by Alexandra Goultiaeva A thesis submitted in conformity with the requirements for the degree of ...

117 Pages·2013·1.26 MB·English

EXPLOITING PROBLEM STRUCTURE IN QBF SOLVING
by
AlexandraGoultiaeva
Athesissubmittedinconformitywiththerequirements
forthedegreeofDoctorofPhilosophy
GraduateDepartmentofComputerScience
UniversityofToronto
c Copyright2014byAlexandraGoultiaeva
(cid:13)
Abstract
ExploitingProblemStructureinQBFSolving
AlexandraGoultiaeva
DoctorofPhilosophy
GraduateDepartmentofComputerScience
UniversityofToronto
2014
Deciding the truth of a Quantified Boolean Formula (QBF) is a canonical PSPACE-complete problem. It
provides a powerful framework for encoding problems that lie in PSPACE. These include many problems in
automaticverification,andproblemswithdiscreteuncertaintyornon-determinism.Twopersonadversarialgames
areanothertypeofproblemthatarenaturallyencodedinQBF.
Itis standardpracticeto useConjunctive NormalForm(CNF) whenrepresentingQBFs. Anypropositional
formula can be efficiently translated to CNF via the addition of new variables, and solvers can be implemented
moreefficientlyduetothestructuralsimplicityofCNF.However,thetranslationtoCNFinvolvesalossofsome
structuralinformation. ThisthesisshowsthatthisstructuralinformationisimportantforefficientQBFsolving,
andshowshowthisstructuralinformationcanbeutilizedtoimprovestate-of-the-artQBFsolving.
First,anon-CNFcircuit-basedsolverispresented. ItmakesuseofinformationnotpresentinCNFtoimprove
itsperformance. Wepresenttechniquesthatallowittoexploitthedualitybetweensolutionsandconflictsthatis
lostwhenworkingwithCNF.Thisdualitycanalsobeutilizedintheproductionofcertificates,allowingbothtrue
andfalseformulastohaveeasy-to-verifycertificatesofthesameform.
Then, it is shown that most modern CNF-based solvers can benefit from the additional information derived
fromdualityusingonlyminormodifications. Furthermore, evenpartialdualityinformationcanbehelpful. We
show that for standard methods for conversion to CNF, some of the required information can be reconstructed
fromtheCNFandgreatlybenefitthesolver.
ii
Contents
1 Introduction 1
2 Background 6
2.1 OverviewofQBF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1.1 ModelsandCountermodels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.2 Resolutionandrefutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 SolvingClausalQBF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.1 BasicAlgorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.2 Unitpropagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.3 Clauselearning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2.4 Cubelearning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 SymbolicapproachestoQBF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4 Non-CNFapproachestosolvingQBF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3 Acircuit-basedQBFsolver 22
3.1 Circuitrepresentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 Thealgorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3 Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.4 Don’tCarePropagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.5 Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.6 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4 Dualpropagationinanon-CNFsolver 32
4.1 Dualpropagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.2 Don’tCarePropagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
iii
4.3 HandlingNNF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.3.1 SimplifyingtheNNF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.4 Phasememorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5 Experimentsonnon-CNFproblems 44
5.1 Earlynon-CNFsolvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.2 Effectofnewtechniquesonthecircuitsolver . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.3 IQTest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.4 GhostQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.5 CNF-basedsolvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
6 Certificategeneration 54
6.1 Q-resolutionfortrueandfalseformulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
6.2 Proofsrepresentstrategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
6.2.1 Updatingtherefutation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
6.2.2 ObtainingwinningmovesfromQ-resolution . . . . . . . . . . . . . . . . . . . . . . . . 63
6.3 Relatedwork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
6.4 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
6.4.1 Strategygenerationexample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
6.4.2 Producingandverifyingcertificates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
7 DualpropagationinCNF-basedsolvers 70
7.1 DualPropagationgivenanon-CNFrepresentation . . . . . . . . . . . . . . . . . . . . . . . . . . 70
7.1.1 SoundnessProperties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
7.1.2 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
7.2 UtilizingpartialinformationextractedfromCNF . . . . . . . . . . . . . . . . . . . . . . . . . . 85
7.2.1 Reconstructingthestructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
7.2.2 Utilizingpartialinformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
7.2.3 Recoveringimplication-basedencodings . . . . . . . . . . . . . . . . . . . . . . . . . . 92
7.2.4 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
7.3 Preprocessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
iv
7.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
8 Conclusion 104
Bibliography 107
v
Chapter 1
Introduction
BooleanSatisfiability(SAT)isthecanonicalNP-completeproblem.Givenabooleanformula,aSATsolverneeds
todeterminewhetherornotthatformulaissatisfiable, i.e., whetherornotthereexistsatruthassignmentunder
whichtheformulaisTRUE.
