest.thepartialorderdenedbyiscalledthe Isarelation. Cost@Part=x:Sum(Assemblingcost(x);Cost(Subparts(x))) Figure1.MethoddenitionsarepresentedinFigure3. Example1.ConsidertheclasshierarchyofFigure2andthetermstandt0in 4.Eachmethodofaritykhasatmostonedenitionatc1;:::;ckforeach of1arefromm0[m1. requirethatm0\m1=;andthatthemethodsthatappearink-terms c1;:::;ck. methodsdenedin0andm1bethesetofmethodsdenedin1.we Fig.2.AnexampleIsahierarchy IntBasepart Part HHHH CompPart PairOfParts schemas. Letusbrieysurveysomeimportantconceptsinconnectionwithmethod

6Head Tail Price Sum @PairOfParts:Part @Basepart:Int @Int;Int:Int GerdHillebrand,ParisKanellakis,andSridharRamaswamy Fig.3.Methods Assemblingcost@Part Cost Subparts @Basepart=x:Price(x) @PairOfParts=t0 :PairOfParts allitsmethodshavearity1.aschemaisrecursion-freeifthereisnodirected dencegrapharethecodedmethodsandthereisanarcfrommtonimoccurs Notethatsimpleschemasareanimportantkindofrecursion-freeschemas. isareasonablesyntacticwayofavoidingrecursion,butnotnecessarilytheonly cycleinthemethoddependencegraph,wheretheverticesofthemethoddepen- one.)aschemaissimpleifonlybasemethodsoccurinitscodeddenitions. insomecodeddenitionofn.(notethat,ourdenitionofrecursion-freeness Syntacticallyrestrictedfamiliesofmethodschemas:Aschemaismonadicif ofbasemethods.asimpleschemaiscovariantifforeachmandforeachpair methods,i.e.,0;1,whichwecallexplicit.itisalsousedtodenemethods ofdenitionsofabasemethodm,wehavethat(8ic0ici))c0c. Forsimpleschemas,covarianceisthefollowingconstraintonthesignatures implicitly,throughinheritance.thisisfortheconvenienceofcodereuse.therefore,inadditiontotheexplicitdenitionsforamethod,denitionsareimplicitly MethodInheritance:Amethodschema(C;;0;1)containsdenitionsof (m;(c01;:::;c0k!c0)) (m;(c1;:::;ck!c)); c01;:::;c0k,thenitsdenitionimplicitlyappliesatc1;:::;ckwherecic0ifor i=1;:::;k. inheritedalongtheclasshierarchy.ifamethodmisexplicitlydenedatclasses itisalsoexplicitlydenedforpairofparts. andgivenclasses,auniquedenition.theresolutionofamethodmatsome itlyinheritedbybasepartandcomppart,andexplicitlyredenedonbasepart; methodcostintheexample.thatmethodisexplicitlydenedonpart,implic- condition4ofthemethodschemadenitiontherecanbeonlyoneexplicitdenition.thiscreatesoverloadingofnames.toillustrateoverloading,considerthinitionsforthesamebaseorcodedmethodatthesameclasses althoughby Resolutionofnameoverloading:Thisconsistsofassigningtoagivenmethod Nameoverloading:Methodinheritanceimpliesthatwecanhaveseveraldef-

(ifsuchauniquecomponent-wiseminimumexists).forexample,costat c1;:::;ckistheexplicitdenitionofmforthe\component-wisesmallest"tuple CompPartisresolvedtobetheexplicitdenitionfromPart,whereasCostat BasepartisBasepart'sexplicitdenition.Ifmhasaresolutionatc1;:::;ck,we c01;:::;c0katwhichmhasanexplicitdenitionandcic0i,fori=1;:::;k saythatmiswelldenedatc1;:::;ck.otherwise,wesaythatmisundenedat FunctionalProgrammingFormalismsforOODBMethods 7 denition)andmisundenedatc1;c1(ambiguityofdenitions).formethods explicitbasedenitionsofmatc1;c2andc2;c1.thenmisundenedatc2;c2(no ofarity1,ambiguity(2)cannotarisebecauseoftheforesthierarchy,butwecan Example2.Letc1;c2beclasseswithc1c2andconsidertheschemawith havenon-denitionbecauseof(1). abovec1;:::;ckor(2)becausethereisambiguityofdenitionsabovec1;:::;ck. c1;:::;ck.non-denitioncancomeeither(1)becausethereisnodenitionofm classesaremappedtodisjointsetsofobjects(setswithpairwiseemptyintersections).foreachc,wedene(c)=sc1c(c1).wewillrefertosc2c(c)as (;)whereisadisjointobjectassignmentforcandatotalfunctionfrom thebasemethodsm0instopartialfunctionssuchthat: 1.Ifmhasarityk,(m)isapartialfunctionfromk-tuplesofobjectsinto mentforcisatotalfunctionfromctonitesetsofobjectssuchthatdistinct ments[2]toprovidesemanticsforclasseswithinheritance.thesemanticsof basemethodsisdenedusingpartialfunctionssatisfyingcertainsignatureconstraints. Denition2.Givenamethodschema(C;;0;1),adisjointobjectassign- objects(o1;o2;:::)disjointfromclassesandmethods.weuseobjectassign- Objectassignments:Weassumetheexistenceofacountablyinnitesetof thesetofobjectsin. Denition3.AninterpretationofaschemaS=(C;;0;1)isapairI= toits\subclasses"c1,wherec1candc16=c.whenabasemethodisdened areinthisclassexclusively(thoseobjectsthatarein(c))andtherestbelong 2.Foreachc1;:::;ck,ifmisundenedatc1;:::;ck,then(m)isundened Intuitively,everyclasscispopulatedbyasetofobjects(c),someofwhich havethat(m)j(c1)(ck)isatotalfunctioninto(c). everywherein(c1)(ck).otherwise,lettheresolutionofmat c1;:::;ckbe(m;(c01;:::;c0k!c))wherecic0ifori=1;:::;k.thenwe objectsin. equivalenttofunctioncomposition. Thatis,givenanobjectexclusivelyinc1,returnanobjectinc. basemethodsandarewritingtechnique.withoutrecursion,thisrewritingis ataclassc1withsignaturec1!c(orisinheritedatc1withsignaturec01!c suchthatc1c01),thenitsmeaningatc1isatotalfunctionfrom(c1)to(c). Thesemanticsofcodedmethodsisdenedusingtheabovesemanticsof

