A Tale Of Two Solvers Eager and Lazy Approaches to Bitvectors Liana Hadarean Kshitij Bansal Dejan Jovanovi Clark Barrett and Cesare Tinelli New York University The University of Iowa SRI Internatio - Pdf
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A Tale Of Two Solvers Eager and Lazy Approaches to Bitvectors Liana Hadarean Kshitij Bansal Dejan Jovanovi Clark Barrett and Cesare Tinelli New York University The University of Iowa SRI Internatio

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The standard method for deciding bitvector constraints is via eager reduction to propositional logic This is usually done after 64257rst applying powerful rewrite techniques While often ef64257cient in practice this method does not scale on problems

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A Tale Of Two Solvers Eager and Lazy Approaches to Bitvectors Liana Hadarean Kshitij Bansal Dejan Jovanovi Clark Barrett and Cesare Tinelli New York University The University of Iowa SRI Internatio






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2Alazysolvercanaddresstheselimitations,explicitlytargetingproblemsthataredifcultforeagersolversandthusprovidingacomplementaryapproach.Thelazyap-proachforbit-vectorswasrstproposedin[8,16].Inthispaper,werevisitthisap-proach,extendingandimprovingitinseveralways.Ourlazysolverintegratesalge-braic,word-levelreasoningwithbit-blasting.Designedforeasyplug-and-playcom-binationwithsolversforothertheories,theprocedureintegratesanon-linelazyTbvsolver(LBV)intotheDPLL(T)framework[20],separatingtheory-specicreasoningfromthesearchovertheBooleanstructureoftheinputproblem.Thisseparationoffersbenetsorthogonaltothoseprovidedbyeagerbit-vectorsolversbutalsoposesinter-estingtrade-offs.Ononehand,ithasthepotentialofincurringadditionaloverheadandlosingimportantconnectionsbetweensubproblems;ontheotherhand,dependingontheBooleanstructureoftheproblem,itoftenallowstheTbvsolvertoreasonaboutmuchsmallerproblemsatatime.Weuseaspecializeddecisionheuristictoreducethesizeofthesesub-problemsevenfurtherbyconsideringonlyliteralsrelevanttothecurrentsearchcontext.Ourapproachisparticularlyusefulonproblemswhosesubproblemsfallintooneoftheefcientlydecidablefragmentofthebit-vectortheory(e.g.,thecoretheoryofconcatenationandextraction[11],thetheoryofbit-vectorinequalities,orfragmentsdecidableusingequationalreasoning).Totargetsuchproblems,ourLBVsolverisbuiltasthecombinationofseveralalgebraicsolversspecializedforsomeofthesefragmentstogetherwithacompletebit-blastingsolver.Thebit-blastingsolverusesadedicatedSATsolverSATbb,distinctfromtheDPLL(T)Booleanenginedrivingthemainsearch(SATmain).TheseparationofthetwoSATenginestscleanlyintotheDPLL(T)frame-workandallowsthesolverstobetunedindependently.Experiments(describedinSection6)conrmourclaimthatthelazyapproachiscomplementarytotheeagerapproach,asthelazysolverefcientlysolvesproblemsthatareeitherimpossibleorverydifcultforeagersolvers.Atthesametime,itisnotrealistictoexpectthelazysolvertodowellonproblemsthatareeasyforeagersolvers(andindeeditisoftenslowerontheseproblems).Forthisreasonweproposeaportfolioapproachthatrunsaneagersolverandalazysolverinparallel.Additionalexperimentsshowthatourportfoliosolveroutperformseagersolversbothintermsofthenumberofproblemssolvedandthetimetakentosolvethem.Therestofthepaperisorganizedasfollows.Section2framesourcontributionsintermsofrelatedwork.Sections3and4providetechnicalpreliminariesandabriefoverviewoftheDPLL(T)framework.Section5describesthecomponentsofourlazysolverLBVincludingsomeoptimizationsenabledbythelazyframework.Wepresentanexperimentalevaluationofthesolverfollowedbyanin-depthanalysisinSection6.Finally,weconcludewithfutureworkinSection7.2RelatedworkThepredominantapproachtosolvingbit-vectorconstraintsisviareductiontoSAT.Boolector,aspecializedsolverforbit-vectorsandarrays,andthewinnerofthe2012SMT-COMPforQF