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1 InclusionConstraintsover MartinMuller1,JoachimNiehren1andAndreasPodelski2 Non-emptySetsofTrees? UniversitatdesSaarlandes,66041Saarbrucken,Germany ImStadtwald,66123Saarbrucken,Germany 2Max-Planck-InstitutfurInformatik, 1ProgrammingSystemLab, Abstract.WepresentanewconstraintsystemcalledINES.Itsconstraintsareconjunctionsofinclusionst1t2betweenrst-orderterms (withoutsetoperators)whichareinterpretedovernon-emptysetsof trees.theexistingsystemsofsetconstraintscanexpressinesconstraintsonlyiftheyincludenegation.theirsatisabilityproblemis NEXPTIME-complete.Wepresentanincrementalalgorithmthatsolves programminglanguage. toapplyinesconstraintsfortypeanalysisforaconcurrentconstraint thesatisabilityproblemofinesconstraintsincubictime.weintend Sets)andpresentanincrementalalgorithmtodecidethesatisabilityofINES 1Introduction WeproposeanewconstraintsystemcalledINES(InclusionsoverNon-Empty constraintsintimeo(n3).inesconstraintsareconjunctionsofinclusionst1t2 betweenrst-orderterms(withoutsetoperators)whichareinterpretedoverthe t16;^t1t2issatisableoverarbitrarysets.notethattheconstraintt6; AnINES-constraintt1t2issatisableovernon-emptysetsifandonlyif innitetrees.allgivenresultscanbeeasilyadaptedtonitetrees. domainofnon-emptysetsoftrees.inthispaperwefocusonsetsofpossibly cannotbeexpressedbypositivesetconstraintsonly[16].theexpressivenessof INESconstraintsissubsumedbythatofsetconstraintswithnegation[9,16].In thecaseofnitetrees,thesatisabilityproblemofsetconstraintswithnegation isknowntobedecidable[1,13];itiscompletefornondeterministicexponential time[9,10].thisresultimpliesthatthesatisabilityproblemofinesconstraints WecharacterizethesatisabilityofINESconstraintsbyasetofaxiomssuchthat aninesconstraintissatisableovernon-emptysetsifandonlyifitissatisable hasnotbeenconsideredbefore. oversetsofnitetreesisdecidable.thecorrespondingproblemforinnitetrees?asummaryhasappearedin:maxdauchet,ed.,proc.ofcaap'97aspartof TAPSOFT'97,TheoryandPracticeofSoftwareDevelopment.April1997,Lille,France.

2 closesagiveninputconstraintunderitsconsequenceswithrespecttotheaxioms. insomemodeloftheseaxioms.theseaxiomsdeneaxpointalgorithmthat Weprovethataconstraint'issatisableifandonlyifthealgorithmwith formulaeinterpretedovertreesandovernon-emptysetsoftreesareclosely SetsversusTrees.Thesatisabilityproblemsofseveralclassesofrst-order willbediscussedlaterinthisintroduction. input'doesnotderive?asaconsequenceof'.allaxioms(forinnitetrees) related.thefollowingtwoinstancesofthisobservationhaveinspiredourchoice ofaxiomsorunderlyourproofs. Equalityconstraintsareconjunctionsofequationst1=t2betweenrst-order oftherst-ordertheoryofequalityconstraintsovertrees[18,19,12]sinceits non-emptysetsoftreescoincide.thisfollowsfromthecompleteaxiomatization ordertheoriesofequalityconstraintsovertreesandofequalityconstraintsover symmetryofsetinclusion(t1=t2$t1t2^t2t1).actually,eventherst- terms.oversets,theycanbeexpressedbyinclusionconstraintsduetoanti- axiomsalsoholdovernon-emptysetsoftrees(butdon'toverpossiblyempty sets). ThereexistsanaturalinterpretationofINES-constraintovertreelikestructures thatwecalltreeprexes.inadierentcontext[6]treeprexesarecalledbohm trees(without-binders).treeprexescomewithanaturalorderingrelation overtreeprexes(wheretheinclusionsymbolisinterpretedastheinverseofthe wheretheemptytreeprexisthegreatestelement.weprovethatanines constraintissatisableovernon-emptysetsoftreesifandonlyifitissatisable formulatedforabinaryfunctionsymbolf). oftheinclusionrelation.wealsoassumethefollowingdecompositionaxiom(here Axioms.Thersttwoaxiomsweneedpostulatethereexivityandtransitivity prexorderingontreeprexes). Thisaxiomholdsovernon-emptysetsoftreesbutnotoverpossiblyemptysets, sinceeveryvariableassignmentwith(x)=;or(y)=;isasolutionof f(x;y)f(x0;y0)!xx0^yy0 f(x;y)f(x0;y0)butnotnecessarilyofxx0^yy0.ananalogousstatement holdsforthefollowingclashaxiom. Forinstance,theunsatisabilityoftheconstraint'givenbyxg(x)^xg(y)^ yz^zaisnotderivablewiththeseaxiomsalone.weneedfurtheraxioms TheseaxiomsdonotsucetocharacterizethesatisabilityofINESconstraints. f(x;y)g(x0;y0)!? forf6=g thatusenon-disjointnessconstraintst16jt2denedast1\t26;.forthenondisjointnessrelationwerequirereexivityandsymmetryandadecompositionaxiomasfortheinclusionrelation. f(y;z)6jf(y0;z0)!y6jy0^z6jz0 2