bereplacedbyanobjecto,asa\procedurecall"tocost.basedontheclassof abase/codedfashion,orbyanerrormessageifitisundened.ingeneral,we termthatisasinglearc.letitsoneinternalnodebelabeledcostanditsinput 8itsargument,wecanreplaceCosteitherbysomeobject/codeifitisdenedin usrstgivesomeintuitionfortherewritingofcodedmethods.considerak- Latebinding:Thisisaresultoftheoperationaldenitionofrewriting.Let GerdHillebrand,ParisKanellakis,andSridharRamaswamy reducetherstmethod(rstinthedepthrstorderingofthek-termweare processing)thathasinstantiatedleavesaschildren.wecontinuethisprocessing untilweeitherobtainaresult(i.e.,anobject),orreachaninconsistency.we mightalsonotterminate.apartialsequenceofrewritingsforexample1is showninfigure4,whereoisinclasspairofpartsando0;o00areinclassbasepart. CostCostCostCostCostPriceCost17Cost17Cost??????? JJ^ Sum @@R Sum @@R? Sum @@R? Sum @@R? Sum @@R whoseintegerlabelshavebeenreplacedbyobjects.letc1;:::;ckbeclasses, Fig.4.Areductionsequence oheadtailo0tailo0tail Moreformally,aninstantiatedtermininterpretationI=(;)isak-term, o o o Tail o o00? termconsistingofasinglevertexlabeledmandkinputslabeledrespectively mbeamethodofarityk,andforeachiletoi2(ci).thentheinstantiated c1;:::;ck.thereductioni(t)oftiniisobtainedasfollows: Leto1;:::;okbethelabelsofthechildrenofwbelongingrespectivelytoclasses tiatedtermini,wbetherstreduciblevertexoft,andmbethelabelofw. term,whichistherootofaredex. Denition4.LetIbeaninterpretationofmethodschemaS,tbeaninstan- o1;:::;okissaidtobearedex.wedenetherstreduciblevertextobethe uniquelydeterminedrstvertexinthedepthrstorderingofaninstantiated 2.Ifmisabasemethodwelldenedatc1;:::;ck,letitsresolutiontherebe 1.Ifmisundenedatc1;:::;ckthenI(t)isaspecialsymbol?. istheinstantiatedtermobtainedbyremovingfromwtheoutgoingedges (m;(c01;:::;c0k!c)).(thisimpliesthatcic0ifori=1;:::;k.)theni(t) t!i(t)andanitesequenceofreductionsfromttot0ast!t0. 3.Ifmisacodedmethodwelldenedatc1;:::;ck,thenI(t)istheinstantiated WhenIisclearfromthecontext,wedenotethereductionofttoI(t)as andchangingitslabelto(m)(o1;:::;ok).(m)isatotalfunctionfrom (c1)(ck)into(c).therefore,(m)(o1;:::;ok)yieldsanobjecto suchthato2(c). termobtainedbysubstituting,withthenaturalrenamingofinputs,the vertexwbythedags,where(m;;s)istheresolutionofmatc1;:::;ck.

ofaritykwelldenedattheseclasses,andforeachiletoi2(ci).thenthe 2.2ConsistencyandSignatureInference Weareinterestedintherewritingsequencesofaparticularsetofinstantiated schemawithinterpretationi=(;),c1;:::;ckbeclassesins,mamethod terms,i.e.,thesearetheinitial\procedurecalls"thatmakesenseaccordingto amethodschema.moreformallywehavethefollowing:letsbeamethod FunctionalProgrammingFormalismsforOODBMethods 9 Denition5.AmethodschemaSisconsistentifforeachinterpretationIofS itisimpossibletorewriteanystart-redexinto?inanitenumberofsteps. ifforsomestart-redextwehavet!?. respectivelyo1;:::;okissaidtobeastart-redex.wehaveaninconsistencyini instantiatedtermconsistingofasinglevertexlabeledmandkinputslabeled sistent?(2)aretherepossible/certaindivergingcomputations?(3)ismethodm on(1)althoughmostofthetechniquesthatwedevelopcanbeusedforstudying monadicarityandabsenceofrecursion. otherproblemssuchas(2)or(3).twopropertiessimplifyconsistencychecking: (atc)possibly/certainly\reachable"fromm0(atc0)[26]?weconcentratehere odseachtimetheschemaisupdated.theproblemistoobtainanincremen- talconsistencycheckingalgorithmthatwouldavoidredoingthesameveri- cations/computations.asolutionthatisadoptedpractically(e.g.,[9])isto Inanobject-orientedcontext,itisnotreasonabletorecompileallmeth- Avarietyofquestionsareimportantinthissetting,e.g.:(1)Isaschemacon- beaectedisthecrucialpart.webrieysummarizesomeresultsfrom[4](on whatmaybeaected)andsomeopenquestions.letnbethesizeofmethod maintainadependencygraphofthemethodsandrecompileonlymethodsthat mayhavebeenaectedbytheupdate.ofcourse,understandingwhatmay denitionsintheinputmethodschemaandcthesizeoftheclasshierarchy. Bothpolyadicmethodsandrecursionarenecessaryforthisresult.Theproofis byreductionfromresultsin[33].thisundecidabilityresultholdsforschemas withmethodsofaritylessthanorequalto3.thedecidabilityofmethodschema consistencywithmethodsofarity2isopen. Inthegeneralcaseofpolyadicrecursiveschemas,consistencyisundecidable. Onecanfocusonthekeyrecursion-freecaseofasinglecodedmethodinthe forconsistencyislogspace-completeinptime.thisindicatesthatveryecient incrementalsolutionsforeventhesimplestsyntacticcasesmaybehardtoobtain. decisionprocedurerunsino(nc3)time. Tohandleoverloading,onecanuseamodicationofatechniqueof[8].The thislanguagehasanon-emptyintersectionwithaparticularregularlanguage. describedusingacontext-freelanguage.aninconsistencymaybereachedif Inthecaseofmonadicandrecursion-freeschemas,checkingawholeschema Inthecaseofmonadicschemas,thesetofpossiblecomputationscanbe logspace-completeinnlogspaceandcanbedonemoreecientlythanthe recursivecaseaboveusingnitestateautomatatechniques(ino(nc2)time). Inheritanceandoverloadingintroducesomenondeterminismintheautomata contextofbasemethods.checkingasinglecodedmethodforconsistencyis

consistencyofmonadic,covariant,recursion-freeschemas.wealsodonotknow notknowwhetherthisradix-treedatastructurecanbeextendedtosolvethe whetherthereareecientincrementalalgorithmsforanyoftheconsistency 10 andcanbecheckedino(n+c)timeusingaradix-treedatastructure.wedo theoreticapproachtoconsistencychecking,butcovariantsignaturesremoveit. Consistencyofasinglecodedmethod,assumingcovariance,isinDLOGSPACE GerdHillebrand,ParisKanellakis,andSridharRamaswamy problems.sinceweknowthattheproblemofconsistencycheckingformonadic, recursion-freeschemasisptime-complete,itwouldbeinterestingtoknowif therearegoodincrementalalgorithmsforthesimple,monadicorthecovariant, monadiccases. completeforasinglecodedmethodwithtwoarguments.somespecialcases doesnothelpintherecursivecase.however,intherecursion-freecovariantcase thereisaptimetestforaxedaritycodedmethod.thisisinterestingin canbeshowntobeinptime,usingtreeautomatatechniques.covariance practice,becauseitmotivatesaheuristicforthegeneralcase. Inthecaseofrecursion-freeschemas,theconsistencyproblemiscoNP- abovetechniquesandresultsaredirectlyapplicablealsotomultipleinheritance argument.webelievethatoncetheresolutionmechanismisdetermined,the itance.also,insomeproposals,itisrequiredthattherstargument,called andattachmenttorstargument. Eachinstanceofamethodisthenviewedas\attached"totheclassoftherst thereceiver,ofamethodmustdeterminetheresolutionofnameoverloading. Insomeproposalstheclasshierarchyisadagandonehasmultipleinher- thissection,wedescribeaverysimplewayofencodingrelationsasinterpreted interpretationofthebasemethodsinordertomakethisquestionprecise.in basemethods,andweshowthatunderthisinterpretation,methodschemas 3TheExpressivePowerofMethodSchemas Ofcourse,sincemethodschemasarejustschemas,onehastoagreeonthe expressexactlytheptimequeries.weassumesomefamiliaritywithdatalog Whenconsideringmethodschemasinthecontextofnitedatabases,itisnatural terminologyandnotationfrom[28,41],e.g.,rules,edbs,idbsetc. toaskwhatkindofqueriescanbeexpressedinthemethodschemaframework. otherrelationooveru.thearitiesandattributenamesoftheinputandoutput relationsaregivenbysomexedrelationalschemas. 3.1I/OConventions Weconsiderqueriesoverniteorderedrelationaldatabases.Theinputofa positiveintegers)andsomerelationsp;q;r;:::overu,andtheoutputisan- queryconsistsofaniteordereduniverseu(w.l.o.ganinitialsegmentofthe elementsoftheuniversearerepresentedbyobjectsofclassnumberandthe orderingamongthemisgivenbyabasemethodpred@number:numberwhich Weencodethesedatainthemethodschemaframeworkasfollows.The

