2Proofbymathematicalinductionplaysacrucialroleinthevericationofprogramtrans-

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1 SubmissiontoJ.FunctionalProgrammingSpecialIssueonTheoremProving&FunctionalProgramming AutomaticVericationofFunctionswith DepartmentofComputing&ElectricalEngineering, AccumulatingParameters UniversityofEdinburgh,80SouthBridge, DepartmentofArticialIntelligence, Heriot-WattUniversity,Riccarton, Edinburgh,ScotlandEH144AS, Edinburgh,ScotlandEH11HN, ANDREWIRELAND ALANBUNDY 1Introduction 2Background 3LimitationsoftheBasicCritic Contents 4SpecifyingSinkTerms 2.3ACriticforDiscoveringGeneralizations 2.1ProofMethodsandCritics 2.2AMethodforGuidingInductiveProof InstantiatingSinkTerms 6OrganizingtheSearchSpace 5.1GuidingSecond-OrderUnication 5.2ListReversalRevisited 5.3TheBenetsofMeta-levelGuidance 4.1PrimarySinkTerms 4.2SecondarySinkTerms Conclusion References 9FutureWork 8RelatedWork 7ImplementationandTesting 7.1ExperimentalResults 7.2ACaseStudy

2 2Proofbymathematicalinductionplaysacrucialroleinthevericationofprogramtrans- formations.thispaperfocusesontheautomaticvericationoftransformationswhich izationstep.inearlierpaperswepresentedatechniqueforautomatingthegeneralization introduceaccumulatingparameters.suchvericationeortstypicallyrequireageneral- Abstract AndrewIrelandandAlanBundy tiontechniquetogetherwithsomepromisingexperimentalresults. stepbyanalysingfailedproofattempts.throughempiricaltestinganaturalgeneralization andextensionofthebasictechniqueemerged.herewedescribeourextendedgeneraliza- reversal: bookexample(henderson,1980).considerthefollowingnaivedenitionoflist 1980;Bird&Wadler,1988;Turner,1991)techniqueforderivingecientfunctional programs.inordertoillustratethebasicideaweuselistreversal,astandardtext Theintroductionofaccumulatorparametersisawelldocumented(Henderson, reverse(nil)=nil 1Introduction where::andappdenotelistconstructionandconcatenationrespectively.anequivalent,butmoreecient,versionisderivedbyintroducinganadditional\accumulator"parameter,i.e.rev(nil;z)=z reverse(x::y)=app(reverse(y);x::nil) Theresultingfunctionrevistail-recursive.Byexploitingthedirectcorrespondencebetweentail-recursionanditerationfurthereciencygainscanbeachieved however,isnotapurelymechanicalprocess.itrequiresustoproveaninductive conjectureoftheform: 8t:list(A):reverse(t)=rev(t;nil) rev(x::y;z)=rev(y;x::z) bypurelymechanicalmeans.establishingthecorrectnessofthistransformation, Inthispaperweareconcernedwithprovingsuchinductiveconjecturesautomatically.Anaiveattemptatproving(1)bystructuralinductiononthelisttfails.The andaninductiveconclusionoftheform: matchingtermstructures,i.e.app(:::;h::nil)ontheleft-hand-sideandh::::: Notethattheconclusionfailstomatchthehypothesisbecauseitcontainsmis- ontheright-hand-side.theproblemisthattheinductionhypothesisisnotstrong failureoccursinthestepcasewherewehaveaninductionhypothesisoftheform: app(reverse(t);h::nil)=rev(t;h::nil)

3 enough,i.e.itonlytellsusaboutthebehaviorofrevwhenitsaccumulatorparameterissettonil.thefailedproofattemptcanbeovercomebygeneralizing theconjecture.thegeneralizationinvolvestheintroductionofanewuniversally quantiedvariableintotheconjecture,i.e. AutomaticVericationofFunctionswithAccumulatingParameters3 aninnitebranchingpointintothesearchspace.itisknown(kreisel,1965)that ofinference.inagoal-directedframework,therefore,ageneralizationintroduces Wewillrefertothisasaccumulatorgeneralization.Thegeneralizedconjectureprovidesastrongerinductionhypothesiswhichenablesthestepcaseprooftosucceed. bymathematicalinduction.ageneralizationstepisunderpinnedbythecut-rule Theneedforgeneralizationrepresentsamajorobstacletotheautomationofproof 8t:list(A):8l:list(A):app(reverse(t);l)=rev(t;l) (2) ductionoftheapp(:::;l)termstructureontheleft-hand-sideof(2)motivated? isrequirediftheprocessistobefullyautomated.forinstance,howistheintro- stronghintastowherethenewuniversalvariableshouldoccurwithinthegeneralizedconjecture.however,evenwiththiselementaryexampleadditionalguidance thecut-eliminationtheoremdoesnotholdforinductivetheories.consequently heuristicsforcontrollinggeneralizationplayanimportantroleintheautomation ofinductiveproof. startingpointisameta-leveldescriptionofthecommonstructurewhichcharacterizesaninductiveproof.whenaproofattemptfailsthisdescriptioncanthenbe Returningtothelistreversalexample,theaccumulatorparameterprovidesa Weaddressthisquestionthroughtheuseofameta-levelreasoningtechnique.Our usedtobridgethegapbetweenthefailureandasubsequentsuccessfulproof.we arguethathavingsuchadescriptionprovidesahandleontheinnitesearchspace Webuilduponthenotionofaproofplan(Bundy,1988)andtactic-basedtheoremproving(Gordonetal.,1979).Whileatacticencodesthelow-levelstructure generatedbythegeneralizationproblem. automateddeduction,aproofplanguidesthesearchforaproof.thatis,givena ofafamilyofproofsaproofplanexpressesthehigh-levelstructure.intermsof collectionofgeneralpurposetacticstheassociatedproofplancanbeusedautomaticallytotailoraspecialpurposetactictoproveaparticularconjecture. 2.1ProofMethodsandCritics 2Background proofplanningframeworkwithanexceptionhandlingmechanismwhichenables (Ireland,1992)wehaveattemptedtoautomatethisprocess.Criticsprovidethe orsupplyingadditionallemmatatotheprover.throughthenotionofaproofcritic odsexpressthepreconditionsfortacticapplication.thebenetsofproofplanscan beseenwhenaproofattemptgoeswrong.experiencedusersoftheoremprovers, suchasnqthm,areusedtointerveningwhentheyobservethefailureofaproof attempt.suchinterventionstypicallyresultintheusergeneralizingtheirconjecture Thebasicbuildingblocksofproofplansaremethods.Usingameta-logic,meth-