BVlogic,employspreprocessingbeforeencodingthebit-vectorformulaintotheAIGformat[7].Z3,aDPLL(T)-styleSMTsolver,appliesbit-blasting 4Table1:Tbvsignaturebv eqsorts[n]n�0constants0;1::[1]equal = ::[n][n]... conconcat  ::[m][n]![m+n]extract [i:j]::[m]![i�j+1] ineqless ::[n][n]less-eq  ::[n][n] ariplus + ::[n][n]![n]neg� ::[n]![n]times  ::[n][n]![n]div = ::[n][n]![n]rem % ::[n][n]![n] booland & ::[n][n]![n]or j ::[n][n]![n]not ::[n]![n]xor  ::[n][n]![n] shiftleftshift ::[n][n]![n]rightshift �� ::[n][n]![n]Wewillwritet[n]forsomexedntodenotethattisabv-termofsort[n].Notethatexceptfortheconstants,thefunctionandpredicatesymbolsinTable1areoverloaded;forexample,+standsforanyofthesymbolsintheinnitefamilyf+::[n];[n]![n]g�n0.Forsimplicity,werestrictourattentiontoasubsetofthebit-vectoroperatorsdescribedintheSMT-LIBv2.0standard[4];themissingonescaneasilybeexpressedintermsofthosegivenhere.TheTbv-satisabilityofconjunctionsofequalitiesbetweentermsoverthecoresub-signatureeq[conisdecidableinpolynomialtime[9,11].However,addingalmostanyoftheadditionaloperators,orallowingforarbitraryBooleanstructure,makestheTbv-satisabilityproblemNP-hard[6].4TheDPLL(T)FrameworkState-of-the-artSMTsolversefcientlydecidethesatisabilityofquantier-freerst-orderformulaswithrespecttoabackgroundtheoryTbyusingtheDPLL(T)frame-work[20].TheframeworkextendstheDavis-Putnam-Logemann-Loveland(DPLL)de-cisionprocedureforSATtohandlereasoninginatheoryTbyrelyingonatheorysolver(T-solver):adecisionprocedurefortheT-satisabilityofT-constraints.Algorithm1givesasimpliedalgorithmicviewoftheDPLL(T)frameworkwithageneralizedtheoryinterface.ThealgorithmtakesasinputaT-formula andreturnssatif isT-satisableandunsatotherwise.VariableCstoresthesetofworkingclausesandAthecurrenttruthassignmentforCasasequenceofT-literals.Weuse[]fortheemptyassignmentand;fortheconcatenationoftwoassignments.Initially,AisemptyandCissimplythesetofclausesobtainedbyconverting toConjunctiveNormalForm(CNF).WesaythatapairhA;CiisinconsistentiftheassignmentAfalsiessomeclauseinC;itisconsistentotherwise.AnassignmentApropositionallysatises if issatisedbyeveryfullassignmentextendingA.InAlgorithm1,theSATandtheorysolverworktogethertoaugmentAandCviaSatSolveandTheoryCheck,respectively.TheinputtoSatSolveisanassignmentandasetofclauseshA;Ci.ThereturnvalueisanewpairhA0;C0iderivedfromthe 6WesayacalltoTheoryCheckisnalwhentheparameternalissettotrue.FinalcallstoTheoryCheckmusteitherensurethatAisT-satisable,orreturnoneormoretheorylemmas.Twoimportantaspectsoftheorysolversarenotcapturedhere.TherstisthatactualimplementationsofTheoryCheckarestateful:theystoreacopyoftheassignmentAinternallyandareinstructedtopushandpopliteralsfromitasAismodiedbythemainloop.Inpractice,itiscrucialthatthetheorysolverbeabletobacktrackefcientlywhenAisshrunk,andreasonincrementallywhenitisextended.Thesecondaspectisthatatheorysolvermustbeabletoprovideanexplanationforeachtheory-propagatedliteralp.Thisisaclauseoftheform:l1__:ln_lforsomesubsetfl1;:::;lngofA,explainingwhytheliteralwasentailed.ExplanationsareneededbySatSolveduringitsconictanalysis.Itisimportantforefciencythatthetheorysolverbeabletocomputeexplanationslazily,onlyasneededbySatSolve.5ALazyBit-vectorSolverWenowproceedtogivethedetailsofourlazybit-vectorsolverLBV,designedtofullltherequirementsoftheTheoryCheckinterfacedescribedabove.5.1SubsolversTheLBVsolverconsistsoffoursub-solvers:theequalitysolverLBVeq,thecoresolverLBVcore,theinequalitysolverLBVineqandthebit-blastingsolverLBVbb.Eachsub-solverisincrementalandprovidesthetheorysolverfunctionalitiesdescribedinSec-tion4.ThearchitectureofLBVwasdesignedtobemodularandextensible:allthebit-vectorreasoningisconnedwithinthesolver,anditiseasytoenhanceitbyaddingmoresub-solvers. Algorithm2:LBVCheck