3 Finally,weassumeaclashaxiomsimilartotheoneforinclusionandrequire nondisjointnesstobecompatiblewithinclusioninthefollowingsense. Nowreconsidertheconstraint'givenaboveandobservethatwecanderive x6jz^xy!y6jz x6jxbyreexivity,thenx6jybydecomposition,andx6jzbycompatibility.this AlgorithmandComplexity.Theaboveaxiomsyieldanalgorithmthatadds yieldsaclashwithxg(x)^za. constraintsoftheformxy,x6jytoagiveninputconstraint'until'isclosed underallaxiomsorimplies?.theinesconstraintxt1^:::^xtnexpresses x1\:::\xk6;(whichcanbeexpressedbytheformula9y(yx1^:::^yxk)) thensetsdenotedbythetermst1;:::;tnhaveanon-emptyintersection.fortunately,itisnotnecessarytoaddk-arynon-disjointnessconstraintsoftheform ofwhichthereareexponentiallymany.instead,ouralgorithmaddsatmost O(n2)constraintstotheinputconstraint',wherenisthenumberofvariables timeo(n).thisyieldsanimplementationofouralgorithmwithtimecomplexity in'.theadditionofasingleconstraintcanbeimplementedsuchthatitcosts TypeAnalysis.OneapplicationforINESconstraintswhichweareinvestigatingin[23]istypeanalysisforconcurrentconstraintprogramming[17,27],in O(n3).Thisimplementationcanbeorganizedincrementally. anerrorifthesetofpossiblerun-timevaluesisemptyforsomevariable.this INESallowsaninterpretationoversetsofpossiblyinnitetrees.Itisconsidered programvariables.sincevaluesinozincludeinnitetrees,itisimportantthat There,INESconstraintsareusedtoapproximatethesetofrun-timevaluesfor particularoz[28].asformalfoundationsweintendtousethecalculiin[24,25]. factwasourinitialmotivationforthechoiceofnon-emptysetsoftreesasthe PlanofthePaper.InSection2,wediscussrelatework.InSection3,we denethesyntaxandsemanticsofinesconstraintsandinsection4,wepresent interpretationdomainforinesconstraints. theaxiomsandthealgorithm.insection5,weprovethecompletenessofour thedetailsoftheproofsintheconferenceversionofthepaper. algorithm.insection6,wecomparetheinterpretationsofinesconstraintsover treeprexesandovernon-emptysetsoftrees.duetospacelimitations,weomit AppendixAgivesanexampleillustratingprogramanalysisforOzwithINES constraints.appendixbcontainstheomittedproofs.appendixcdetailshow ofatomicsetconstraints(standardsetconstraintswithoutsetoperatorsand toimplementthealgorithmwithincrementalo(n3)complexity.inappendixd, negation)isinvariantwithrespecttothechoiceofniteorinnitetrees. withexplicitnon-emptinessconstraintsx6;.wealsoprovethatsatisability ofstandardsetconstraints(interpretedoverpossiblyemptysetsofnitetrees) weadaptthealgorithmtothenite-treecase,andinappendixetoasubclass 3

4 2RelatedWork StandardSetConstraints.Setconstraintsasin[2,5,10,15]areinclusions Ouralgorithmcanbeadaptedsuchthatitsolvesasubclassofsetconstraints betweenrst-ordertermswithsetoperatorsinterpretedoversetsofnitetrees. withoutsetoperatorsincubictime(seeappendixe).thegeneralcaseisnondeterministicallyexponentialtimecompleteasprovedin[1,13].thesubclass thatwecansolveincubictimesyntacticallyextendstheinesconstraintswith explicitnon-emptinessconstraintx6;(seeappendixe).notethatthesatisabilityofthesesetconstraintsdependsonthechoiceofniteorinnitetrees (considerxf(x)^x6;),whichisincontrasttostandardsetconstraintswithout negation.ouralgorithmaccountsfornitenessthroughtheoccurcheck. AtomicSetConstraints.HeintzeandJaarconsiderso-calledatomicset constraints[15]whichsyntacticallycoincidewithinesconstraintsbutareinterpretedoverpossiblyemptysetsofnitetrees.thesatisabilityproblemfor SetConstraintsforTypeAnalysis.Aikenetal.[3,4]useconstraints resultsof[14]and[15].anexplicitproofisgiveninappendixeofthispaper. overspecicsetsoftreescalled\types"forthetypeanalysisoffl.thereisa atomicsetconstraintsisalsoo(n3).thisresultisimplicitinthecombined minimaltype0which{intermsofconstraintsolving{behavesjustlikethe emptysetinstandardsetconstraints(althoughitisnotanemptysetfromthe typespointofviewbutcontainsavaluedenotingnon-termination).incontrast followingconstraintsimplicationrulebydroppingthedisjunctsinbrackets[4]. intersection.oneoftheoptimizationsusedbyaikenetal.istostrengthenthe totheconstraintsofthispaper,theirsetconstraintsprovideforunionand Asstatedin[4],thisoptimizationdoesnotpreservesoundness(f(a;0)f(b;0) holdsbutab^00doesnot).itmightbepossibletojustifyitbyusingnonemptysetsasinterpretationdomain.thisislefttofurtherresearch. Podelski[11]giveanalgorithmwhichdecidestheentailmentproblembetween INESconstraintswheninterpretedoversetsofnitetrees.Theyalsodecidethe EntailmentandIndependenceforInesConstraints.Charatonikand in[11]donotincludeanyoftheresultspresentedheresincetheyuseasan satisabilityofinesconstraintswithnegationinthenitetreecase.theresults explicitprerequisitethefactthatsatisabilityofinesconstraintsisdecidable. f(x;y)f(x0;y0)!xx0^yy0[_x0_y0] thetwosatisabilityproblemsareratherdierentproblemssincetarskianset TarskianSetConstraints.MacAllesterandGivan[21]giveacubicalgorithm whichdecidessatisabilityforaclassoftarskiansetconstraints[22],andwhich alsocontainsanon-disjointnessconstraint.apartfromthissyntacticsimilarity, constraintsarenotinterpretedoverthedomainoftrees(thisisalsoobserved in[22]).arelatedopenquestioniswhetherouraxiomsdenealocaltheory[20, 8],whichwouldalsoproofthecubiccomplexityboundofouralgorithm. 4