4 4thepartialsuccessofaproofplantobeexploitedinsearchforaproof.Themech- anismworksbyallowingproofpatchestobeassociatedwithdierentpatternsof describedbelow.2.2amethodforguidinginductiveproof ofpatchinginductiveproofsbaseduponthepartialsuccessoftheripplemethod preconditionfailure.wepreviouslyreported(ireland&bundy,1996)variousways AndrewIrelandandAlanBundy Inthecontextofmathematicalinductiontheripplemethodplaysapivotalrolein berepresentedasfollows: ofrewriterulesinordertoprovestepcasegoals.schematicallyastepcasegoalcan guidingthesearchforaproof.theripplemethodcontrolstheselectiveapplication {z} hypothesis`p[c1(a);b] bec3(b)whichgivesrisetoagoaloftheform: wherec1(a)denotestheinductionterm.toachieveastepcasegoaltheconclusion Notethatinordertoapplytheinductionhypothesiswemustrstinstantiateb0to mustberewrittensoastoallowthehypothesistobeapplied: 8b0:P[a;b0]`c2(P[a;c3(b)]) {z} conclusion Inductionandrecursionarecloselyrelated.Theapplicationofaninductionhypothesiscorrespondstoarecursivecallwhiletheinstantiationofaninductionhypothesis correspondstothemodicationofanaccumulatorparameter.theneedtoinstantiateinductionhypothesesiscommonplaceininductiveproof.ourtechnique,as willbeexplainedbelow,exploitsthisfact. Syntacticallyaninductionhypothesisandconclusionareverysimilar.Moreformally,thehypothesiscanbeexpressedasanembeddingwithintheconclusionmizesthechancesofapplyinganinductionhypothesis.Thisisthebasicideabehind causethemismatchbetweenthehypothesisandconclusion.converselyanyterm givenabovetakestheform: sentwave-frontsandwave-holesrespectively,e.g.anannotatedversionofthegoal meta-levelannotationscalledwave-frontstodistinguishthetermstructureswhich structurewithintheconclusionwhichcorrespondstothehypothesisiscalledskeleton.ingeneral,embeddedwithineachwave-frontwillbepartsoftheskeletonterm structure,theseareknownaswave-holes.weuseaboxandanunderlinetorepre- theripplemethod.theapplicationoftheripplemethod,orrippling,makesuseof P[a;c3(b)]`c2(P[a;c3(b)]) Restrictingtherewritingoftheconclusionsoastopreservethisembeddingmaxiterm.Thearrowsareusedtoindicatethedirectioninwhichwave-frontscanbe Wewillrefertoawave-frontanditsassociatedwave-hole,e.g.c1(a)",asawave- 8b0:P[a;b0]`P[c1(a)";bbc]

5 rippling-out:f1(:::(fn(c1(:::)")):::) AutomaticVericationofFunctionswithAccumulatingParameters5 rippling-in: rippling-sideways: f1(c1(:::)";:::;fi(:::);:::) before before f1(:::;:::;ci(fi(:::))#;:::) cn(f1(:::(fn(:::)):::))" after Anoutwardrippleinvolvesthemovementofwave-frontsintolessnestedtermtreepositions.Asidewaysripplesmoveswave-frontsbetweendistinctbranchesinthetermtree whileinwardripplesmovementofwave-frontsintomorenestedtermtreepositions.in general,awave-rulemaycombineallthreeforms. cn(f1(:::fn(:::):::))# before f1(:::fn(c1(:::)#):::) after after i.e.b:::c.aswillbeexplainedbelowsinksplayanimportantroleinidentifyingthe movedthroughthetermstructure.atermstructurewiththeannotationsremoved bematchedbyuniversalvariablesinthehypothesisweuseannotationscalledsinks, needforaccumulatorgeneralization.asuccessfulapplicationoftheripplemethod iscalledtheerasure.inordertodistinguishtermswithintheconclusionwhichcan Fig.1.Thethreebasicripplingpatterns canbecharacterizedasfollows: wave-frontannotationisnolongerrequired.ripplingrestrictsrewritingtoasyntacticclassofrulescalledwave-rules.wave-rulesmakeprogresstowardseliminating theripplegivenabovetakestheformy: 8b0:P[a;b0]`c2(P[a;bc3(b)c])" Notethatthetermc3(b),i.e.theinstantiationforb0,occurswithinasinksothe wave-frontswhilepreservingskeletontermstructure.awave-rulewhichachieves whicharesummariesedschematicallyingure1.thepreconditionsforapplying Wave-rulesarederivedautomaticallyfromdenitionsandlogicalpropertieslike requiremultiplewave-ruleapplications.therearethreebasicpatternsofrippling substitution,associativityanddistributivityetc.ingeneral,asuccessfulripplewill P[c1(X)";Y])c2(P[X;c3(Y)#])" etal.,1993;basin&walsh,1994).toillustrateoneofthebasicpatternsofrippling wave-rulesaregiveningure(2).foracompletedescriptionofripplingsee(bundy (3) aninductiveproofofconjecture(2)ispresented.structuralinductiononthelistt yweuse)todenoterewriterulesand!todenotelogicalimplication.