sentedbyanobjectofclasszero,whichisasubclassofnumber.wecapture FunctionalProgrammingFormalismsforOODBMethods computesthepredecessorfunction.theinitialelementoftheuniverseisrepre- thenitenessoftheuniversebyprovidingitssizeaspartoftheinput.more precisely,weprovideaspartoftheinputanobjectnofclassnumbersuchthat jectofclasszeroandthatthereisatleastonenon-zeroelementintheuniverse. NdoesnotappearintherangeofPred.Weassumethatthereisauniqueob- 11 languageismswithorder. Whenourinterpretationssatisfythesesemanticconditionswewillsaythatour classnumberifthetuplerepresentedby(x1;:::;xk)belongstorandanobject r@number;:::;number:number,suchthatr(x1;:::;xk)returnsanobjectof ofclasszeroifitdoesnot.wewillfrequentlywriter(~x)insteadofr(x1;:::;xk) ifthearityofthemethoddoesnotmatter. Ak-aryrelationRisrepresentedbyak-arybasemethodrofsignature q@number;:::;number:number.ifq(~x)reducestoanobjectofclassnumber, oftheoutput. andzero=f0g,wewouldinterpretpredaspred(2)=1,pred(1)=pred(0)=0, thenthetuplerepresentedby~xisconsideredpartoftheoutput;ifq(~x)reduces andwewouldinterpretrasr(1;1)=r(1;2)=1,r(x;y)=0forallotherx;y. f0;1;2g,wewouldpopulatetheclassesnumberandzeroasnumber=f1;2g toanobjectofclasszero,thenthetuplerepresentedby~xisnotconsideredpart Queriesarerepresentedbycodedmethodsqwithasignatureoftheform Forexample,toencodetherelationR=f(1;1);(1;2)govertheuniverseU= oftheseconcepts).theresultsofimmerman[27]andvardi[42]thenimplythat Weshowthatundertheinput/outputconventionsgivenabove,methodschemas cancomputeinationaryxpointsofdatalog:programs(see[28]forasurvey methodschemascancomputeallptimequeriesoverniteordereddatabases. 3.2ExpressingPTIMEQueries anumberobject,orfalsewhenitreturnsthezeroobject. Pred(N),wherePred@Number=x:Pred(Pred(x))andPred@Zero= x:x.)wewillofteninformallysaythatamethodreturnstruewhenitreturns functions.werepresentthetruthvaluestrueandfalsebytheclassesnumber singleobject0ofclasszerorepresentsfalse.(notethatwecancompute0as0= andzero.anyobjectofclassnumber(n,forexample)representstrue,andthe TheBooleanfunctionscanbecodedas: Tobeginwith,letusillustratehowtocodesimpleBooleanandarithmetical And@Zero;Number=xy:0 And@Zero;Zero Not@Number Not@Zero And@Number;Zero=xy:0 And@Number;Number=xy:N(andsimilarforOr) Equal@Zero;Zero =x:0(where0:=pred(n)) =x:n

12 denemethodspredkisuchthatpredki(x1;:::;xk)istheithcomponentofthe WealsoneedtoextendthePredfunctiontok-tuplesofobjects.Below,we Equal@Zero;Number=xy:0 Equal@Number;Zero=xy:0 Equal@Number;Number=xy:Equal Pred(x);Pred(y) GerdHillebrand,ParisKanellakis,andSridharRamaswamy lexicographicalpredecessorofthetuple(x1;:::;xk),for1ik. Number;:::;Number;k i+1 i 1 ourpurposes,wecanthinkofadatalog:programasarelationalexpression Withthismachineryinplace,wecannowcodeDatalog:programs.For Predki@ z } ki { z} { Zero;:::;Zero=x1:::xk:N Zero;:::;Zero=x1:::xk:Pred(xi) andproject.thesemanticsoftheprogramaregivenbytheleastxpointof E(O;P;Q;R;:::)involvingadesignatedoutputrelationO,inputrelationsP, canbecomputedbyiteratingthismappingnktimes,wherenisthesizeofthe Q,R,:::,andtheoperatorsUnion,Intersection,Complement,Times,Select, themappingo7!o[e(o;p;q;r;:::).itiswellknownthattheleastxpoint =x1:::xk:xi(i6=k) universeandkisthearityofo. relationsrandsasdescribedabove,then: Therelationaloperatorscanbecodedasfollows.Assumethatrandscode Projectionisabitmoredicult,becauseitinvolvesquantierelimination. Intersection(r;s)~x:And r(~x);s(~x) Times(r;s)~x~y:And r(~x);s(~y) Complement(r)~x:Not r(~x) Selecti=j(r)~x:And r(~x);equal(xi;xj) Union(r;s)~x:Or r(~x);s(~x) TheprojectionofRontoitsrstk 1columnsisencodedas: r(x1;:::;xk 1;z)isTrue.r0canbecodedasfollows: wherer0(x1;:::;xk 1;y)isTrueithereexistsaz2f0;:::;ygsuchthat r0@ Project(r)x1:::xk 1:r0(x1;:::;xk 1;N); x1:::xk 1y:Or r(x1;:::;xk 1;y);r0(x1;:::;xk 1;Pred(y)) z Number;:::;Number;Zero=x1:::xk 1y:r(x1;:::;xk 1;y) Number;:::;Number;Number= } {