5 asetoffunctionsymbolsf;gandtheirrespectivearityn0.constants(i.e. Weassumeasetofvariablesrangedoverbyx;y;zandasignaturethatdenes 3SyntaxandSemanticsofInesConstraints functionsymbolsofarity0)aredenotedwithaandb. Trees.Webasethedenitionoftreesonthenotionofpathssincewewishto includeinnitetrees.pathswillturnoutcentralforourproofsinsection5.a pathpisasequenceofpositiveintegersrangedoverbyi;j;n;m.theemptypath isdenotedby".wewritethefree-monoidconcatenationofpathspandqaspq; wehave"p=p"=p.givenpathspandq,qiscalledaprexofpifp=qp0for somepathp0. Letbeasetofpairs(p;f)ofpathspandfunctionsymbolsf.Wesaythat isprexclosed,if(p;f)2andqisaprexofpimpliesthatthereisagsuch that(q;g)2.itispathconsistent,if(p;f)2and(p;g)2impliesf=g. Wecallarityconsistent,if(p;f)2,(pi;g)2impliesthati2f1;:::;ng providedthearityoffisn.finally,iscalledaritycomplete,if(p;f)2, wherethearityoffisn,impliesforalli2f1;:::;ngtheexistenceofagwith (pi;g)2. A(possiblyinnite)treeisasetofpairs(p;f)thatisnon-empty,prexclosed, aritycomplete,pathconsistent,andarityconsistent.thesetofall(possibly InesConstraints.AnINESconstraintt1t01^:::^tnt0nisaconjunctionof treesbyp+(tree). innite)treesoverisdenotedbytreeandthesetofallnon-emptysetsof inclusionsbetweenrst-ordertermstdenedbythefollowingabstractsyntax. implicitlythatthelengthoftcoincideswiththearityoff.weinterpretines Hereandthroughoutthepaper,tstandsforasequenceoftermsandweassume constraintsoverthestructurep+(tree)ofnon-emptysetsoftrees.inthisstructure,afunctionsymbolfofisinterpretedaselementwisetreeconstructor andtherelationsymbolassubsetrelation.wecallarst-orderformulaover INESconstraintsatisableifitissatisableinthestructureP+(Tree).TworstorderformulaeoverINESconstraintsarecalledequivalentiftheyareequivalently interpretedinp+(tree). t::=xjf(t) inclusionsxf(y)andf(y)x(thisisamatteroftaste).andthird,weneed insteadofpossiblydeeptermst.second,weuseequalitiesx=f(y)ratherthan binarynon-disjointnessconstraintsx6jy.theirsemanticsisgivenbytheequivalencetotheformulax\y6;oversetsoftrees.overnon-emptysetsoftrees,x6jstraintsyntaxinthesequel.first,werestrictourselvestoattermsf(x)andx FlatInesConstraints.Foralgorithmicreasons,weuseanalternativecon- 5

6 isequivalentto9z(zx^zy).crucially,however,nondisjointnessconstraints x6jyavoidexplicitexistentialquanticationinouralgorithm. junction,i.e.,weconsider'asamultisetofinclusionsxy,equalitiesx=f(y), WeidentifyatINESconstraints'uptoassociativityandcommutativityofcon- ThesethreestepsleadustoatINESconstraints'denedasfollows. andnon-disjointnessconstraintsx6jy. '::='1^'2jxyjx=f(y)jx6jy Fromnowon,wewillconsideronlyatINESconstraintsandcallthemconstraintsforshort.ThisisjustiedbythefollowingProposition.Letthesize ofaconstraint'bethenumberoffunctionsymboloccurrencesplusvariable Proposition1.ThesatisabilityproblemsofINESconstraintsandofatINES occurrencesin'. 4AxiomsandAlgorithm constraintshavethesametimecomplexityuptoalineartransformation. WepresentasetofaxiomsvalidforINES-constraintsinterpretedovernon-empty solvesthesatisabilityproblemofinesconstraints.thecorrectnessandthe setsoftrees.inasecondstep,weinterprettheseaxiomsasanalgorithmthat A1.xxandxy^yz!xz complexityofthisalgorithmwillbeprovedinsection5. A2.x=f(y)^xx0^x0=f(z)!yz A5.x=f(y)^x6jx0^x0=f(z)!y6jz A3.xy!x6jyandxy^x6jz!y6jzandx6jy!y6jx A4.x=f(y)^x6jx0^x0=g(z)!?forf6=g Table1containsverulesA1-A5representingsetsofaxioms.1Theunionof Table1.AxiomsofINESconstraintsovernon-emptysetsofinnitetrees thesesetsisdenotedbya.forinstance,arulexxrepresentstheinniteset 1Notethattheseaxiomsdierfromtheonesgivenintheintroduction.Theconstraints usedtherearenotatandthevariable-variablecasexyandx6jyareomitted. Indeed,theaxiomsintheintroductionaresemanticallycomplete,althoughthisis non-trivialtoseeanddependsonthecorrectnessofthealgorithmpresentedhere. 6

7 ofaxiomsthatisobtainedbyinstantiationofthemetavariablexwithconcrete Proposition2.ThestructureP+(Tree)isamodeloftheaxiomsinA. constraints'! variables.notethatanaxiomiseitheraconstraint',animplicationbetween,oranimplication'!?. Proof.Byaroutinecheck.Wenotethatthenon-emptinessassumptionof P+(Tree)isessentialforaxiomsA2andA3:1. TheAlgorithm.ThesetofaxiomsAcanbeconsideredasa(nave)xed pointalgorithmathat,givenaninputconstraint',iterativelyaddslogical 2 consequencesofa[f'gto'.moreprecisely,ineverystepainputsaconstraint' possibleif?takesplaceifthereexists andeitherterminateswith?oroutputsaconstraint'^ 02'suchthat 0!?2A.Outputof'^.Terminationwith Example1.Arsttypeofinconsistencydependsonthetransitivityofsetinclusion.Hereisatypicalexample: 2Aorthereexists 0in'with 0! with?bya4. AlgorithmAmayaddxzbyA1:2,thenx6jzwithA3:1,andthenterminate x=a^xy^yz^z=b!? fora6=b disjointnessrequirements.forillustration,weconsider: Example2.Asecondtypeofinconsistencycomeswithimplicitorexplicitnon- AlgorithmAmayaddz6jxbyA3:1,thenx6jzviaA3:3,thenx6jywithA3:2,and nallyterminatewith?viaa4. x=a^zx^zy^y=b!?for fora6=b reasoningwitha2.consider: Example3.Inconsistenciesoftheabovetwotypesmaybedetectedbystructural AlgorithmAmayaddxxbyA1:1,thenxzwithA2,thenx6jzbyA3:1,and nallyterminatewith?witha4. x=f(x)^x=f(z)^z=a!? Example4.WeneedanotherstructuralargumentbasedonA5forderivingthe unsatisabilityofthefollowingconstraint. AlgorithmAmayaddx6jyafterseveralstepsasshowninExample2.Thenit mayproceedwithx6jx0viaa5andterminatewith?viaa4. x=f(x)^zx^zy^y=f(x0)^x0=a!? 7