6 6Inputsequent:H`G[f1(c1(:::)";f2(b:::c);f3(c2(:::)"))] Methodpreconditions: 1.thereexistsasubtermTofGwhichcontainswave-front(s),e.g. AndrewIrelandandAlanBundy 3.thewave-ruleconditionfollowsfromthecontext,e.g. 2.thereexistsawave-rulewhichmatchesT,e.g. C!f1(c1(X)";Y;Z))c5(f1(X;c3(Y)#;c4(Z)#))" Outputsequent: 4.resultinginwarddirectedwave-frontsarepotentiallyremovable,e.g. :::c3(f2(b:::c))# {z} (sinkable):::or:::c4(f3(c2(:::)"))# H`C H`G[c5(f1(:::;c3(f2(b:::c))#;c4(f3(c2(:::)"))#))"] (cancellable)::: {z } hypothesistakestheform: structuresmustmatchḟig.2.preconditionsforapplyingwave-rules givesrisetoatrivialbasecase.wefocushereonthestepcasewheretheinduction Notethatinorderforawave-ruletobeapplicablebothobject-levelandmeta-levelterm andtheannotatedconclusiontakestheform: Theproofofthestepcaserequiresthedenitionsofreverse,revandapp,aswellas theassociativityofapp.thesedenitionsgiveriseto49wave-ruleswhichinclude: app(reverse(h::t");blc)=rev(h::t";blc) 8l0:list(A):app(reverse(t);l0)=rev(t;l0) (5) (4)

7 AutomaticVericationofFunctionswithAccumulatingParameters7 rule(6)appliesontheleft-hand-sideof(5)togive: Notethatallwave-rulesareavailableduringtheprocessofplanningaproof.Wave- app(app(x;y)";z))app(x;app(y;z)#) reverse(x::y"))app(reverse(y);x::nil)" rev(x::y";z))rev(y;x::z#) (7) (8) (6) Wave-rule(8)appliesontheleft-hand-sidegiving: Applyingwave-rule(7)ontheright-hand-sideof(9)gives: app(app(reverse(t);h::nil)";blc)=rev(h::t";blc) app(app(reverse(t);h::nil)";blc)=rev(t;bh::lc) (9) simpliestogive:app(reverse(t);bh::lc)=rev(t;bh::lc) Notethatthetermstructuredelimitedbythesinkannotationontheleft-hand-side Amatchbetween(10)and(4)isachievedbyinstantiatingl0tobeh::l.This completesthestepcaseproof. app(reverse(t);bapp(h::nil;l)c)=rev(t;bh::lc) 2.3ACriticforDiscoveringGeneralizations whereddenotesatermwhichdoesnotcontainanysinks.wecalltheoccurrenceof Intermsofthepreconditionsforapplyingwave-rules,theneedforanaccumulator ofwave-rule(3).theidenticationofablockagetermtriggersthegeneralization dablockagetermbecauseitblocksthesidewaysripple,inthiscasetheapplication generalizationcanbeexplainedbythefailureofprecondition4,i.e.amissingsink (seegure2).schematicallythisfailurepatterncanbecharacterisedasfollows: topartiallyspecifytheoccurrencesofasinkvariable.intheschematicexample critic.theassociatedproofpatchintroducesschematictermsintothegoalinorder presentedabovethisleadstoapatchedgoaloftheform: P[a;d]`P[c1(a)";d] 8l0:P[a;M(l0)]`8l:P[c1(a)";M(blc)]

8 8whereMdenotesasecond-ordermeta-variable.Notethatwave-rule(3)isnow applicable,givingrisetoarenedgoaloftheform: 8l0:P[a;M(l0)]`8l:c2(P[a;c3(M(blc))#])" AndrewIrelandandAlanBundy TheexpectationisthataninwardripplewilldeterminetheidentityofM. Notethattheoccurrenceofnilontheright-hand-sideisablockagetermbecause jecture(1)givesrisetothefollowingfailurepattern: Relatingthisproofpatchtothelistreversalexampleaninductiveproofofcon- itpreventstheapplicationofwave-rule(7).thepatchedgoaltakestheform: reverse(t)=rev(t;nil)` M2(reverse(h::t");blc)=rev(h::t";M1blc)(12) app(reverse(t);h::nil)"=rev(h::t";nil) {z} blocked (11) Usingwave-rule(6)thegoalbecomes: Wave-rule(7)isnowapplicableandgivesrisetoagoaloftheform: 8l0:list(A):M2(reverse(t);l0)=rev(t;M1(l0))` M2(app(reverse(t);h::nil)";blc)=rev(h::t";M1(blc)) Thebasiccriticdescribedinx2.3hasprovedverysuccessful(Ireland&Bundy, willbedetailedinx5.wewillrefertotheabovegeneralizationasthebasiccritic. Ourapproachtotheproblemofconstrainingtheinstantiationofschematicterms M2(app(reverse(t);h::nil)";blc)=rev(t;h::M1(blc)#) 1996).Throughourempiricaltesting,however,anumberoflimitationshavebeen observed: 2.Thebasiccriticwasdesignedinthecontextofequationalproofs.Asink 1.Certainclassesofexamplerequiretheintroductionofmultiplesinkvariables. Thebasiccriticonlydealswithsinglesinkvariables. 3LimitationsoftheBasicCritic variableisassumedtooccuronbothsidesofanequation.onthesideopposite totheblockagetermitisassumedthatintheresultinggeneralizedterm structurethesink(auxiliary)willoccurasanargumentoftheoutermost functor.