SATsolvingoffersasimpleanduniformrepresentationthatcanbeusedforawidevarietyofproblems. SAT
solversusewell-developedlightweightandefficientdatastructuresandbenefitfrompowerfulbutsimplelearning
techniques. AlthoughtherearenoalgorithmsforSATsolvingthatareefficientintheworstcase,inpracticethere
areawiderangeofareaswhereSATsolversaresuccessfullyemployedtosolvelargeindustrialproblems.
SATsolverscanbeusedtosolvemanyinterestingproblemsbyencodingtheseproblemsasbooleanformulas,
andtheninterpretingtheanswer. Forexample,onemightencodeastatementsuchas“thisbooleancircuitdoes
notimplementthecorrectfunction.” Then,iftheformulais FALSE,thecircuitcanbeconsideredcorrect,andif
theformulaisTRUE,thesatisfyingassignmentwouldgiveanexamplecasewherethecircuitfails. Thevariables
here are implicitly existentially quantified, i.e., the formula actually encodes the statement “there exists a set of
inputssothatthecircuitgivestheincorrectoutput.” Ingeneral,SATseekstofindasinglesatisfyingassignment
totheformula—itisaskingifthereexistsasettingofthevariablesthatmakestheformulaTRUE.
QuantifiedBooleanFormulas(QBFs)areanextensionofSATthatallowsbothuniversalandexistentialquan-
tification. In particular, a QBF is specified by a sequence of alternating universally and existentially quantified
variablesfollowedbyabooleanformulaoverthesevariables. SATcanthenbeviewedasthespecialcasewhere
allofthevariablesareexistentiallyquantified. TheQBFdecisionproblemistheproblemofdecidingwhetheror
notaQBFisTRUE. ThisproblemisPSPACEcomplete.
WewillpresentQBFsmoreformallyinSection2.1, butsemanticallyaQBFcanbeviewedasatwo-player
game. The two players, one controlling the existentially quantified variables, and the other controlling the uni-
1
CHAPTER1. INTRODUCTION 2
versally quantified variables, take turns setting their variables. The sequence of quantified variables, i.e., the
quantifier, dictatestheorderinwhichthemovesareplayed. Ateachpoint, theowneroftheoutermostvariable
assigns it a value, and the formula is simplified accordingly. The goal of the existential player is to make the
formulaTRUE,whilethegoaloftheuniversalplayeristomaketheformulaFALSE. Theexistentialplayerhasa
winning strategy if it can force the formula to become TRUE no matter how universal sets its variables (in this
case,wesaythattheQBFformulaisTRUE). Similarlytheuniversalplayerhasawinningstrategyifitcanforce
the formula to become FALSE no matter how existential sets its variables (in this case, we say that the QBF
formulaisFALSE).
ManyproblemshaveanaturalrepresentationinQBF.Forexample,QBFnaturallyrepresentsfinitestatetwo-
player games, where the existential and the universal variables represent the possible moves, and the formula
representsthegamemechanicsandthewinningcondition. SupposethetwoplayersareP andP ,andthegame
1 2
isthreemoveslong. TocheckwhetherP hasawinningstrategy,wemightencodetheformulaasfollows: “there
1
existsamoveM forP ,suchthatM isavalidmove,andforallmovesM byP ,eitherM isinvalid,orthere
1 1 1 2 2 2
existsavalidmoveM forP suchthatP winsthegame.”
3 1 1
QBFisnotlimitedtosimpletwo-playergames,however. Anybehaviorthatcanbecastasadversarialcanbe
encoded. Anexampleofthisistheencodingofconformantandcontingentplanningproblems.
Conformant and contingent planning involve planning problems that allow incomplete information and/or
nondeterminism. Theplannerneedstocomeupwithaplanthatwouldachievethegoalnomatterhowthenon-
determinism plays out. Since the plan must always work, we could view the world as an adversary, each valid
move of which the plan must be able to handle. We would let the adversary determine the unknown and non-
deterministicevents,andtrytocreateastrategythatworksnomatterwhicheventsoccur. Inotherwords,wecan
viewthesethesetypesofplanningasatwo-playergameandcastthemasQBFs[53].
AnotherareawhereQBFcanbeusedisinsolvingvariousproblemsthatariseincircuitverification,design,
and debugging. One example is configuring Field Programmable Gate Arrays (FPGAs). Field Programmable
GateArrays(FPGAs)areprogrammablecircuits. Thequestionofwhetheracertainfunctioncanbeprogrammed
byaparticularFPGAcanbecastasaQBF:doesthereexistasettingoftheFPGAcontrollinessuchthatforall
inputstheFPGAoutputsthecorrectfunctionvalue[40].
Recently, QBF solvers have been applied to an interesting problem of automated reduction finding [36]. In
short,theformulaencodesastatement“thereexistsareductionrfromproblemAtoproblemBsuchthatforall
inputsx,x Aiffr(x) B.” Althoughthisapplicationislimitedtoboundedinputlengthreductionsinacertain
∈ ∈
class,itcanstillprovideusefultoolsforcomplexitytheoreticreasoning.