8 Termination. straintsx6jyandxyto'whicharenotcontainedin'.wealsorestrictre- addingasimplecontrol.givenaninputconstraint',weaddonlysuchcon- exivityofinclusionxxtosuchvariablesxoccurringin'.givenasubsets ofa,aconstraint'iscalleda0-closed,ifalgorithmaunderthegivencontrol AlgorithmAcanbeorganizedinaterminatingmannerby notcontain?bydenition.)thisdenesthenotionofa-closednessbutalsoof andrestrictedtotheaxiomsina0cannotproceed.(notethatconstraintsdo A1-closedness,A2-closedness,etc.,whichwillbeneededlateron. Example5.Ourcontroltakescareofterminationinpresenceofcycleslike x=f(x).forinstance,thefollowingconstraintisa-closed. Inparticular,A2andA5donotloopthroughthecyclex=f(x)innitelyoften. Proposition3.If'isaconstraintwithmvariablesthenalgorithmAwith x=f(x)^xy^y=f(x)^xx^yy^x6jx^y6jy^x6jy^y6jx Proof.SinceAdoesnotintroducenewvariables,itmayaddatmostm2nondisjointnessconstraintsx6jyandm2inclusionsxy. 2 TheproofofthisstatementisthesubjectofSection5.There,weconstructthe Proposition4.EveryA-closedconstraint'issatisableoverP+(Tree). input'terminatesundertheabovecontrolinatmost2m2steps. (oineandonline)wherenistheconstraintsize. generaldonothaveasmallestsolution(considerxf(xy)). Theorem5.ThesatisabilityofINESconstraintscanbedecidedintimeO(n3) greatestsolutionforasatisableconstraint(lemma9).notethatconstraintsin control(proposition3),thisyieldsaeectivedecisionprocedure.thecomplexitystatementisprovedinproposition14.themainideaisthateverystepof implementationofalgorithma.itexploitsthatalgorithmaleavestheorder unspeciedinwhichaxiomsinaareapplied. algorithmacanbeimplementedintimeo(n)andthatthereareo(n2)steps (Proposition3).2IntheproofofProposition14,wepresentanincremental ThereisaclassofconstraintsonwhichalgorithmAindeedtakescubictime, namelytheinclusionscyclesx1x2^:::^xn?1xn^xnx1wheren1.the closureunderaisthefulltransitiveclosurevfxixjji;j2f1:::nggplusthe 2 with?.proposition4provesthat'issatisableifastartedwith'terminates Proof.Proposition2showsthat'isunsatisableifAstartedwith'terminates withaconstraint.sinceaterminatesforallinputconstraintsundertheabove correspondingnon-disjointnessconstraints. 2EverystepofalgorithmAcoststimeO(n)onlywithrespecttoanamortizedtime analysis,whichwedonotmakeexplicitinourcomplexityproof. 8

9 inproposition4.wehavetoconstructasolutionforeverya-closedconstraint. 5Completeness ThegoalofthisSectionistoprovethecompletenessofouralgorithmasstated TheideaistoconstructsolutioninasubstructureofP+(Tree)thestructureof treeprexes. TreePrexes.Atreeprexisasetofpairs(p;f)thatisprexclosed,path overtreeprexessuchthatprexbecomesastructure.functionsymbolsf2 alltreeprexesisdenotedbyprex.wecannaturallyinterpretinesconstraints consistent,andarityconsistent.notethateverytreeisatreeprex.thesetof areinterpretedastreeprexconstructors(generalizingtreeconstructors).the inclusionsymbolisinterpretedastheinvertedsubsetrelationontreeprexes thatwedenotewith(i.e.,12i12).therelation16j2holdsover Prexi1[2ispathconsistent(andhenceatreeprex). dingtrees:prex!p+(tree)givenby: Proposition6.PrexisasubstructureofP+(Tree)withrespecttotheembed- Proof.ThemappingTreesisahomomorphismwithrespecttofunctionsymbolsf2andtherelationsymbolsand6j. Trees()=f0j0isatreesuchthat0g Corollary7.IfaconstraintissatisableoverPrexthenitissatisableover P+(Tree). 2 Aconjunctionofsuchconstraintsissatisableifallconjunctsaresatisable.2 PathReachability. Proof.Forconstraintsxy,x=f(y),andx6jy,thisfollowsfromProposition6. constraint',wedeneabinaryrelation';p,wherex';pyreadsas\yis thenotionofpathconsistencywithrespecttoconstraints.forallpathspand reachablefromxoverpathpin'": Weintroducethepathreachabilityrelations';pand x';"yifxyin' Wedenerelationsx';pfmeaning\fcanbereachedfromxviapathpin'": x';iyiifx=f(y1:::yi:::yn)in'; x';pqyifx';puandu';qy: Forexample,if'istheconstraintxy^y=f(u;z)^z=g(x)thenthefollowing reachabilityfromxrelationshipshold:x';"y,x';2z,x';21x,x';21y,etc., aswellasx';"f,x';2g,x';21f,etc. x';pfifx';pyandy=f(u)in'; 9

10 Denition8PathConsistency.Wecallaconstraint'pathconsistentifthe followingtwoconditionsholdforallx,y,p,f,andg. Lemma9.EveryA1-A2-closedandpathconsistentconstraintissatisableover 1.Ifx';pg,xx,andx';pfthenf=g. Prex. 2.Ifx';pg,x6jy,andy';pfthenf=g. ProofofProposition4.WehavetoshowthateveryA-closedconstraint'is andhencesatisableinp+(tree)bycorollary7. satisable.'ispathconsistentbylemma10,satisableinprexbylemma9, Lemma10.EveryA3-A5-closedconstraintispathconsistent. 6Non-EmptySetsversusTrees 2 Theorem11.GivenanINESconstraints',thefollowingthreestatementsare interpretationovertrees. emptysetsoftrees.forthefragmentofequalityconstraintswealsoconsideran WediscussinterpretationsofINESconstraintsovertreeprexesandovernon- equivalent: 1.'issatisable(overP+(Tree)). Proof.1)to3).If'issatisableoverP+(Tree),thenitissatisableinsome 3.'issatisableinsomemodeloftheaxiomsinA. 2.'issatisableoverPrex. 3)to2).Let'besatisableinsomemodelofA.AlgorithmAterminateswhen modelofa,sincep+(tree)isamodelofabyproposition2. 2)to1).If'issatisableoverPrexthenitissatisablebyCorollary7. startedwith'byproposition3.itoutputsaconstraint isequivalentto'inallmodelsofa. PrexbyLemmata9and10. isa-closedandhencesatisableover (andnot?)that ingisantisymmetric(x=y$xy^yx). P+(Tree),equalitiescanbeexpressedbyinclusionssincetheinclusionorder- Anequalityconstraintisaconjunctionofequalitiesx=yandx=f(y).Over 2 Theorem12.Thethreerst-ordertheoriesofequalityconstraintsovernonemptysetsoftrees,overtreeprexes,andovertreescoincide(i.e.,ofthestructuresP+(Tree),PrexandTree).3 3Independently,A.ColmerauerobservedthisforP+(Tree)andTree(pers.comm.). 10