9 >Fromtheseobservationsanumberofnaturalextensionstothebasiccriticemerged. Theseextensionsaredescribedinthefollowingsections. 3.Sinktermoccurrenceswhicharemotivatedbyblockagetermsaremoreconstrainedthanthosewhicharenot.Thisisnotexploitedbythebasiccritic AutomaticVericationofFunctionswithAccumulatingParameters9 duringthesearchforageneralization. aboveweextendthemeta-levelannotationstoincludethenotionsofprimaryand tobeprimary.allotherwave-frontsaredesignatedtobesecondary.toillustrate, secondarywave-fronts.awave-frontwhichprovidesthebasisforasidewaysripple considerthefollowingschematicconclusion: Inordertoexploitthedistinctionbetweendierentsinktermoccurrenceshintedat butwhichisnotapplicablebecauseofthepresenceofablockagetermisdesignated 4SpecifyingSinkTerms Assumingthattheoccurrenceofdin(13)denotesablockagetermthenwave-rule andthefollowingwave-rules: f(c1(x;y)";z))f(x;c2(z;y)#) g(x;c1(y;z)"))c3(g(x;y);z)" g(f(c1(a;b)";d);c1(a;b)") (15) (13) i.e. Usingsubscriptsztodenoteprimaryandsecondarywave-frontsthentheanalysis (14)isnotapplicable.Wave-rule(15)isapplicableandenablesanoutwardsripple, presentedabovegivesrisetothefollowingclassicationofthewave-frontsappearing in(13): c3(g(f(c1(a;b)";d);a);b)" Notethattheripplingofthesecondarywave-frontsisundone.Thisincreasesthe numberofgeneralizationswhichmaybesubsequentlydiscovered.relatingthenotionofprimaryandsecondarywave-frontstoblockedgoal(11)givesriseto: reverse(h::t"2)=rev(h::t"1;nil) g(f(c1(a;b)"1;d);c1(a;b)"2) (16) znotethatwave-rulesmustalsotakeaccountoftheextensiontothewave-front annotations.

10 10 sinktermisafunctionoftheblockagetermandiscomputedasfollows: theseasprimarysinkterms.thepositionofaprimarysinktermcorrespondsto Foreachprimarywave-frontanassociatedsinktermisintroduced.Wereferto thepositionoftheblockagetermwithintheconclusion.thestructureofaprimary AndrewIrelandandAlanBundy pri(x)=8><>:mi(blic) 4.1PrimarySinkTerms thesameobject-levelvariable.thisrepresentsachoicepointintheconstruction levelvariable.ingeneraldistinctprimarysinktermsmayormaynotneedtoshare NotethatMidenotesahigher-ordermeta-variablewhilelidenotesanewobject- ofprimarysinkterms.assumingddenotesaconstantthenpri(d)evaluatesto F(pri(Y1);:::;pri(Yn)) Mi(X;blic) wherexf(y1;:::;yn) otherwise ifxisawave-front ifxisaconstant Relatingthegeneralnotionofprimarysinktermstothespeciclistreversalexamplegives: M1(bl1c).Substitutingthissinktermfordin(16)givesaschematicconclusionof theform: reverse(h::t"2)=rev(h::t"1;m1(bl1c)) g(f(c1(a;b)"1;m1(bl1c));c1(a;b)"2) 4.2SecondarySinkTerms (17) computeasecondarysinktermasfollows: Foreachsecondarywave-frontweeagerlyattempttoapplyasidewaysrippleby introducingoccurrencesofthevariablesassociatedwiththeprimarysinkterms. theprimarysinkterms.toillustrate,consideragaintheschematicconclusion(17). eachsubterm,x,oftheconclusionwhichcontainsasecondarywave-front,we Theseoccurrencesarespeciedagainusingschematictermstructuresandarecalled TakingXtobec1(a;b)"2thentheprocessofintroducingsecondarysinkterms wherel1;:::;lmdenotethevectorofvariablesgeneratedbytheconstructionof secondarysinkterms.theconstructionofsecondarysinktermsareasfollows.for givesrisetoanewschematicconclusionoftheform: g(f(c1(a;b)"1;m1(bl1c));m2(c1(a;b)"2;bl1c)) sec(x)=mi(x;bl1c;:::;blmc) NotethattheselectionofXrepresentsachoicepointintheconstructionofsecondarysinkterms.Inthecaseof(17),anotheralternativeinstantiationforXexists, i.e. g(:::;c1(a;b)"2) (18)