xpointofewithrespecttooisgivenby: representingo,andletebethecodedmethodcorrespondingtoe.thenthe bearelationalexpression.letkbethearityofo,letobethebasemethod FunctionalProgrammingFormalismsforOODBMethods Finally,wehavetoshowhowtocomputexpoints.LetE(O;P;Q;R;:::) Fix(o;e)x1:::xk:o0(x1;:::;xk;k z} { N;:::;N); 13 O7!O[E(O;P;Q;R;:::),~nbeinginterpretedasabase-(N+1)number. o0canbedenedasfollows: whereo0(~x;~n)istruei~xisintheimageofthe~nthiterateofthemapping wherethemethodearisesfromebyreplacingeveryoccurrenceofo(~x)ineby o0(~x;predk1(~n);:::;predk(~n)). o0@number;:::;number;k z} { Thus,foreveryDatalog:program(interpretedinaninationaryfashion ~x~n:or o0(~x;predk1(~n);:::;predk(~n));e(~x) ; z Number;:::;Number= Zero;:::;Zero k { =~x~n:0 canexpresseveryptimequery. Lemma6.UndertheinputoutputconventionsofSection3.1,MSwithorder spondenceofniteorderedrelationaldatabasesandmethodschemawithorder interpretations.wecanconcludethat, input-outputconventionsofsection3.1,thereisaneasytoseeone-to-onecorre- Firstnotethatthismethodschemaisconsistent.Alsonotethat,underthe theinput-outputconventionsofsection3.1)thatcomputesthesamequery. overanorderedinput)onecanconstructamethodschemawithorder(under Inthissection,weshowthattheexpressivepowerofmethodschemasover methodschemasexpressexactlytheptimequeries. orderedniteinterpretationsisnomorethanthatofdatalog:overordered input.this,combinedwiththepreviousreduction,allowsustodeducethat 3.3SimulatingMethodSchemasinDatalog thatorderstheinput. theinputsofthemethodandthe(k+1)thargumentistheoutput.basemethods meansofaunaryedbpredicate.thefactthattheinputisorderediscaptured inthedatalogsimulationbymeansofaspecialinterpretedbinaryedbpredicate correspondnaturallytoedbsandcodedmethodstoidbs.thedisjointobject arityk+1.theunderstandingisthattherstkargumentsofthepredicateare input/outputcharacteristicsofamethodofaritykbymeansofapredicateof assignmenttoclassesinaninterpretationofamethodschemaissimulatedby Wewillrstexplaintheintuitionbehindthesimulation.Wecapturethe willbeinputdatabasestothedatalogprogramthatdonotcorrespondtoany Twotechnicalproblemsarisewiththisapproach.Therstisthatthere

14 theinputdatabasedoesnotcorrespondtoavalidinterpretation.thesecond interpretationforthemethodschema.inordertohandlethiscase,weintroducerulesthatwill\ood"thecomputationbyderivingeverypossiblefactif technicalproblemistheissueofinconsistencies.inthecaseofmethodschemas, aninconsistencycorrespondstoanabnormalterminationofcomputation.in thedatalogsimulation,wesimplyfailwhenaninconsistencyoccursinthecorrespondingmethodexecution.thetwocasesaresimilarinthesensethatno outputisproducedfortheoendingmethodcall. plexity,wewillassumethatthesizeofthemethodschemaisxed.wewillalso schemas,wewillconstructadatalogprogrampssuchthatthecomputations assumethattherearenoambiguousmethoddenitions.(thisisasyntactic propertythatcanbecheckedinconstanttime.)further,wewillalsoassume thatallthemethoddenitionsareexplicitatalltheclassesatwhichtheyare weassumethatsisofxedsize,thisdoesnotmatter.fortheinputmethod dened.thismightincreasethesizeoftheschemabyafactorofckwherecis thesizeoftheclasshierarchy,andkthemaximumarityofthemethods.since ConsideraschemaS=(C;;0;1).Sinceweareinterestedindatacom- GerdHillebrand,ParisKanellakis,andSridharRamaswamy performedbysandpsareequivalentinasensewhichwillbeexplainedbelow. 1.Foreachclassc2C,aunaryEDBpredicateIsac.TheIsacpredicatesare PShasthefollowingpredicates: 5.AnIDBpredicateNotequalthatcapturesinequalitybetweeninputobjects. 4.Foreachcodedmethodm21ofarityk,a(k+1)-aryIDBpredicatem. 3.Foreachbasemethodm20ofarityk,a(k+1)-aryEDBpredicatem. 2.AspecialinterpretedbinarypredicatePredthatorderstheinputbyprovidingthepredecessorfunction.Therewillbeobjectso0(zero)andoN(highest valueofx.predcorrespondsnaturallytotheinterpretedbasemethodpred object)suchthatpred(o0;o0)isafactandpred(x;on)isnottrueforany meanttocapturethemeaningofthedisjointobjectassignment. inmethodschemas. correspondingidbpredicatemhasthefollowingrule.thevariablesintherule formcorrespondto(1)theinputstothiscodeformethodmand(2)theresult andintermediateresultsofthiscodeformethodm. 6.AunaryIDBpredicateFloodthatwillbeusedtocheckthevalidityofthe Foreachmethoddenitionm@c1;c2;:::;ck=x1;:::;xk:m1()inS,the inputdatabase. ItiscodedintermsofPred. built-ininterpreterinmethodschemas.)theargumentstothesemethodsare Theunderstandinghereisthattherstkpredicatesactas\guards"making arrangedastheywouldbeinamethodschemaexecution. surethatthe\code"thatgetsexecutedhastheargumentsinthecorrectclasses. Afterthat,wehavethemethodnamesthatappearinthecodeofmsortedin left-to-righttopologicalorder.(thisistheorderofexecutionprovidedbythe m(c1;c2;:::;ck;x): Isac1(C1);:::;Isack(Ck); mi1(:::);mi2(:::);:::;m1(:::;x);

dierentwaysinwhichtheinputdatabasecanbeinvalid.theyare:(1)the (4)abasemethodmightproducearesultofaninvalidoutputclass.Wehandle FunctionalProgrammingFormalismsforOODBMethods disjointobjectassignmentcouldbeinvalid;(2)abasemethodcouldbeundened atsomepointwhereitissupposedtobedened(basemethodscorrespondto totalfunctions);(3)abasemethodcouldbemultiplydenedatsomepoint; WenowwritedownthedenitionsfortheFloodpredicate.Therearefour 15 theseonebyonebymeansofthefollowingrules.beforethat,weneedtocode enforcethedisjointobjectassignment: inequalityamongobjects.thisiseasilydonewiththefollowingrules: Foreverypairofclassesc1andc2(c16=c2),weusethefollowingruleto Notequal(X;Y): Pred(X;X0);Pred(Y;Y0);Notequal(X0;Y0); Flood(X): Isac1(C);Isac2(C); Notequal(X;Y): :Pred(X;X);Pred(Y;Y); Notequal(X;Y): Pred(X;X);:Pred(Y;Y); thereisanobjectthatdoesnotbelongtoexactlyoneclass,flood(x)isderived wherec1;c2;:::;clarealltheclassesinc.theserulesensurethatwhenever forallvaluesofx. everyidbpredicatep: Weoodthecomputationbywritingdownrulesofthefollowingformfor Tomakesurethateveryobjectbelongstoaclass,weusethisrule: Flood(X): :Isac1(C);:Isac2(C);:::;:Isacl(C); thefollowingrulesthatoodthecomputationifmisnotdenedeverywherein (c1)(ck): Next,ifabasemethodminShasthesignaturem@c1;:::;ck:ck+1,wewrite Next,forabasemethodminSwiththesignaturem@c1;:::;ck:ck+1,we Flood(X): Isac1(C1);:::;Isack(Ck);:m(C1;:::;Ck;Ck+1); p(x1;:::;xk): Flood(X); writethefollowingrulethatoodsthecomputationifmismultiplydened thefollowingrulethatoodsthecomputationifmisimproperlydened,i.e., somewherein(c1)(ck). Finally,forabasemethodminSwithsignaturem@c1;:::;ck:ck+1,wewrite Notequal(Ck+1;C0k+1); m(c1;:::;ck;ck+1);m(c1;:::;ck;c0k+1);