11 equality. isimmediatesincetheyarealreadycontainedinawithinclusionreplacedfor Proof.Thisfollowsfromthefactthatallaxiomsofthecompleteaxiomatization oftrees[18,19,12]arevalidfornon-emptysetsoftrees.thisholdsfortheaxioms oftheform8y9!x(x1=f1(xy)^:::^xn=fn(xy)).validityoftheotheraxioms structuresp+(tree)andprex.aformulathatholdsoverprexbutnotover Incontrast,rst-orderformulaeoverinclusionconstraintscandistinguishthe P+(Tree)isgivenby 2 partialordersin[6]). wherea6=b.anotherformuladistinguishingbothstructurescomeswitha constraint-basedreformulationofthecoherenceproperty(denedforcomplete 8x(ax^bx!8y(yx)) Wesaythatanorderingrelationsatisesthecoherencepropertyifitsatisesthe thegivenordering).vi;j2i9z(zxi^zxj)!9z(vi2izxi) followingformulaeforallnitesetsi(whereinclusionsymbolisinterpretedas propertydoesnothold.thereitstatesthenon-emptinessofann-intersection Thisformulastatesthatforallvariableassignmenttheelementsfromthe (xi);(xj)have(i;j;2f1;:::;ng).forinclusionovernon-emptysetsthis nitesetf(xi)ji2ighaveacommonlowerboundifeverytwoofitselements t1\:::\tnifallpairwiseintersectionsti\tjarenon-empty(i;j2f1:::ng), whichisrefutedbytheexamplei=f1;2;3gand(x1)=fa;bg,(x2)=fa;cg, (x3)=fb;cgfordistinctconstantsa;b;c. Proposition13.Thetreeprexorderingsatisesthecoherenceproperty. solutionof9z(vi2izxi). Proof.ForsomeniteindexsetJIandvariableassignmentintoPrex, pathconsistentsuchthattheunionsi2i(xi)ispathconsistent.henceisa isasolutionofall9z(zxi^zxj)thenallpairwiseunions(xi)[(xj)are notethatisasolutionof9z(vi2jzxi)isi2j(xi)ispathconsistent.if Acknowledgements.WewouldliketothankDavidBasin,DenysDuchier,Witold astheanonymousrefereesforvaluablecommentsondraftsofthispaper.theresearch Charatonik,HaraldGanzinger,GertSmolka,RalfTreinenandUweWaldmann,aswell 2 reportedinthispaperhasbeensupportedbythetheespritworkinggroupcclii (EP22457)andtheDeutscheForschungsgemeinschaftthroughtheGraduiertenkolleg KognitionswissenschaftandtheSFB378attheUniversitatdesSaarlandes. References 1.A.Aiken,D.Kozen,andE.Wimmers.DecidabilityofSystemsofSetConstraints withnegativeconstraints.informationandcomputation,

12 2.A.AikenandE.Wimmers.SolvingSystemsofSetConstraints.InProc.7thLICS, pp.329{340.ieee, A.AikenandE.Wimmers.TypeInclusionConstraintsandTypeInference.In Proc.6thFPCA,pp.31{ A.Aiken,E.Wimmers,andT.Lakshman.SoftTypingwithConditionalTypes. InProc.21stPOPL.ACM, L.Bachmair,H.Ganzinger,andU.Waldmann.SetConstraintsaretheMonadic Class.InProc.8thLICS,pp.75{83.IEEE, H.P.Barendregt.TheTypeFreeLambdaCalculus.InBarwise[7], J.Barwise,ed.HandbookofMathematicalLogic.Number90inStudiesinLogic. North{Holland, D.BasinandH.Ganzinger.AutomatedComplexityAnalysisBasedonOrdered Resolution.In11thLICS.IEEE, W.CharatonikandL.Pacholski.Negativesetconstraintswithequality.In Proc.9thLICS,pp.128{ W.CharatonikandL.Pacholski.SetconstraintswithprojectionsareinNEXP- TIME.InProc.35thFOCS,pp.642{ W.CharatonikandA.Podelski.TheIndependencePropertyofaClassofSet Constraints.InProc.2ndCP.LNCS1118,Springer, H.ComonandP.Lescanne.Equationalproblemsanddisunication.Journalof SymbolicComputation,7:371{ R.Gilleron,S.Tison,andM.Tommasi.SolvingSystemsofSetConstraintswith NegatedSubsetRelationships.InProc.34ndFOCS,pp.372{ N.Heintze.SetBasedAnalysisofMLPrograms.TechnicalReportCMU{CS{93{ 193,SchoolofComputerScience,CarnegieMellonUniversity.July N.HeintzeandJ.Jaar.ADecisionProcedureforaClassofSetConstraints (ExtendedAbstract).InProc.5thLICS,pp.42{51.IEEE, D.Kozen.Logicalaspectsofsetconstraints.InProc.CSL,pp.175{ M.J.Maher.Logicsemanticsforaclassofcommitted-choiceprograms.InJ.-L. Lassez,ed.,Proc.4thICLP,pp.858{876.TheMITPress, M.J.Maher.CompleteAxiomatizationsoftheAlgebrasofFinite,Rationaland InniteTrees.InProc.3rdLICS,pp.348{457.IEEE, A.I.Malc'ev.AxiomatizableClassesofLocallyFreeAlgebrasofVariousType. InTheMetamathematicsofAlgebraicSystens:CollectedPapers ,ch.23, pp.262{281.north{holland, D.McAllester.AutomaticRecognitionofTractabilityinInferenceRelations. JournaloftheACM,40(2),Apr D.McAllesterandR.Givan.TaxonomicSyntaxforFirst-OrderInference.Journal oftheacm,40(2),apr D.McAllester,R.Givan,D.Kozen,andC.Witty.TarskianSetConstraints.In Proc.11thLICS.IEEE, M.Muller.TypeAnalysisforaHigher-OrderConcurrentConstraintLanguage. DoctoralDissertation.UniversitatdesSaarlandes,TechnischeFakultat,66041 Saarbrucken,Germany.Inpreparation. 24.J.Niehren.FunctionalComputationasConcurrentComputation.In23rdPOPL, pp.333{343.acm, J.NiehrenandM.Muller.ConstraintsforFreeinConcurrentComputation.In Proc.1stASIAN,LNCS1023,pp.171{186.Springer, TheOzProgrammingSystem.ProgrammingSystemsLab,UniversitatdesSaarlandes.Availableathttp:// 12