Input:hA,nalihPeq;Leq;completei LBVCheckeq(A,nal);ifcompletethen returnhPeq;Leqi; hPineq;Lineq;completei LBVCheckineq(A;Peq,nal);ifcompletethen returnhPeq;Pineq;Leq[Lineqi; hPbb;Lbbi LBVCheckbb(A;Peq;Pineq,nal);returnhPeq;Pineq;Pbb;Leq[Lineq[Lbbi Algorithm2showstheimplementationofLBVCheck,theTheoryCheckfromAl-gorithm1correspondingtotheLBVsolver.LBVCheckcallsthesubsolversinincreasingorderofcomputationalcost.Foreachi2feq;ineq;bbg,LBVCheckireturnsasequence 10 Algorithm3:LBVCheckbb Input:hA,nalihP;Li BvSatBCP(A);ifnalandL=;then L BvSatSolve(A); returnhP;Li; 5.2LazyTechniquesThelazyDPLL(T)frameworkenablesseveraltechniquesthataredifcultorimpossibletousewitheagersolvers.Inthissectionwediscusstwoofthesetechniques:applyingword-levelrewritesduringsolving(inprocessing)andreducingtheproblemsizebyonlyreasoningaboutatomsrelevantinthecurrentsearchcontext(relevancy-baseddecisionheuristics).InprocessingTechniquesBeforeengaginginpotentiallyexpensiveSATreasoning,LBVbbreliesontheinprocessingmoduletocheckiftheproblemcanbesolvedorsig-nicantlysimpliedbyword-levelsimplicationtechniques.Thisisdonebyaprocess,describedinAlgorithm4,thathastheavorofGaussianelimination.Itworksbyiter-atingoveraworklistoftheoryliteralsWwhilemaintainingasubstitutionmap.Initially,WisinitializedtothesetofliteralsAassignedtotrueinthecurrentsearchcontext.Foreachworklistassertionw2W,werstapplythesubstitutionmap,andthenrewriteitusingword-levelsimplicationtechniques(Simplify).TheSolveEqprocedurethenattemptstosolvetheupdatedassertionwtoobtainanewsubstitution.Alternatively,itcanalsolearnnewequalitiesentailedbywandaddthesetothework-inglist.8TheworkinglistWandthesubstitutionmapareupdatedwiththisnewinformation,andtheprocessisrepeatedtoaxpoint.9IfanyoftheassertionsinWreducestofalse,wehaveaconict.IftherearenosuchobviousinconsistencieswecanruntheLBVCheckbbroutineonthesimpliedsetofassertionsW.Wedothisheuristically,iftheproblemhasbeenreducedenoughintermsofthecircuitsize.Wefoundcheckingthesimpliedassertionswhentheyarelessthan50%ofthesizeoftheoriginalassertionstobeagoodheuristic.Relevancy-AwareDecisionHeuristicsTheideaofrelevancyisbestunderstoodwithasimpleexample.Let =:a^(b_')withassignmentA=[:a;b].NotethatApropositionallysatises regardlessofhowmanyunassignedliteralsarein'.Theliteralsin'areirrelevant.TheDPLL(T)frameworkmakesiteasytoaddadecisionheuristicthatavoidssplit-tingonirrelevantliterals.Inparticular,wecan(i)detectwhenanassignmentAbe-comespropositionallysatisfyingandstopearlyinordertoreducethenumberofliterals 8Inourimplementation,wesolvexorequationsandsliceequationsbetweenconcatenationexpressionstogetnewequalities.9Thedata-structuresareenhancedwithextrabook-keepinginformationtokeeptrackofexpla-nations.Weomitthesedetailsforsimplicity. 12 (a)cvcLzvscvcLz-J (b)cvcLzvscvcLz-P (c)cvcLzvscvcLz-AlgFig.2:Impactofvariousfeaturesofthelazysolver.Allplotsareonalogarithmicscale.SMT-LIBv2.0.Instead,weselected3786ofthembyfocusingonexamplescomingfromvericationapplications:weexcludedtheanswer-setprogrammingaspfamilyaswellasthecheck2andcraftedfamiliesthatcontaintoyexamples.Topreventverylargefamiliessuchassage(26K)andspear(1694)fromdominatingtheresults,weusedarandomizedprocesstoselectarepresentativefractionofthebenchmarksfromthem.Becausemanyofthesageproblemsareveryeasy,weconsideredonlybenchmarksthattakemorethan10secondstosolve.Fromthespearfamilyweincludedallsmallsub-families,andrandomlyselectedafractionofthelargestsubfamily.Forbrevity,wemergeherethefourfamilieswithabrummayerbiereprexintobrummayerbiere*,uclidanduclid-contrib-smtcomp09intouclid*,andstpandstp-samplesintostp*.WeusecvcEtorefertotheimplementationoftheeagersolverinCVC4,cvcLzforthelazyLBVsolverandcvcPllfortheparallelsolver.TheletterspreceededbyaminussignrepresentwhichfeatureofcvcLzhasbeenturnedoff:Jforthejusticationheuristic,PforLBVbbpropagation,Algforallofthealgebraicsub-solvers(LBVeq,LBVcore,LBVineq)plustheword-levelin-processingtechniques.ThescatterplotsinFigure2comparetheruntimeperformanceofthefullfeaturedlazysolverwithaversionwithoutoneofthefeaturesabove.Figure2ashowstheimpactofthejusticationheuristic.Whileoverallthejusticationheuristicimprovesperfor-mance,ithasanegativeimpactonbenchmarksinthemcmfamily.Theseproblemsconsistofconjunctionsoflargedisjunctions.Onsuchproblemsthejusticationheuris-ticforcesSATmaintochooseanaivepatternofdecisionsbyalwaysinitiallydecidingontherstdisjunctofeachconjunct.Figure2bshowsthatLBVbbpropagationisessen-tialtosolvingdifcultbenchmarks,althoughitaddssomeoverheadtotheeasierones.Figure2cshowstheimpactofalltheword-leveltechniquesenabledbythelazyap-proach.Theplotshowsarelativelysmalloverheadwhenthesetechniquesdonothelp,butdramaticimprovementswhentheydoapply.Table2comparestheperformanceofcvcE,cvcLzandthatoftheonlyotherbit-vectorsolverthatsupportslazybit-blasting:mathsatL(smtcomp2012versionwithlazysolvingenabled).TheeagersolvercvcEperformsbetteronfamiliesthatinvolvebit-levelmanipulations,suchasthebrummayerebiere*families.ThelazysolvercvcLzex- 14celsonfamiliescalypto,tacas07,lfsr,coreandsimple processorsthatbenetfromal-gebraicreasoning.Furthermore,cvcLzsolves6problemsthatnoneoftheothersolversweconsideredcouldsolveinthegiventimelimit.Theunique-solverowatthebottomofTable2andTable3showsthisgureforallothersolvers.Finally,inTable3wecomparecvcPllwithotherstate-of-the-artbit-vectorsolvers:yices(2.1.1),stp2(r1673),z3(r0e74362),boolector(1.6),sonolar(smtcomp2012)andmathsat(smtcomp2012witheagersolver).FortheparallelsolvercvcPllwereportwallclocktime.TheportfoliosolvercvcPllsolvesthelargestnumberofproblems.Weat-tributethisincreaseinperformancetothecomplementarynatureofthetwoapproaches.ToillustratethatthelazycvcLzapproachcomplementseagersolvers,wealsosimulatedrunningcvcLzinparallelwithtwoofthemostefcenteagerbit-vectorsolvers:boolec-torandz3.Wedidthisbychosingthebestresultfromeithersolverforeachproblem.Evenforthesesolvers,cvcLzgreatlyimprovesontheirperformance:thecombinedboolector+cvc4Lsolves57moreproblemsinaquarteroftheoriginalboolectortotaltimeandz3+cvcLsolves42moreproblemsinjustoverhalfthetotaltime.DiscussionWenowprovideamoredetailedanalysisofthetradeoffsbetweenthetwoapproaches,basedonourexperimentalresults.TheeagersolvercvcEisparticularlyefcientonhardwareequivalencecheckingbenchmarksthatverifytheequivalenceofabit-levelimplementationtoitsword-levelspecication.Insuchcasesthecorrectnessoftheproofoftendependsonbit-levelprop-ertiesthatbenetfromefcientpropositionalanalysismorethanthekindofalgebraicreasoningdoneinthelazysolver.Thisisespeciallyobviousinthedifferenceintheper-formanceofcvcEandcvcLzonthebrummayerbiere*family,ascanbeseeninTable2.Maintainingtheword-levelstructureduringthecomputationinLBVrequireses-tablishingacommonlanguagebetweenSATmain,theSATsolverdrivingthemainDPLL(T)search,andSATbb.Inourapproach,thislanguageconsistsoftheTbv-atomsandrepresentsafrontierthatpartitionstheproblembetweenthetwosolvers.LBVcon-ictscanbeseenasinterpolantsbetweenthepartoftheproblemdescribingthecontrolow(theBooleanabstraction)andthedatapath.RestrictingtheconictlanguagetoTbv-atomslimitsthegranularityoftheconicts:wecannotexpressbit-levelconicts.Insomecasesthiscanproveinefcient.Considerthefollowingexample.Example2.Thefollowingassertionsareunsatisable.Allpathsthroughthedisjunctionforcethelastbitofthexivariablestobe0[1].Thereforetheirdisjunctionmustalsohavetheleastsignicantbitequalto0[i]whichmakestheequalityfalse.n_i=0xi=y1[1]^n^i=0(xi=ti0[1]_xi=si0[1])InExample2,aneagersolvermaypotentiallylearnthatthelastbitofxihastobe0.Thelazysolverontheotherhand,willhavetotryallpossiblepathsthroughthedisjunctionandlearnaconictforeachoneofthem.Forproblemswithexpensivearithmeticoperators,thebenetsofmaintainingtheword-levelstructureoutweighthislimitation.Whileeagersolvershavesophisticated 