11 Againrelatingthegeneralnotiontothespeciclistreversalexamplegivesriseto 2alternativepatchesoftheform: givingrisetoaschematicconclusionoftheform: AutomaticVericationofFunctionswithAccumulatingParameters11 reverse(m2(h::t"2;bl1c))=rev(h::t"1;m1(bl1c)) M2(g(f(c1(a;b)"1;M1(bl1c));c1(a;b)"2);bl1c) Notethatthesecondofthesecorrespondstothepatchedgoal(12). M2(reverse(h::t"2);bl1c)=rev(h::t"1;M1(bl1c)) (19) 5InstantiatingSinkTerms (20) beinstantiatedthroughtheunicationwithwave-rules.belowweshowindetail x7).inthisapplication,however,weonlyrequiresecond-orderunication.the Theprocessofinstantiatingthesinktermsintroducedbythegeneralizationcritic rulesinthepresenceofschematictermstructurerequireshigher-orderunication. Ourimplementationthereforeexploitsahigher-orderunicationprocedure(see howthemeta-levelannotationscanbeusedtoconstraintheunicationprocess goal-termrequiresnarrowing,i.e.rewritingwherefreevariablesintheredexcan applicationofwave-rulesinthepresenceofsecond-ordermeta-variableswithinthe isguidedbytheapplicationofwave-rules.ingeneral,theapplicationofwave- Ourmethodforconstrainingtheapplicationofrewriteruleswithinthecontext ingure3.second-orderunicationwill,ingeneral,leadtoanon-terminatingsequenceofwave-ruleapplications.forthisreasonprojectionsareusedtoeagerly terminateinwardripples.aprojectionisappliedwhenevertheimmediatesuper- ofskeletontermstructurewhichcontainssecond-ordermeta-variablesispresented termofasinktermisaninwarddirectedwave-front.analternativetooureager 5.1GuidingSecond-OrderUnication anddiscussthebenetsofthisapproach. sinktermsisthatitonlydealswithwave-frontswhichcontainsinglewave-holes. tracksandattemptsfurtherrippling.alimitationofourmethodforinstantiating respondingtobasecases.ondetectinganon-theoremthecriticmechanismback- conjecture.thecheckerevaluatesgroundinstancesoftheconjecture,typicallycor- counterexamplecheckerisusedtoltercandidateinstantiationsoftheschematic variablesmayofcoursegiverisetoanover-generalization,i.e.anon-theorem.a instantiationstrategyisdiscussedinx9.thestrategyofeagerinstantiationofmeta-

12 12 Goal-term(before): Wave-rule:f(bWc;c1(X;Y)"N;Z))f(bc2(Y)c;X;c3(Z;Y)#N) AndrewIrelandandAlanBundy Guidance: 1.Unifyallwave-termswithintheleft-hand-sideoftheselectedwave-rulewithwavetermswithintheselectedgoal-term.Successrequiresamatchbetweenwavedirections,wave-holesandwave-terms.Theprocessisrecursiveandworksfrom theinsideout,e.g. goal-term,e.g. f(m2(c1(a;b)"2;bl1c);c1(a;b)"2;m3(c1(a;b)"2;bl1c))) f(bwc;c1(a;b)"2;z))f(bc2(b)c;a;c3(z;b)#2) M1(c1(a;b)"2;bl1c) 2.Unifytheerasureoftheleft-hand-sideofthewave-rulewiththeerasureofthe Goal-term(after):f(bc2(b)c;a;c3(M3(c1(a;b)"2;bl1c);b)#2) 3.Matchallsinkswhichappearwithintheleft-hand-sideoftheselectedwave-rule withsinkswithintheselectedgoal-term. f(bl1c;c1(a;b)"2;m3(c1(a;b)"2;bl1c)))f(bc2(b)c;a;c3(m3(c1(a;b)"2;bl1c);b)#2) f(bc2(b)c;a;c3(m3(c1(a;b)"2;bl1c);b)#2) Notethatinstep3theremayexistmultiplesinksgivingrisetoalternativeprojections. egyduringtheplanningprocessenableseachalternativerippletobeexplored.although Eachprojectionresultsinadistinctripple.Ouruseofaniterativedeepeningsearchstrat- applicabletoinwarddirectedwave-fonts. illustratedintermsofanoutwarddirectedwave-front,theprocessdescribedaboveisalso Fig.3.AnnotatedSecond-OrderUnication

13 wave-rules(6)and(7)togive: Returningtothelistreversalexample,consideragainpatch(19)whichripplesby 8l0:list(A):reverse(M2(t;l0))=rev(t;M1(l0))` AutomaticVericationofFunctionswithAccumulatingParameters13 reverse(m2(h::t"2;blc))=rev(t;h::m1blc#2) 5.2ListReversalRevisited Notethatthewave-frontontheleft-hand-sideisnowclassiedassecondary.When- givesriseto: siedassecondary.sincebothwave-frontsareclassiedassecondarytheneither canberippledatthisstage.considerthewave-termontheleft-hand-side: everthemovementofawave-frontrequiressecond-orderunicationthenitisclas- Usingtheannotatedunicationprocessdescribedabovewave-rule(7)appliesand 8l0:list(A):reverse(rev(M3(t;l0);t)=rev(t;M1(l0))` reverse(rev(h::m3(h::t"2;blc)#2;t))=rev(t;h::m1blc#2) M2(h::t"2;blc) (21) givesrisetoagoaloftheform: instantiation,mentionedinx5.1,m1becomesx:xandm3tobex:y:y.this NotethatM2isinstantiatedtobex:y:rev(M3(x;y);x).Bytheprocessofeager Alternatively,ifpatch(20)isselectedthenwave-rules(6)and(7)applytogive: resultinggeneralizationcorrespondsto: Theinductionhypothesiscannowbeappliedbyinstantiatingl0tobeh::l.The 8l0:list(A):reverse(rev(l0;t)=rev(t;l0)` 8l0:list(A):M2(reverse(t);l0)=rev(t;M1(l0))` 8t:list(A):8l:list(A):reverse(rev(l;t))=rev(t;l) reverse(rev(bh::lc);t))=rev(t;bh::lc) Nowconsiderthewave-termontheleft-hand-sideoftheform: Usingtheannotatedunicationprocesswave-rule(8)nowappliestogive: M2(app(reverse(t);h::nil"2);blc)=rev(t;h::M1blc#2) app(reverse(t);app(h::nil;m3(app(reverse(t);h::nil)"2;blc))#2)=rev(t;h::m1blc#2) M2(app(reverse(t);h::nil)"2;blc) 8l0:list(A):app(reverse(t);M3(t;l0))=rev(t;M1(l0))` (22)