16 theresultbelongstoaninvalidclass.letck1;ck2;:::;ckjbethesubclassesof ck+1,includingck+1itself. Flood(X): Isac1(C1);:::;Isack(Ck); m(c1;:::;ck;ck+1); :Isack1(Ck+1);:::;:Isackj(Ck+1); GerdHillebrand,ParisKanellakis,andSridharRamaswamy indatalog:bymeansoftheisac,m,andprededbpredicatesisthesamesize and\valid"inputdatabasesforps. astheoriginalmethodschemainterpretation. Lemma7.EveryinterpretationformethodschemaScorrespondstoaninput ofinterestisthesizeoftheinputdatawhichwemeasurebythesizeoftheinterpretationi.itisimmediatelyclearthatourrepresentationofaninterpretation Wenowestablishaone-to-onecorrespondencebetweeninterpretationsforS Sinceinthissimulationweareinterestedindatacomplexity,theparameter sensethatflood(x)willneverbederivedforanyvalueofxforsuchadatabase. Lemma8.Every\valid"inputdatabaseforPS(oneinwhichFlood(X)is databaseforps.further,thisinputdatabaseisa\valid"inputdatabaseinthe neverderivedforanyvalueofx)correspondstoaninterpretationfors. Lemma10.Ifafactm(o1;o2;:::;ok;ok+1)canbederivedinPSwithavalid bederivedforanyvalueofxinsuchaninputdatabase. performthesamecomputations. Lemma9.Acalltomethodmwithargumentso1;o2;:::;okinaninterpretationIterminateswitharesultok+1ifandonlyifm(o1;o2;:::;ok;ok+1)canbe derivedinthecorrespondinginputdatabaseinps.further,flood(x)willnever Now,wecanusestructuralinductiontoshowthatSandPSdoindeed witharesultok+1inthecorrespondinginterpretationforp. inputdatabase,thenacalltomethodmwithargumentso1;o2;:::;okterminates expressesexactlyptime. 4TheTypedLambdaCalculuswithEquality(TLC=) 4.1LanguageDenition Theorem11.UndertheinputoutputconventionsofSection3.1,MSwithorder Weputeverythingtogetherinthefollowingtheorem: Typed-termsaregivenbythegrammarExj(EE)jx:T:E,wherexranges overasetofexpressionvariables.asusual,!!standsfor!(!) andpqrstandsfor(pq)r. overasetoftypevariables.thus,isatype,asare(!)and(!(!)). ThesyntaxoftypesisgivenbythegrammarTtj(T!T),wheretranges

where isafunctionfromexpressionvariablestotypes,and [x:]isthe FunctionalProgrammingFormalismsforOODBMethods function 0augmenting with 0(x)=: (Var) Well-typednessofexpressionsisdenedbythefollowinginferencerules, (x)= `x: 17 (!-Elim) (!-Int) `e:!0 `e0: `x::e:!0 [x:]`e:0 max(1+o(0);o(00)).wealsorefertotheorderofatyped-termastheorder ofitstype. termofthattype,isdenedaso(t)=0foratypevariablet,ando(0!00)= isderivablebytheaboverules,forsome and. Forwell-typed-termse;e0,wewritee>e0(-reduction)whene0canbe Theorderofatype,whichmeasuresthehigher-orderfunctionalityofa- Inwhatfollows,wewrite\typed-termE"tomeanEiswelltyped: `E: `ee0:0 derivedfromebyrenamingofaboundvariable,forexamplex::y::y> x::z::z.wewritee>e0(-reduction)whene0canbederivedfromeby =:o!o!!!.forconstantsa;b:o,a6b,weaddto>the constantsoftypeo,andforsomexedtype,introducinganequalityconstant above,byenrichingthesimply-typed-calculuswithacountablyinnitesetof replacingasubtermineoftheform(x::e)e0bye[e0=x](ewithe0substitutedforallfreeoccurrencesofxine).thefactthatthesubtermanditrem[12,theorem4.2.8].wewrite>forthereexive,transitiveclosureof replacementhavethesametypeisreferredtoasthesubjectreductiontheo- >and>. existsa-terme000suchthate0>e000ande00>e000. Theorem12(Church-Rosserproperty).Ife>e0ande>e00,thenthere reductionrules(=aa)>(x::y::x)and(=ab)>(x::y::y). Inthispaper,weconsideramildmodicationofthestandardpresentation reductions. Theorem13(Strongnormalizationproperty).Foreache,thereexistsan integernsuchthatife>e0,thenthederivationinvolvesnomorethann- Themodiedsystemenjoysthefollowingwell-knownproperties: discussesthechurch-rosserpropertyfortheuntyped-calculusenrichedwith equality.wereferto[25]forhowtoremoveequalityfromthelanguage. i.e.theproblemofdecidingwhetheragiven(untyped)termcanbetyped,seefor ForresultsconcerningthetypeinferenceproblemforTLCanditsextensions, Formoredetails,seeforexample[19,11].Inparticular,[11,Section15.3]

of[35]withoutletandwithoutanyexplicitxpointconstruct. calculus.ithasnotypepolymorphismandisasubsetofthecore-mllanguage unication. 18 example[29].thetypeinferenceproblemfortlc=isequivalenttorst-order Notethat,thelanguagepresentedaboveisthe\simplest"typedlambda GerdHillebrand,ParisKanellakis,andSridharRamaswamy 5TheExpressivePowerofTLC= Lisatypabletermnomatterwhattypewechoose(thoughonexedterm howitworksanditslimitations. Theprincipaltechnicaltoolforthissectionislistiteration.Letusbrieyreview isa-termoftype(!!)!!,foranytype inotherwords, mustbechosen).weabbreviatethislistconstructionas[x1;x2;:::;xk];the variablescandnabstractoverthelistconstructorsconsandnil.listiteration implementsprimitiverecursion. Lc:!!:n::cx1(cx2:::(cxkn):::) Letfx1;x2;:::;xkgbeasetof-terms,eachoftype;then andfalsex::y::y,bothoftypebool!!.denetheexclusive oras andtheparityofalistofbooleanvaluesas Xorp:Bool:q:Bool:x::y::p(qyx)(qxy) ParityL:(Bool!Bool!Bool)!Bool!Bool:LXorFalse: Forexample,astandardcodingofBooleanlogicusesTruex::y::x Unlikecircuitcomplexity,thesizeoftheprogramcomputingparityisconstant, becausetheiterativemachineryistakenfromthedata,i.e.,thelistl. wheresuccn:(!)!!:s:!:z::ns(sz)andzeros:! :z::zcodesuccessorandzeroonthechurchnumerals(oftypeint(! )!!). LengthL:(!Int!Int)!Int!Int:L(x::Succ)Zero; Ontheotherhand,tocomputethelengthofalist,wedene Bool!!substitutedfor,whichwewriteasB[:=Bool].Similarly,!)!!foranytype.TotypeParityL,wemusttypeLwith totypelengthl,weneedtotypel:b[:=int].thinkingofthetypeofl tothisproblem. simplytypedlambdacalculus:itslackofpolymorphism.therearetwofacets asanoutputtypevariable,inbothinstancesweseethattheoutputtypemust be\contaminated"or\raised"toprovidetheprimitiverecursiveiterator.ifwe (1)AlistLofBooleanscanbecodedasatermoftypeB(Bool! Thesetwosimpleexamplespointalreadytoarestrictionimposedbythe