13 27.V.A.Saraswat.ConcurrentConstraintProgramming.TheMITPress, G.Smolka.TheOzProgrammingModel.InJ.vanLeeuwen,ed.,ComputerScienceToday,LNCS1000,pp.324{343.Springer,1995. AInes-ConstraintsforProgramAnalysis constraintprogramminglanguages[17,27]suchasoz[28](see[24,25]forformalfoundationsofoz).duringtheexecutionofprogramsintheselanguages, WeareinvestigatingtheapplicationofINESconstraintsforprogramanalysis. Morespecically,weintendtoconstructatypeanalysissystemforconcurrent thesetofpossiblevaluesisemptyforsomeprogramvariable. thepossiblevaluesofprogramvariablesareapproximatedbyconstraints.for analysisaddedincomments(usingthespecialfunctionsymbolproc).4 Forillustration,considerthefollowingOzprogramwithitsconstraint-based programswithoutsearch(backtracking),itisconsideredaprogrammingerrorif proc{px}x=aend proc{qy}y=bend {PZ}{QZ} %9x(p=proc(x)^x=a)^ %9y(q=proc(y)^y=b)^ argumentsxandy,respectively,aswellastwoprocedureapplicationswiththe TheprogramcontainsthedenitionoftwoproceduresPandQwithformal %proc(z)p^proc(z)q andz=bwillbeemittedwhichareinconsistentwitheachother. TheprogramvariablesP,Q,X,Y,andZaremappedtoconstraintvariablesp, sameactualargumentz.onexecutionoftheseapplications,theconstraintsz=a indicatedinthecomments.theconjunctionoftheseconstraintsischeckedfor q,x,y,andz,andtheprogramsubexpressionsaremappedtoconstraintsas AprogramanalysisintermsofINES-constraintscandetectthiserrorasfollows. satisabilityandtheprogramisrejectedifthistestfails.theaboveprogramis rejectedsinceitsanalysisimpliesza^zbwhichisunsatisable. useitexperimentallyforozprograms.thefulldescriptionofthetypeanalysis systemisoutofthescopeofthispaperandwillbereportedin[23]. WehaveimplementedatypeanalysissystembasedonINES-constraintsand constraintshavethethesametimecomplexityuptoalineartransformation. BOmittedProofs Proposition1.ThesatisabilityproblemsofINESconstraintsandofatINES 4Thisexamplealsoappearedinthefollow-uppaper[11]withtheexplicitstatement thatitisborrowedfromhere. 13

14 equivalenttoarst-orderformulaoverinesconstraints. Proof.WithrespecttothestructureP+(Tree),everyatINESconstraintis INESconstraints. Conversely,everyINESconstraintisequivalenttoarst-orderformulaoverat x=f(y)$xf(y)^f(y)x x6jy$9z(zx^zy): straintsintoatinesconstraintsandviceversa.hence,foreveryinesconstraint TheseequivalencescanbeinterpretedasconstrainttransformersfromINEScon- f(t)x$9y9z(tz^f(z)=y^yx) xf(t)$9y9z(xy^y=f(z)^zt) tt0$9x(tx^xt0) thereexistsasatisfactionequivalentconstraintandviceversa.itiseasytoorganizethetransformationssuchthattheypreservethesizeofconstraintsuptoa Prex. Lemma9.EveryA1-A2-closedandpathconsistentconstraintissatisableover factorof2.hence,thecomplexityofthesatisabilityproblemsispreserved.ut Proof.Let'beA1-A2-closedandpathconsistent.Wedeneavariableassignmentprex'intoPrexasfollows: Thepathconsistencyof'(condition1)impliesthepathconsistencyof p).wenowverifythatprex'isasolutionof'. prex'(x).thusprex'(x)isatreeprex(onecanshowthisbyinductionover prex'(x)=f(p;f)jx';pfg {Considerx=f(y1:::yn)in'.Ifi2f1:::ngandyi';pgthenx';ipg. {Letxyin'.Ify';pgthenx';pgbythedenitionofpathreachability. Thus,prex'(y)prex'(x). Thus,f(prex'(y1):::prex'(yn))prex'(x).Fortheconverseinclusion, werstshowthat'satisesthefollowingtwopropertiesforallgandi: ForprovingP1weassumex';"g.Sincex=f(u)in'wehavexxin'by P2 P1 ifi2f1:::ngandx';ipgthenyi';pg. ifx';"gthenf=g. A1:1-closedness.Thusx';"fwhichimpliesf=gsince'ispathconsistent (condition1)anda1:1-closed,i.e.p1holds. ForprovingP2,weassumei2f1:::ngandx';ipg.Bydenitionofpath reachabilitythereexistsx0,f0,andvsuchthat x';"x0; x0=f0(y01:::y0i:::y0n); 14 y0i';pg:

15 TheA1:2-closednessof'andx';"x0implyxx0in'.Thepathconsistency Wenallyshowprex'(x)f(prex'(y1):::prex'(yn)).Given(p;g)2 closednessensuresyiy0iin'suchthatyi';pgholds.thisprovesp2. of'(condition1)andthea1:1-closednessof'impliesf=f0.hence,a2- {Letx6jyin'.Wehavetoshowthatthesetprex'(x)[prex'(y)ispath f=gandhence(";g)2f(prex'(y1):::prex'(yn)).ifp=iqthenx';iqg prex'(x),wedistinguishtwocases.ifp=",thenx';"gsuchthatp1implies suchthatp2yieldsyi';qgandhence(p;g)2f(prex'(y1):::prex'(yn)). Lemma10.EveryA3-A5-closedconstraintispathconsistent. consistent.if(p;g)2prex'(x)and(p;f)2prex'(y)thenx';pgand y';pf.thepathconsistencyof'(condition2)impliesf=g. 2 2ofDenition8andA3:1-closedness.Theproofofcondition2inDenition8is Proof.Let'beA3?A5-closed.Condition1ofDenition8followsfromcondition byinductiononpathsp.weassumex,y,f,andgsuchthatx';pf,x6jyin', andx';pg. Ifp=",thenthereexistn;m0,x1;:::;xn,y1;:::ym,u,andvsuchthat: A3-closednessimpliesthatxn6jymin'(A3:2yieldsx6jy1in',:::,x6jymin'. Thusym6jxin'byA3:3-closednesssuchthatA3:2-closednessyields yy1^:::^ym?1ym^ym=g(y0)in': xx1^:::^xn?1xn^xn=f(x0)in'; ym6jx1in',:::,ym6jxnin').hence,a4-closednessimpliesf=g.ifp=iq, thenthereexistf0,g0,x0,y0,u,vwith: Sincex6jx0in',wehavex06jy0in'byA3-closedness(thishasbeenprovedfor y';"y0;y0=g0(y01:::y0i:::y0n)in';y0i';pg: x';"x0;x0=f0(x01:::x0i:::x0n)in';x0i';pf; thecasep=").thus,a4-closednessyieldsf0=g0suchthata5-closedness impliesx0i6jy0iin',andhencef=gholdsbyinductionassumption. CComplexity 2 WeelaboratetheproofofthecomplexityandincrementalitystatementinTheorem5bypresentinganimplementationofalgorithmA. itterminatesintimeo(n3)wherenisthesizeoftheinputconstraint. Proposition14.AlgorithmAcanbeimplemented(onlineandoine)suchthat 15