14 Simplifyingthesinkontheleft-hand-sideandinstantiatingl0tobeh::lenables NotethatM2isinstantiatedtobex:y:app(x;M3(x;y)).Againbytheprocess ofeagerinstantiationm1becomesx:xandm3tobex:y:ygiving: 148l0:list(A):app(reverse(t);l0)=rev(t;l0)` app(reverse(t);bapp(h::nil;l)c)=rev(t;bh::lc) AndrewIrelandandAlanBundy annotationstoguidetheunicationprocess.wecomparethebranchingrateswhen Usingthelistreversalexamplewenowconsiderthebenetsofusingmeta-level theapplicationofinductionhypothesis.notethattheresultinggeneralizationcorresponds(2).5.3thebenetsofmeta-levelguidancamplegivesriseto49wave-rules. applyingannotatedandunannotatedrewriterules.asmentionedinx2.2thisex- eliminatesallbutthefollowing4wave-rules: Firstly,consideragain(21),forthisgoal-termtheannotatedunicationprocess reverse(x::y"))app(reverse(y);x::nil)" Thisshouldbecomparedwithunannotatedunicationwhichgivesriseto18applicablerewriterules. Secondly,inthecaseof(22)theannotatedunicationprocessagaineliminates rev(y;x::z"))rev(x::y#;z) app(x::y";z))x::app(y;z)" rev(x::y";z))rev(y;x::z#) allbut4wave-rules,i.e. app(app(x;y)";z))app(x;app(y;z)#) app(x;app(y;z)"))app(app(x;y);z)" wave-rulesshouldthenbecomparedwiththeresultsofunannotatedunication whichagaingivesriseto18applicablerewriterules. Notethatonlytherstthreeofthesewillactuallyapplysincethethirdisruled-out byprecondition4oftheripplemethod,i.e.sink-ability.the3remainingapplicable app(reverse(y);x::nil)")reverse(x::y#) X::app(Y;Z)")app(X::Y;Z)" cationtheyalsoconstrainthenumberofuniers.toillustrate,consideragain Whiletheannotationsreducethenumberofwave-rulesconsideredforuni-

15 theunicationofgoal-term(22)andtheleft-hand-sideofwave-rule(8).secondorderunicationgeneratestwopossibleuniers,i.e.x:y:app(x;m3(x;y))and skeletonpreservation(seex2.2),soisrejectedbytheannotatedunicationprocess. x:y:app(h::t;m2(x;y)).notethattherstisbaseduponprojectionthesecond usesimitation.theimitation,however,violatesthekeypropertyofrippling,i.e. AutomaticVericationofFunctionswithAccumulatingParameters15 Incontrollingthesearchforageneralizationweplaceanumberofconstraintson theproofplanningprocess: Planninginthecontextofschematictermstructuresrequiresabounded tobethesequence(s)oftermtreepositionswhichcanbereachedbythe ofripplepaths.givenawave-front,itsassociatedripplepathsaredened searchstrategy.weuseaniterativedeepeningstrategybaseduponthelength 6OrganizingtheSearchSpace Backtrackingovertheconstructionofsecondarysinktermsdealswiththe Sinceprimarysinktermsaremoreconstrainedthansecondarysinkterms choicepointissueraisedinx4.2. bethenumberofwave-ruleapplicationsusedinitsconstruction. applicationofwave-rules.thelengthofaparticularripplepathisdenedto Theextensionstothebasiccriticdescribedabovedirectlyaddressthelimitations highlightedinx3: priorityisgiventotheripplingofprimarywave-fronts. 2.Theissueofpositioningauxiliarysinkvariablesisdealtwithbytheability 1.Thelinkageofblockagetermswiththeintroductionofprimarysinkterms torevisetheconstructionofsecondarysinkterms. withintheschematicconjectureaddressestheissueofmultiplesinkvariables. 7ImplementationandTesting planner(bundyetal.,1990).theimplementationmakesuseofthehigher-order featuresof-prolog(miller&nadathur,1988).belowwedocumentthetestingof OurextendedcritichasbeenimplementedandintegratedwithintheCLAMproof ourimplementation.7.1experimentalresults 3.Byextendingthemeta-logictoincludethenotionsofprimaryandsecondary wave-frontsweareabletoexploittheobservationthatcertainsinktermsare moreconstrainedthanothersduringthesearchforgeneralizations. whichthebasiccriticmissed.moreover,anumberofnewexamplesweregeneralizedbytheextendedcriticforwhichtheapplicationofthebasiccriticresultedin bytheextendedcritic.theextendedcritic,however,discoveredgeneralizations Theresultspresentedin(Ireland&Bundy,1996)forthebasiccriticwerereplicated