codings,forexample FunctionalProgrammingFormalismsforOODBMethods wanttheoutputtypeofinputltobexed,andtwodierentoutputtypesare neededtocarryoutthecomputation,thisposesaproblem. LengthL:(!!)!!:s:!:z::L(x::s)z; Wepointoutthatproblem(2)cansometimesbehandledbyalternateen- (2)Ifwewanttheoutputtypetoremain,thisposesyetanotherdiculty. 19 wherelisusedtoiterateoverobjectsoflowertype.problem(1)canbesolved Int,andusingthistypingtoiterativelygeneratetwocopiesofthelist,with bydeningtheoutputtypetobeasuitablydenedcrossproductofbooland setsofrelations,whichareencodedaccordingtotheschemebelow. respectiveoutputtypesboolandint. followingsections. 5.1I/OConventions Theparadigmweuseiscomputationonnitestructures,i.e.,a-calculusfor nitemodeltheory.inputsandoutputsofacomputationaredatabases,i.e., Thesetechniques,withelaboratevariants,areexploitedrepeatedlyinthe basetypeo.wealsox,onceandforall,atypevariable. Weabbreviatethetype!!asBool. True:!!u::v::u; False:!!u::v::v: Booleanvaluesarerepresentedby Westartoutwithacountablyinnitesupplyo1;o2;:::ofconstants,ofsome 1;:::;k,then Weabbreviatethetype(1!!k!)!as1kor,ifi hx1;:::;xki:(1!!k!)! Tuplesarerepresentedinthestandardway:ifx1;:::;xkare-termsoftype then for1ik,ask.inparticular,okisthetypeofak-tupleofconstants. [x1;:::;xk]:(!!)!! Listsarerepresentedasdescribedabove:ifx1;:::;xkare-termsoftype, f:1!!k!:fx1:::xk oftermsofaxedtype. thetypeofak-aryrelationisfokg.notethat,inallthisdevelopmentweuse termsofvaryingtype,whereasalistrepresentsacollectionofavariablenumber betweenlistsandtuples:atuplerepresentsacollectionofaxednumberof Weabbreviatethetype(!!)!!asfg.Notethedierence Relationsarerepresentedasduplicate-freelistsoftuplesofconstants.Thus, c:!!:n::cx1(cx2:::(cxkn))

the-calculus;butweshowhowtoeliminateduplicates. whenappliedtoencodingsr1;:::;rkofinputrelations,reducetoanormalform 20 listsinsteadofnitesets,becausenitesetsarenotavailableinthesyntaxof whichistheencodingofthedesiredoutputrelation.wemeasurethetime complexityofcomputationbythelengthofreductionsequences,accordingtoa Queriesarerepresentedas-termsoftheformQR1:::Rk:M,which GerdHillebrand,ParisKanellakis,andSridharRamaswamy eratorwhichisalistofalltheobjectsintheuniverse(theequivalentofthe isequivalenttothe\relationsascharacteristicfunctions"conventionadopted insection3.1,providedthatinthelattercase,theinputcomeswithanit- \sizeparameter"ninthemethodschemascase).clearly,withoutanyapriori knowledgeoftheuniverse,itisimpossibletoobtainalistingofarelationfrom itscharacteristicfunction. xedevaluationstrategy,asafunctionofthedatabasesize. universe,thentranslationsbetweenthetwoconventionsareeasilyfurnished. However,ifweassumetheexistenceofaniteratorU:fogrepresentingthe Itisinterestingtoobservethatthe\relationsasiterators"conventionabove Indeed,ifR:fokgisaniteratorencodingarelationr,then r:k z} { o!!o!bool functionofarelationr,then (whereequalk,theequalitypredicatefork-tuples,isdenedbelow)computes thecharacteristicfunctionofr. x1:o:::xk:o: R:fokg Ontheotherhand,ifr:o!!o!Boolencodesthecharacteristic u::v:: c:ok!!:n:: R(r:ok:T::(Equalkrhx1;:::;xki)uT)v isaniteratorrepresentingr. U(x1:o:T1:: U(x2:o:T2:: U(xk:o:Tk:: order3.the(moresimple)encodinggivenbelowusestypesoforder4. ispossibletoexploitthis\ordereconomy"ofcharacteristicfunctionstoprovide thesensethatitstypeisoforder1,whereasthetypeofaniteratorisoforder2.it anencodingofallptimecomputablequeriesinthetypedlambdacalculusof Notethatthecharacteristicfunctionrepresentationofarelationissimplerin (rx1:::xk)(chx1;:::;xkitk)tk)tk 1):::T1)n

FunctionalProgrammingFormalismsforOODBMethods 5.2ExpressingRelationalQueries Inthissection,weshowhowtoexpresstherelationalalgebraoperatorsof[20] inthesimplytypedlambdacalculuswithequality.everyoperatoriscodedas a-termthattakesoneortworelationsinthelist-of-tuplesformatdescribedin Section5.1asinputandproducesanotherrelationinthesameformatasoutput. Thehomogeneityoftheinputandoutputformatallowstheresultofoneoperator 21 termuv:uifthecomparisoncomesouttrueanduv:votherwise.thecode dependsonthearityofthetuplesinvolved,soweusethesubscriptktoindicate thatthisparticulartermworksfork-tuples. f:fb1:::bkforequality.theresultofthecomparisonisabool,i.e.,the operators. applicationtobeusedasinputtoanother,sothatarbitraryrelationalalgebra expressionscanbecodedbynestingthe-termscorrespondingtotheindividual Equalk:ok!ok!Bool Webeginwitha-termEqualkthatteststwok-tuplesf:fa1:::akand t:ok:s:ok: u::v:: t(x1:o:::xk:o: s(y1:o:::yk:o: ((Eqx1y1) the\body" Oncetandsareinstantiatedwithtuplesf:fa1:::akandf:fb1:::bk,all (Eqxiyi)termsreducetoeitherTrueu:v:uorFalseu:v:v.Hence Here,Eqdenotestheequalitypredicateforconstants,oftypeo!o!Bool. ((Eqx2y2) u::v:: ((Eqxkyk)uv)v)v))) t(x1:o:::xk:o: s(y1:o:::yk:o: ((Eqx1y1) ((Eqx2y2) alistcn:ct1(ct2(ctmn))ofk-tuplesbycomparingthetupletoevery twok-tuplesinaprocedurallanguage. pendentofthereductionstrategy),whichmatchesthetimeneededtocompare oftheequalkpredicatereducestou:v:uifai=biforalliandtou:v:v otherwise.notethatthenalresultisreachedaftero(k)reductionsteps(inde- Thefollowingtermcheckswhetherak-tuplef:fa1:::akisamemberof ((Eqxkyk)uv)v)v)))