16 oremptymultisetsrepresentedby>.initially,thepool'istheinputconstraint called'0(whichmaybeinputedincrementallyintheonlinecase)andthestore where'iscalledpooland Proof.WeorganizealgorithmAasareductionrelationonpairs('; isempty. store.thestoreandthepoolareeitherconstraints )or?, of apair('; Reductionpreservestheinvariantthat'^ ('0; withrespecttoalgorithma(andrestrictedtovariablesoccuringin'0).if )reducesto?then'^ isequivalentto?.if('; containsallone-stepconsequences nalstoreisa-closedandequivalenttotheinitialconstraint'0. orwithanemptypool.inthelattercase,theaboveinvariantsensurethatthe 0)then'^ isequivalentto'0^ 0.Reductioneitherterminateswith? )reducesto implementedbyrecursivelyexecutingthefollowingsequenceofinstructions: Letabasicconstraintbeoftheformxy,x6jy,orx=f(y).Reductioncanbe 1.Selectabasicconstraint'0fromthepool.If'0iscontainedinthestore 2.Else,forallaxiomsinAoftheform'0^ deleteiffromthepoolandnish. '00tothepool.Ifthereexistsanaxiomoftheform'0^ 0!'00with 0inthepooladd 3.Add'0tothestoreanddeleteitfromthepool. isnotcontainedinthestorethenaddittothestore. 0inthepoolthenreduceto?.If'0containsavariablexsuchthatxx 0!?inAwith arestrictedcase.inasecondstepweshowthattheserestrictionscanbeomitted. Werstdiscussthenecessarydatastructuresforimplementingthereductionin R3'0containsatmostoneequalitypervariable. R2Thearityofconstructorsin'0isboundedbyaconstant,sayk. R1Thealgorithmisoine,i.e.theinputconstraint'0isstaticallyknown. Letmbethenumberofvariablesin'0.Thepoolcanbeimplementedsuchthat itprovidesforthefollowingoperations(forinstanceasaqueue). {selectanddeleteabasicconstraintfromthepoolino(1). mostonepervariable)andatableofsize2m2fortheconstraintsxyandx6jy Thestorecanbeimplementedasanarrayofsizemfortheequalitiesx=f(y)(at foralloccuringvariables.thestorecansupportthethefollowingoperations: {addabasicconstrainttothepoolino(1). {testthepresenceofanequalityforxino(1). {givenavariablexwithx=f(y)2',retrievethefunctionsymbolfandthe {testthemembershipxy2 sequenceyintimeo(1). andx6jy2 intimeo(1). 16

17 {givenavariablex,retrievethesetofallysuchthatxy2'intimeo(m) {addabasicconstraintintimeo(1). Asshowninthenextparagraph,thereductionrelationcanbeimplementedsuch thatalloperationsonthestoreandthepoolareinvokedadmosto(m2)times. (analogouslyforx6jy). SinceeveryoperationcostsatmostO(m)timeandmn,thisyieldsanO(n3) ThereareatmostO(m2)distinctbasicconstraintsthatmaybeaddedtothe implementation. storeandeverybasicconstraintmaybeaddedatmostonce.hencethereareat mosto(m2)addoperationsonthestore.constraintsareaddedtothepoolonly ifsomebasicconstraintisaddedtothestore.inthiscase,atmosto(k)basic constraintsareaddedtothepoolbyr2.hence,thereareatmosto(km2)add operationsonthepool. Wenallydiscusshowtogetridoftheaboverestrictions. R2IfthearityofconstructorsisunboundedthenwestillknowthateveryoperationcostatmostO(n)wherenisthesizeof'0.Theonlyproblemis thatthenumberofbasicconstraintsthatmaybeaddedtothepoolisno tothepoolatmostonce,i.e.byrememberingthoseconstraintsthathave beenaddedtopool(andpossiblydeleted)before.thiscanbedonewitha quadratictableasforthestore. moreboundedbyo(n2).thiscanbecircumventedbyaddingconstraints R1Foranonlinealgorithm,wecanaddtheinputconstraint'0incrementally R3Ifwereplaceallequalitiesx=f(y)in'0byxx0^x0x^x0=f(y)wherex0 isafreshvariablerespectivelythentheresultingconstraintdoesnotcontain twoequationsforthesamevariable. tothepool.theproblemisthatthenumberofvariablesin'0isnotknown statically.wehavetoreplaceourstatictablesandarraysbydynamichash DFiniteTrees tablessuchthatnewvariablescanbeinserted. 2 niteorinnitetrees. Example6.Forinstance,theconstraintxf(x)issatisableoversetsofinnite ThesatisabilityofINESconstraintsdependsontheinterpretationoversetsof treesbyx7!ff(f(f(:::)))g,butnon-satisableoversetsofnitetrees. TheresultsofSectionCcarryovertothenitetreecasewhenweaddthe \occurs-check"axioma6fromtable2toaxiomseta.inparticular,lemma9 andtheorem5canbeadapted.calltreenthesetofnitetrees. 17