16 proverssuchasnqthm(boyer&moore,1979).therelativeperformanceofthe failure.ourresultsaredocumentedinthetablesgiveninappendixc.theexampleconjecturesforwhichtheextendedcriticimprovesupontheperformanceofthe basicandextendedcriticsontheexampleconjecturesisrecordedintableii.the AndrewIrelandandAlanBundy 16 basiccriticarepresentedintablei.alltheexamplesrequireaccumulatorgeneralizationandthereforecannotbeprovedautomaticallybyotherinductivetheorem lemmatausedinmotivatingthegeneralizationsarepresentedintableiiiwhilethe actualgeneralizedconjecturesaregivenintableiv.allthesegeneralizationsare discoveredautomatically,i.e.nouserintervention. funatendxnil=(x::nil) funsplitxy=split11xnily; valatend=fn:'a->'alist-> siderthefunctionsdenedingure4.rewriterulesderivedfromthesedenitions Wenowillustrateourgeneralizationtechniqueusingamorerealisticexample.Con- atendx(y::z)=y::(atendxz);valsplit=fn:nat->'alist 7.2ACaseStudy funsplit1vwxnil=(x::nil) valapp=fn:'alist->'alist then if(v>w) else split1vwx(y::z)= x::(split12w(y::nil)z)valmap=fn:('a->'b)->'alist funmapxnil=nil funappnilz=z mapx(y::z)=(xy)::(mapxz); app(x::y)z=x::(appyz); ->'alistlist valsplit1=fn:nat->nat (split1v+1w(atendyx)z);funreducexnil=nil ->'alist->'alist reducex(y::z)= areamongthosegiveninappendixa.usingthesedenitionswecanspecifyan ->'alistlist Fig.4.Examplelistprocessingfunctions valreduce=fn:('a->'blist-> 'blist)->'alist-> (xy(reducexz)); aconjectureoftheform: equivalencebetweenasingleandadistributedapplicationofthemapfunctionby Thisconjecturewasprovidedbyanindependentresearchgroupworkingonthe astomakesitesofpotentialparallelismexplicit.conjecture(23)issuchatransformation.provingthecorrectnessoftransformationsiscurrentlyundertakenby 8t:list(A):8f:A!B:8n:IN: developmentofparallelsystemsfromfunctionalprototypes(michaelson&scaife, 1995).Theirdevelopmentprocessinvolvestransformingfunctionalprototypesso map(f;t)=reduce(x:y:app(x;y);map(x:map(f;x);split(n;t)))(23)

17 techniquesweresuccessfulinautomaticallyndingaproofof(23). handandrepresentsatimeconsuminghurdletotheresearchproject.havingfailed ofrewriterule(26)givesrisetoarenedgoaloftheform: toproveconjecture(23)byhanditwaspassedtousasachallengetheorem.our Inordertoprove(23)wemustrstunfoldthedenitionofsplit.Anapplication AutomaticVericationofFunctionswithAccumulatingParameters17 derforaninductiveproofattempttosucceed.anaccumulatorgeneralizationis required.thegeneralizedconjecturetakestheform: Aproofof(24)requiresinduction.However,(24)mustrstbegeneralizedinor- 8t:list(A):8f:A!B:8n:IN: map(f;t)=reduce(x:y:app(x;y);map(x:map(f;x);split1(1;n;nil;t)))(24) roleourextendedcriticplaysinautomatingthediscoveryof(25),therequired Notethetwonewuniversallyquantiedvariablesl1andl2.Wefocusuponthe 8t:list(A):8f:A!B:8n:IN:8l1:IN:8l2:list(A): derivedfromdenitions.7.2.1firstproofattempt: generalization,anditsproof.thewave-rulesrequiredforthisproofaregivenin appendixb.withtheexceptionofwave-rules(31)and(32)allthewave-rulesare map(f;app(l2;t))= reduce(x:y:app(x;y);map(x:map(f;x);split1(l1;n;l2;t)))(25) Aninductiveproofof(24)requiresinductiononthestructureofthelistt.The basecasegoalistrivial.wefocushereonthestepcasegoalwhichgivesrisetoan inductionhypothesisoftheform: 8f0:A!B:8n0:IN: andaninductionconclusionoftheform: map(bfc;h::t")= map(f0;t)=reduce(x:y:app(x;y);map(x:map(f0;x);split1(1;n0;nil;t))) theblockageterms1andnilwhichoccurintherstandthirdargumentpositionsofsplit1.triggeredbytheseblockagetermstheextendedgeneralizationcritic 8f0:A!B:8n0:IN:8l01:IN:8l02:list(A) Wave-rule(29)isapplicable,however,wave-rules(28)and(27)arenotbecauseof generatesaschematichypothesisoftheform: reduce(x:y:app(x;y);map(x:map(bfc;x);split1(1;bnc;nil;h::t") blocked {z })) map(f0;m3(t;l01;l02))= reduce(x:y:app(x;y);map(x:map(f0;x);split1(m1(l01);n0;m2(l02);t)))

18 map(bfc;m3(h::t"2;bl1c;bl2c))= whiletheschematicconclusiontakestheform: 18reduce(x:y:app(x;y); map(x:map(bfc;x);split1(m1(bl1c);bnc;m2(bl2c);h::t"1))) AndrewIrelandandAlanBundy Notethattheblockageterms1andnilhavebeenreplacedbyprimarysinkterms sideofthegoalequationisclassiedassecondaryandconsequentlyitisassociated withasecondarysinktermwhichcontainsoccurrencesofl1andl2. M1(bl1c)andM2(bl2c)respectively.Notealsothatthewave-frontontheleft-hand- ripplingofprimarywave-frontssothereisnochoiceastowhichwave-rulesshould beinitiallyapplied.theintroductionofsinktermsm1(bl1c)andm2(bl2c)enable Theripplemethodisnowappliedtotheschematicgoal.Priorityisgiventothe wave-rules(27)and(28)tobeapplied.jointlytheymotivateacasesplitonm1(l1) andn Secondproofattempt: Usingwave-rule(27)theright-hand-sideoftheconclusionripplestogive: Notethattheconstraintsoftheannotatedunicationprocess(seegure3,step :::=reduce(x:y:app(x;y); map(x:map(bfc;x);l2::split1(b2c;bnc;bh::nilc;t)"1) Case:M1(l1)>n: rule(30)gives: Notethatthewave-fronthasbeen-reduced.Afurtheroutwardrippleusingwave- 3)instantiateM1andM2tobex:x.Bywave-rule(29)theconclusionripples furthertogive: :::=reduce(x:y:app(x;y); Theleft-hand-sideoftheconclusioncontainsasecondarysinktermsorippling :::=app(map(f;l2);reduce(:::;map(:::;split1(b2c;bnc;bh::nilc;t))))"1 map(f;l2)::map(:::;split1(b2c;bnc;bh::nilc;t))"1 involvesmoresearch.asmentionedinx6aniterativedeepeningsearchstrategyis employed.usingannotatedunicationwave-rule(31)appliesgivingriseto: app(map(f;l2);map(bfc;app(bh::nilc;t)))"2=:::

19 rule(33)theripplingoftheconclusioniscomplete: map(bfc;app(bh::nilc;t))= Theapplicationof(31)instantiatesM3tobex:y:z:app(z;x).Finally,bywave- Theinductionhypothesiscanbeappliedbyinstantiatingl01tobe2andl02to reduce(x:y:app(x;y);map(x:map(bfc;x);split1(b2c;bnc;bh::nilc;t))) AutomaticVericationofFunctionswithAccumulatingParameters19 remainingbranchofthecasesplit.case:l1n: Usingwave-rule(28)asidewaysripplecanbeappliedtotheright-hand-sideofthe beh::nil.theinstantiationsform1,m2andm3arepropagatedthroughthe conclusion: Theleft-hand-sideoftheconclusionisoftheform: :::=reduce(x:y:app(x;y); map(x:map(bfc;x);split1(bl1+1c;bnc;batend(h;l2)c;t))) Theripplinginthisbranchofthecasesplitiscomplete: towhichwave-rule(32)appliesgiving: map(bfc;app(batend(h;l2)c;t))= map(bfc;app(batend(h;l2)c;t))=::: map(bfc;app(bl2c;h::t"1))=::: canbeconstructedbyclamcompletelyautomatically. automaticallygenerated(25),therequiredgeneralizationof(24).aproofof(25) Theinductionhypothesiscanbeappliedbyinstantiatingl01tobel1+1andl02to beatend(h;l2). Tosummarize,theripplemethodinconjunctionwiththeextendedcritichave reduce(x:y:app(x;y);map(x:map(bfc;x); split1(bl1+1c;bnc;batend(h;l2)c;t))) generalizationsbaseduponthefailureofanunfoldingstrategy.basicallyheused InAubin'sthesis(Aubin,1976)hepresentsatechniquefordiscoveringaccumulator ofwhatwecallprimarysinks.withregardtosecondarysinks,aubinappealstoa notionofanequationbeing\balanced".thatis,asinkshouldoccuronbothsides ofanequality. themismatchbetweentheconclusionandhypothesistosuggesttheintroduction 8RelatedWork eralizationinthecontextofproofplanningandrippling.herapproach,however, Heskethinherthesis(Hesketh,1991)tackledtheproblemofaccumulatorgen-

20 didnotdealwithmultiplesinks.byintroducingtheprimaryandsecondaryclassicationofwave-frontswebelievethatourapproachprovidesgreatercontrolin thesearchforgeneralizations.thisbecomescrucialasthecomplexityxofexamplesincreases.inaddition,weusesinkannotationsexplicitlyinselectingpotential projectionsforhigher-ordermeta-variables. Jane'swork,however,wasmuchbroaderthanoursinthatsheuniedanum- AndrewIrelandandAlanBundy 20 areembeddedwithintheactualunicationalgorithm. berofdierentkindsofgeneralization.moreover,shewasalsoabletosynthesize Kohlhase,1997)whereessentiallythestructurepreservationconstraintsofrippling 1992). tail-recursivefunctionsgivenequivalentnaiverecursivedenitions(heskethetal., Analternativetoourstrategyofannotatedunicationispresentedin(Hutter& tothatofaninductionhypothesiswithinastepcaseproof.asaresult,theripple tures.thisisreectedatthelevelofproofwheretheinvariantplaysarolesimilar belowwhereweoutlinethekeyareaswherewearedevelopingthiswork. Ourresultsfortheextendedcritichavebeenpromising.Webelievethatourtechniqueisnotrestrictedtoreasoningaboutfunctionalprograms.Thisisdiscussed Thereexistsastrongconnectionbetweenloopinvariantsandinductiveconjec- 9FutureWork typicallyseenasthemajoreurekastepintheprocessofverifyinganimperative usingthecriticmechanism(ireland&stark,1997).discoveringaloopinvariantis ants.ourworkonproofcriticsisbeingtransferredacrosstothisnewdomain.in particular,wehavebeguntoinvestigatetheautomaticdiscoveryofloopinvariants techniquecanbeusedtoguidethesearchforaproofwhenverifyingloopinvari- samepatternofsidewaysripplingwhichoccurswithininductiveproofwheresinks areexploited.initialexperimentshavedemonstratedthatourcriticforaccumulator notionofatailinvariant(kaldewaij,1990)representsonesuchwayofderivingan program.acommonstrategyfordiscoveringinvariantsistostartwithadesired generalizationcanalsoplayaroleinthediscoveryoftailinvariants. invariant.usingripplingtoguidethevericationoftailinvariantsgivesrisetothe post-conditionfromwhichtheinvariantisderivedbyaprocessofweakening.the interaction.ashighlightedinx5,thecritic'smechanismmaygenerateapartialgeneralization.aninteractiveversionofthecriticmechanismhasbeenimplementeever,thatthecriticmechanismalsoprovidesabasisfordevelopingeectiveuser (Irelandetal.,1997)whichinvitesausertocompletetheinstantiationofsuch partialgeneralizations. proverwhichwasmorerobustthanconventionalprovers.thehigh-levelrepresentationprovidedbyaproofplanenabledustoachievethisgoal.webelieve,how- Thecriticmechanismwasmotivatedbyadesiretobuildanautomatictheorem xthatis,asthenumberofdenitionsandlemmataavailabletotheproverincreases. Wealsobelievethatourtechniqueisapplicableinthecontextofhardwareveri-

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