22 elementofthelist. Memberk:ok!fokg!Bool t:ok:r:fokg: u::v:: R(r:ok:T::(Equalkrt)uT)v GerdHillebrand,ParisKanellakis,andSridharRamaswamy Forexample,(Member2h1;2i[h3;4i;h5;6i])reducesto oftheinputlist,andthisagainmatchesthetimerequiredbyastraightforward proceduralimplementation. ofreductionsneededtoreachthenormalformiso(mk),wheremisthelength whichinturnreducestou:v:v,becauseboth(equal2:::)termsevaluate (whichreducesaredex(x:m)nonlyafterfullyreducingn),thenthenumber tofalse.ifthereductionsaredoneaccordingtoaneagerreductionstrategy u:v:(equal2h3;4ih1;2i)u((equal2h5;6ih1;2i)uv); sibletoreduceanexpression(memberktr),whererisoflengthm,toaterm substitutingit. witho(km)redexesbyneverevaluatingthefourthargumenttoequalkbefore multipliesbyk.sincethishappensateverylevelofequalknesting,itispos- unresolvedredexesissubstitutedforv(e.g.,theterm((equal2h5;6ih1;2i)uv) intheexampleabove),thenthenumberofredexesinthewholeexpression tialnumberofreductionsteps.thisisduetothefactthatthefourthparameterv oftheequalkpredicateoccursktimesinthebodyofthepredicate.ifatermwith Notethattherearealso\bad"reductionstrategieswhichleadtoanexponen- occursins,itisincludedintheoutput,otherwiseitisignored. by\walkingdown"randtestingeachtupleformembershipins.ifthetuple atorsoftherelationalalgebra: Intersection:Tointersecttwok-aryrelationsRandS,webuildanewrelationT Intersectionk:fokg!fokg!fokg WiththeaidofEqualandMember,wecannowcodetheset-theoreticoper- R:fokg:S:fokg: c:ok!!:n:: cludedintheoutputifitdoesnotoccurins. Setdierence:Thisworkslikeintersection,exceptthatatuplefromRisin- whichfurtherreducestoc:n:ch1;2in[h1;2i]. Forexample,(Intersection2[h1;2i][h1;2i;h3;4i])reducesto c:n:(member2h1;2i[h1;2i;h3;4i])(ch1;2in)n; R(r:ok:T::(MemberkrS)(crT)T)n Setminusk:fokg!fokg!fokg

andaddingthosetuplesofrthatdonotoccurins. FunctionalProgrammingFormalismsforOODBMethods Union:Theunionoftwok-aryrelationsRandSisformedbystartingwithS R:fokg:S:fokg: c:ok!!:n:: R(r:ok:T::(MemberkrS)T(crT))n 23 (IfweknewthatRandSweredisjoint,wecouldformtheirunionbyjust Unionk:fokg!fokg!fokg R:fokg:S:fokg: c:ok!!:n:: atorconcatenatesak-tupleandanl-tupletoforma(k+l)-tuple,andthe weneedthemorecomplextermabovetoensureaduplicate-freeoutput.) oriented"operators.weneedtwoauxiliarytermsrst:theconcatk;loper- concatenatingthemasfollows:c:n:rc(scn).inthegeneralcase,however, Havingdealtwiththe\set-oriented"operators,wenowexaminethe\tuple- R(r:ok:T::(MemberkrS)T(crT))(Scn) columnsi1;:::;iloftheinput. Rearrangek;i1;:::;iloperatortakesak-tupleandreturnsanl-tupleconsistingof Concatk;l:ok!ol!ok+l Rearrangek;i1;:::;il:ok!ol t:ok:s:ol: t:ok:f:o!!o!: f:o!!o!: t(x1:o:::xk:o: t(x1:o:::xk:o: s(y1:o:::yl:o: fxi1xil) fx1xky1yl)) Crossproduct:TheCartesianproductofk-aryrelationRandl-aryrelationS isformedbyastraightforwardnestediteration,inwhicheverytupleofris concatenatedwitheverytupleofsandappendedtotheoutput. Timesk;l:fokg!folg!fok+lg R:fokg:S:folg: c:ok+l!!:n:: R(r:ok:T:: S(s:ol:U:: c(concatk;lrs)u)t)n

24 Forexample,(Times1;1[h1i;h2i][h3i;h4i])reducesto c:n:c(concat1;1h1ih3i) (c(concat1;1h1ih4i) (c(concat1;1h2ih3i) GerdHillebrand,ParisKanellakis,andSridharRamaswamy Selection:Toselecttuplesfromak-aryrelationRaccordingtoacertaincondition,saycolumni=columnj,itsucestoiterateoverRandincludethose tuplesintheoutputthatsatisfythecondition. whichfurtherreducesto[h1;3i;h1;4i;h2;3i;h2;4i]. Selectk;i=j:fokg!fokg(c(Concat1;1h2ih4i)n))) columnsi1;:::;ilofak-aryrelationr, Projection:Thisisslightlytricky.Thestraightforwardattempttoprojectonto R:fokg: SimpleProjectk;i1;:::;il:fokg!folg c:ok!!:n:: R:fokg: R(r:ok:T:: c:ol!!:n:: r(x1:o:::xk:o:(eqxixj)(crt)t))n inrwhoseprojectionist0: weincludetheprojectiont0oftupletintheoutputonlyiftisthersttuple hasthedisadvantagethattheoutputmaycontainduplicatetuples.toxthis, Projectk;i1;:::;il:fokg!folg R:fokg: c:ol!!:n:: R(r:ok:T::c(Rearrangek;i1;:::;ilr)T)n; where(firstrr):boolisanabbreviationfor u::v:: R(r0:ok:T0:: (Equall(Rearrangek;i1;:::;ilr) R(r:ok:T:: (Rearrangek;i1;:::;ilr0)) (FirstrR)(c(Rearrangek;i1;:::;ilr)T)T)n; ((Equalkrr0)uv)T0)u:

FunctionalProgrammingFormalismsforOODBMethods Toseewhythisworks,considerforexamplethequery(Project2;1[h1;2i;h1;3i]), whichprojectstherelation[h1;2i;h1;3i]ontoitsrstcolumn.theexpression (Firsth1;2i[h1;2i;h1;3i])reducesto u:v:(equal1h1ih1i) 25 reducesto whichfurtherreducestou:v:utrue,whereas(firsth1;3i[h1;2i;h1;3i]) u:v:(equal1h1ih1i) ((Equal1h1ih1i)((Equal2h1;2ih1;3i)uv)u); ((Equal2h1;2ih1;2i)uv) andthentou:v:vfalse.hence,thewholequeryreducesto c:n:(firsth1;2i[h1;2i;h1;3i]) ((Equal1h1ih1i)((Equal2h1;3ih1;3i)uv)u) ((Equal2h1;3ih1;2i)uv) thateverytermgivenaboveiswell-typedand,whenappliedtoone(resp.,two) andthentoc:n:ch1in[h1i]. relationscodedintheformatdescribedinsection5.1,reducestoanormalform whichistheencodingofthedesiredoutputrelation.moreover,whentheterms Thiscompletesthecodingofrelationalalgebra.Itisstraightforwardtoverify ((Firsth1;3i[h1;2i;h1;3i])(ch1in)n) (ch1i((firsth1;3i[h1;2i;h1;3i])(ch1in)n)) typedlambdacalculuswithequalitysuchthatforeveryinstance(r1;:::;rl)ofs, arereducedaccordingtoaneagerreductionstrategy(whichreducesaredex tionalschemas=(r1;:::;rl).thenthereexistsaterm typicallymatchestherunningtimeofanaiveproceduralimplementationofthe polynomialinthesizeoftheinputs.(infact,thelengthofthereductionsequence theexpression( operatorinquestion.)therefore,wehavethefollowingtheorem. (x:m)nonlyafterfullyreducingn),thelengthofthereductionsequenceis Theorem14.Let(R1;:::;Rl)bearelationalalgebraexpressionoverrela- ( ingofrelationrasdescribedinsection5.1.moreover,thenormalformof sizeofr1;:::;rl. evenabiteboul'sandbeeri'scomplexobjectalgebra[1].thistakesusfrom r1rl)canbecomputedinanumberofreductionstepspolynomialinthe Itispossibletoextendtheschemeabovetoanencodingofxpointsand r1rl)reducesto(r1;:::;rl),whererdenotestheencod- ofthesimply