18 A6.'!?ifx';pxforsomepathp6=" Lemma15.Apathconsistentconstraint'closedunderA1-A3andA6issatisableinP+(Treen). Table2.Theoccurscheckaxiom ableinp+(tree)wehavedenedtheprexprex'(x)=f(p;f)jx';pfg.since Proof.ToshowaA1-A3andA6-closedandpathconsistentconstraint'satis- mustbeaniteprexforallx.hence,'issatisableinp+(treen). 'isniteandtheassumptionaboutaxioma6excludescyclicpaths,prex'(x) Theorem16.ThesatisabilityofINESconstraintsovernon-emptysetsofnite treescanbedecided(oineoronline)intimeo(n3). 2 everystep.thisisconstantiftheclosureofthereachabilityrelationbetween variablesis(justlike)implementedbyatableofsizequadraticinthenumber termination.thisislinearinthesizeofthenalconstraintandcubicinthesize ofthestartconstraint.theonlineversionmustscheduletheoccurs-checkafter Proof.Theoineversionofouralgorithmmayperformtheoccurs-checkupon EStandardSetConstraints ofvariables. 2 emptinessconstraintsx6;(\xdenotesanon-emptyset").weshowthatthe cubicalgorithmforinesconstraintscanbeadaptedtothisfragmentofstandard INESconstraintsoverpossiblyemptysetsoftreesandallowingforexplicitnon- Inthissection,wetakeaalternativeapproachtoachievetheexpressivenessof setconstraintsatthecostofadditionalaxioms. INESconstraints.Weconsideraclassofstandardsetconstraintsbyinterpreting WeinterpretetheseconstrainteitherinthestructureofsetsoftreesP(Tree) lows. Weextendourconstraintsyntaxwithexplicitnon-emptinessconstraintsasfol- satisabilityofsetconstraintsdiersdependingonthechoiceofniteorinnite orinthestructureofsetsofnitetreesp(treen).duetotheconstraintx6;, '::='1^'2jx=f(y)jxyjx6jyjx6; (1) trees.thisisnotthecasewithoutx6;aswewillshowbelow(corollary20). Example6adaptsasfollows. Example7.Theconstraintx6;^xf(x)issatisableoversetsofinnitetrees bythevariableassignmentx7!ff(f(f(:::)))g,butnon-satisableoversetsof nitetrees. 18

19 A1.xxandxy^yz!xz A3'.x6;^xy!x6jyandxy^x6jz!y6jzandx6jy!y6jx A4.f(y)x^x6jx0^x0g(z)!?forf6=g A2'.x6;^x=f(y)^xx0^x0=f(z)!yz A6'.x6;^'!?ifx';pxforsomepathp6=" A5.x=f(y)^x6jx0^x0=f(z)!y6jz B8.x6jy!y6; B7.x6;^x=f(y)!y6;andy6;^x=f(y)!x6; InTable3,wepresentthesetofaxiomsB,whichadaptsthesetAforthenew Table3.Axiomsforinclusionconstraintsover(possiblyempty)setsofnitetrees emptinesspremisesexplicit,andb7andb8havebeenadded.b7propagatesnon- emptinessthroughterms.foreveryconstantsymbola2,b7:2postulatesthat musthaveanon-emptydenotationthemselves.itiseasilycheckedthatallthese constraints.theaxiomsetsa20,a30,anda60arechangedtomakeimplicitnon- P(Tree). axiomsarevalidinp(treen)andthatallaxiomsapartfroma6arevalidin x=a!x6;.b8statesthatvariablesinvolvedinanon-disjointnessconstraint Proposition17.EveryB-closedconstraintissatisableoverP(Treen).Every constraintthatisb-closedapartfromtheoccurs-checkaxioma6'issatisable overp(tree). Proof.GivenaB-closedconstraint',wedenethesetofvariablesin'which areconstrainedtobenon-empty. Thepart of'containingonlyvariablesvar' 6;def =fxjx6;in'g Forthenitetreecase,assume'tobeB-closed.ByProposition15thereexists straintsisaclosedinesconstraint. avariableassignmentintop+(treen)whichsatises 6;andnonon-emptinesscon- Denethevariableassignmentby(x)=;forx62Var' elsewhere.weshowthatsatises'.letx62var' in'containingx. 6;.Weconsidertheinclusions inp(treen). 6;and(x)=(x) 19

20 {Ify=f(:::x:::)in',theny6;cannotbein'duetoB7:2.Hence(y)= {Constraintsxyaretriviallysatisedby. {Ifx=f(y1:::yn)in',thenyi6;cannotbein'forsomeyiduetoB7:1. {Ifyxin',theny6;cannotbein'duetoA30andB8.Hence(y)= ;(f(:::x:::)). ;(x). Hence(x)=;=(f(:::yi:::)). Fortheinnitetreecaseassume'tobeB-closedwiththeexceptionofA6'.Then bylemma9,thereexistsasatisfyingvariableassignmentintop+(tree).apart fromthat,theaboveargumentisunchanged. Theorem18.Thesatisabilityofconjunctionsofinclusionconstraintsand non-emptinessconstraintsoversetsofnitetreescanbetestedino(n3). 2 AtomicSetConstraints.INESconstraintsinterpretedoverallsetsofnite Proof.TheaxiomsinTable3againinduceaxedpointalgorithmforthesatisabilitytest.Bycarryingoverthetechniquesforthecomplexityresultsfrom treesp(treen)arealsocalledatomicsetconstraints[15].theorem18implies SectionC,weobtainthesamecomplexitybound. timecomplexityo(n3)fortheirsatisabilityproblem.furthermore,weshow 2 thattheoccurscheckaxioma60isnotneededtodecidesatisabilityofatomic setconstraints. Proof.Ifx';pxforsomep6=",thenalsox';pnxforeverypathpn=pp:::p (n-foldconcatenation).thus,foreveryprexqofsuchapathpn,thereexistsa doesnotimply?accordingtoaxioma60. Lemma19.Lettheconstraint'beB-closedwiththeexceptionofA60.Then' Butthenthereexistn1andaprexqofpnleadingtoaleafint.Thus, non-constantfunctionsymbolf2andatermf(y)suchthatx';qf(y). (q;a)2tforsomeconstantsymbola2.ifx';qf(y),wecanshowby Ifx6;2'then'containsaconjunctionexpressingthattxforsome groundtermt. (2) inductionoverqthatthereexistz;z0suchthatz=a,zz0,andz0=f(y)in'. FromLemma19andTheorem18wehavethefollowingCorollary[.Notethat thisisincontrasttoexamples6and7. whichcontradictstheassumption. FromB7:2,andA30weobtainz6jz0andhence,?isaconsequenceofAxiomA4 2 Corollary20.Thesatisabilityofatomicsetconstraintsisinvariantwithrespecttotheinterpretationoversetsofniteorinnitetrees. 20