15rewritetechniques,suchtechniquesareusuallyonlyapplicableatthetoplevel.Equiva-lencecheckingproblemsbetweenhigherleveldesignscanrequireprovingtheequiva-lenceofresultsobtainedbytakingdifferentcontrol-owpaths.Thesecanbeencodedaslargeite(if-then-else)termtreeswithasimilarstructure,asinthefollowingexample.Example3.Theformulabelowisunsatisable.Theconditionsonallpathsthroughtheitetreesforcetheleavestobeequal.ite(x0=y0;x0(ite(x1=y1;2x1;2));2)6=2ite(x0=y0;y0(ite(x1=y1;y1;1));1)Collectingtheassertionsdownanyitepathintheexample,andapplyingsimpleequalitysubstitutionsrenderseachsuchpathtriviallyunsatisable.Nomultiplicationreasoningisrequired.However,bitblastingthisexpressionresultsinadifcultSATproblemasthelargecircuitsrequiredtomodeltheproductsobscurethetrivialincon-sistency.Thecalypto,lfsrandsimple processors(Table2)exhibitthistypeofstruc-ture.Onthesefamilies,ourLBVin-processingmodulecanoftensimplifyeachcalltoTheoryChecktofalseorasignicantlysimplercircuit.Othervericationproblems,suchascheckingthecorrectnessofsortingalgorithms,relyonthearithmeticpropertiesofatotalorder.Theequality,coreandinequalitysubsolverscandecidesuchproblems,oftenwithoutanybit-levelreasoningatall.7FutureWorkForfuturework,weplantobothimprovetheperformanceofthelazysolverandin-vestigateheuristicsforautomaticallyselectingbetweentheeagerandlazysolvers.InSection6wegavesomeintuitionforwhichofthetwoapproachesisbestsuitedforwhichproblemstructure.Itwouldbeinterestingtoseeifitispossibletostaticallydeterminewhichsolverislikelytoperformbetter.Thelazysolvercanbeimprovedbyaddingmoresub-theorysolvers,suchasasub-solvercompleteforsomefragmentofmodulararithmetic.Theinprocessingmodulecurrentlyonlyhandlesequalityreasoning,xorsolvingandslicing.Althoughitisal-readyremarkablyefcient,theSolveEqroutinecouldbegeneralizedtoothertypesofequationsolving.Anotherwaytoimprovetheperformanceofthelazysolveristominimizethecon-ictsobtainedfromthebit-blastingsubsolver.Theconictsreturnedbythatsubsolverwithassumptionsinfrastructurearenotguaranteedtobeminimal.Indeed,inourexpe-riencetheyareoftennon-minimal,insomecaseslargerthanminimalonesbyafactorof10.ThechallengehereistominimizetheconictinanefcientlysincesatisabilityqueriesinTbvarepotentiallyveryexpensive.Onewaytoexpandthescopeofthelazybit-vectorsolver,andovercomesomeofitslimitation,wouldbetoincreasethekindofconictsitcanreturn.Currently,thesolvercanonlyreturnconictsintermsofbit-vectoratoms.Itwouldbeinterestingtoexperimentwithexpandingthisvocabularydynamically,byaddingconictsthatrefertoindividualbitsoftheterms.Thiscouldpotentiallybesupportedbyusingthesplittingondemandframework[3].

A Tale Of Two Solvers Eager and Lazy Approaches to Bitvectors Liana Hadarean Kshitij Bansal Dejan Jovanovi Clark Barrett and Cesare Tinelli New York University The University of Iowa SRI Internatio - Description

The standard method for deciding bitvector constraints is via eager reduction to propositional logic This is usually done after 64257rst applying powerful rewrite techniques While often ef64257cient in practice this method does not scale on problems ID: 7376 Download Pdf

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