26 manipulating\at"relationstomorecomplicateddatastructures,namelyarbitrarynitetreesbuiltfromtupleandsetconstructors.thealgebraunder considerationisextremelypowerful infactitexpressesany\generic"(see[16]) elementarytimecomputationonnitestructures. Theorem15.Let(R1;:::;Rl)beacomplexobjectalgebraexpressionoverthe GerdHillebrand,ParisKanellakis,andSridharRamaswamy typedlambdacalculuswithequalitysuchthatforeveryinstance(r1;:::;rl)ofs, sizeofr1;:::;rl. 6ConclusionsandSomeOpenProblems theexpression( relationalschemas=(r1;:::;rl).thenthereexistsaterm ofrelationr.moreover,ifthepowersetoperatorisnotused,thenormalform of( r1rl)canbecomputedinanumberofreductionstepspolynomialinthe r1rl)reducesto(r1;:::;rl),whererdenotestheencoding ofthesimply ourexpositionisthattheseformalismscanberelatedtotraditionaldatabase expressingdatabasequeriesorqueriesovernitestructures.themainpointof lus/algebra,datalog,etc. Wehaveexploredtwofunctionalformalisms,MSandTLC=,inthecontextof querylanguagesandcanexpressallthatisexpressibleusingrelationalcalcu- formofsignatureinferenceandhavehighlightedanumberofecientlydecidable Thefollowingopenproblemshighlightsometopicsforfutureresearch. tionalsetting,whilepreservingsomeofthebasictheoremsofdatabasetheory. explorationasanattempttochangethevocabularyfromarelationaltoafunc- formsclasses,methods,inheritance,nameoverloading,latebinding,andfor thembettervehiclesfortheformalstudyofoodbs.thus,onecanviewour TLC=higher-orderfunctionsandlist,set,andtuplestructuresviatyping)make MS:Wehaveinvestigatedtheconsistencyproblemofmethodschemasasa Webelievethattheadditionalfeaturesofthesefunctionalformalisms(e.g., ofincrementalconsistencychecking.inourformalizationofmethodschema consistency,allbasemethodsareuninterpreted.aswesawfromtheanalysisof cases.thedecidabilityofthearity2caseisstillopen,aswellasmanyaspects andcdrfunctionssothatwecan\glue"and\unglue"arguments.however,the problemofcheckingconsistencywithmonadiccodedmethodsandpolyadicbase methodswithoutequalityisopen. becomesundecidablewithmonadiccodedmethodsandpolyadicbasemethods whenaugmentedwithequality.withequality,wecansimulatethelispcar twoargumentsthatitissuppliedwitharethesame.consistencychecking consistencythen?forexample,letussupposethatwehaveequality,withits normalinterpretation,asabasemethod.thismethodwillsimplytellifthe expressivepowersomedegreeofsemantics,e.g.,order,isuseful.whatabout thecomplexobjectalgebraof[1].inourembeddingsweuselistsoftuplesand calculuswithequalityisthesimplestsyntaxtodate,thatcanbeusedtodescribe withoutaddedconstructsoraddedpolymorphism.forexample,thetyped- thatwepresenthereareinterestingforanumberofreasons.theyare\pure," TLC=:Theembeddingsofdatabasequerylanguagesinthetyped-calculus

shouldbeaddedtothetyped-calculustocaptureexactlythemorepowerful constructs,andsetiterationreplaceslistiterationin[15,13].oneopenquestion computationthatisnotelementaryisnotcapturedbyourframework.what FunctionalProgrammingFormalismsforOODBMethods Notalldatabasequerylanguagescanbeembeddedinthetyped-calculus.Any simulatesetsbyeliminatingduplicates.in[1,15,13]setsareusedasbasic is:howcanthe-calculussyntaxandsemanticsbeaugmentedwithsetiteration? 27 timeinsteadoftuple-at-at-timeevaluation,wouldbetheappropriateformalism investigated. theobject-orientedfeaturesofmsandtheexpressibilityanddataabstraction (orveryclosetoit)foroodbquerying.suchaformalismwouldhaveall theory,nitemodeltheory,andthetyped-calculus,thatshouldbefurther databasequerylanguagessuchasthecomputablequeriesof[16,2]?theuse capabilitiesoftlc=.giventheorganizationofdataintocollectionsonewould ofreductionstrategiesintheseembeddingsprovidesalinkbetweencomplexity orienteddialectsoflisp[7]illustratesuchlinguisticsyntheses(withoutsetsand withalesscleartypesystem).thechallengeistocomeupwithasimply typed(andelegant)formalsynthesis,thatalsofacilitatesthemanipulationof liketheevaluationmechanismtoemphasizesets,asin[1,15,13].object- MS+TLC=:WebelievethatasynthesisofMSandTLC=,withset-at-a- programminglanguagemodels.forexample,let-orml-polymorphism[35]isa veryusefulfeaturetoincludeinafunctionaldatamodel.thelanguagemlhas andsimplytypedcalculi.thisisarststeptowardsmorecomplexdatabase collectionsofobjects. beenthebasisformanyprogramminglanguageinvestigationsonstructureand butforpointerstotherecentliteratureandforaprogramminglanguageview methodinheritance.theliteratureistooextensivetoevenattemptasurvey, ofsomeoftheissuesraisedherewerecommend[36,37]. Inourexpositionwehaveexploredtherelationshipwithprogramschemas References 3.S.AbiteboulandP.Kanellakis.DatabaseTheoryColumn:QueryLanguagesfor 1.S.AbiteboulandC.Beeri.OnthePowerofLanguagesfortheManipulationof 2.S.AbiteboulandP.Kanellakis.ObjectIdentityasaQueryLanguagePrimitive. InProceedingsACMSIGMOD,pp.159{173,1989.(AlsoINRIATech.Rep.1022, 1989.) ComplexObjects.INRIAResearchReport846,1988. 5.S.Abiteboul,G.Lausen,H.Upho,andE.Waller.MethodsandRules.InProceedingsACMSIGMOD,pp.32{41,1993. JCSS,43(1991),pp.62{124. ComplexObjectDatabases.SIGACTNews,21(1990),pp.9{18. 4.S.Abiteboul,P.Kanellakis,S.Ramaswamy,andE.Waller.MethodSchemas. 6.S.AbiteboulandV.Vianu.DatalogExtensionsforDatabaseQueriesandUpdates. 7.N.AdamsandJ.Rees.Object-OrientedProgramminginScheme.InProceedings BrownUniversityTech.Rep.CS-92-33,1992.(AnearlierversionappearedinProceedings9thACMPODS,1990.) ACMLispandFunctionalProgramming,pp.277{288,1988.

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