PSPACE is widely believed to contain harder problems than NP, and thus QBF being PSPACE complete is
likelytohardertosolvethanSAT.InpracticeQBFhasshownitselftobeharderthanSAT.Ontheflipside,QBF
CHAPTER1. INTRODUCTION 3
is more expressive, and can succinctly represent some problems whose natural SAT encoding is exponentially
sized. Inpractice,QBFhasbeenappliedtoproblemsthatgrowtoolargeforaSATsolvertohandle. Forexample,
some verification problems traditionally solved by SAT solvers contain structural repetition, and can be more
efficientlyrepresentedasaQBF.ThisefficiencyinrepresentationcanallowQBF-basedapproachestooutperform
SAT-basedapproacheswhentheproblemsbecomelarge[45].
Despiteitsworst-casecomplexity,QBFsolversarebeingactivelydevelopedandaregainingefficiency. This
thesisisaimedatimprovingQBFsolvingtechniquesbyalleviatingalong-standingprobleminQBF:theinability
of QBF solvers to reason equally efficiently for both the existential and the universal players. The roots of this
problemstemfromtheformularepresentationthatisbeingused.
Conjunctive Normal Form (CNF) is a representation for boolean formulas which has become ubiquitous in
SAT solving. Any propositional formula can be efficiently translated to CNF via the addition of new variables,
and solvers can be implemented more efficiently due to the structural simplicity of CNF. Many effective data
structures and techniques have been developed to work with CNF. Clause learning, which is probably the most
importantbreakthroughisSATsolvingtechnology,isaneffectivereasoningtechniquespecifictoCNF.
QBFsolving,havinginheritedmanytechniquesfromSAT,hasalsoadoptedCNF.Themostwidespreadformat
forQBFsolvingisPrenexCNF,whereaseriesofquantifiersisfollowedbyabooleanformulaexpressedinCNF.
QBF solvers are able to benefit from the well-engineered data structures and techniques aimed at working with
CNF.However,theCNFrepresentationposessomeproblemsforQBFsolving.
CNFisextremelywell-suitedtoreasonaboutconflicts(variableassignmentsthatfalsifytheformula), how-
ever, itisquiteinefficientwhenitcomestoreasoningaboutsolutions(variableassignmentsthatsatisfythefor-
mula). Prenex CNFallowstheQBFsolverto reasonefficientlyfortheexistentialplayer(which needstoavoid
conflicts),butpreventseffectivereasoningfortheuniversalplayer(whichneedstoavoidsolutions). Thisisnota
problemforSATsolvers,since,asmentionedpreviously,aSATformulaisaQBFwithonlyexistentialquantifica-
tion. So,aSATsolverneverneedstoreasonfortheuniversalplayer. Itonlyencountersasatisfiedformulaonce,
whenthesearchisover—soefficiencyindetectingandreasoningaboutwhatsatisfiestheformulaisnotneeded.
Ontheotherhand,aSATsolverneedstoanalyzemanyconflicts,soitneedstoreasonaboutthoseefficiently.
TheineffectivenessofCNFforreasoningwithsolutionsslowsdownQBFsolverswhichoftenmustprocess
anexponentialnumberofsolutions. InadditiontoincreasingthesolvingtimethebuiltinbiasofCNFtowards
conflictscausesotherproblems. Onehastodowithcertificatesandstrategies.
Acertificateisanadditionalpieceofinformationthatthesolvermightprovidetogetherwithitsresult. The
certificateallowstheusertoverifythattheprovidedresultiscorrect. ThereisaCNF-basedproofsystemcalled
resolution which can be efficiently verified and can certify that the formula is false for any variable setting (is
UNSAT).So,aresolutionproofiscommonlyusedasacertificateofunsatisfiability. Toverifythataformulais
CHAPTER1. INTRODUCTION 4
satisfiable,itisenoughtosupplythesatisfyingassignmentandverifythatitsatisfiestheformula.Sothesatisfying
assignmentisaneffectivecertificateofsatisfiability.
The resolution proof system has been extended to QBF, forming a proof system called Q-resolution. A Q-
resolution proof is an easily verifiable certificate for QBFs which are FALSE. However, because CNF is biased
towards conflicts, there is no straightforward way to extend Q-resolution to provide certificates for true QBFs.
Instead, the attempts to certify true QBFs have typically tried to mimic the SAT case by focusing on obtaining
strategies.
Recall that in SAT solving, in addition to providing an easy way to verify that the formula is satisfiable,
a satisfying assignment also carries useful information. For example, it might encode an actual plan, a failure
scenario,orthesolutiontoapuzzle. InQBFsolvingonemightalsobeinterestednotonlyintheanswer(which
playercanbeguaranteedtowinthegame),butalsoinhowthevictorycanbeobtained.However,inQBFasimple
variableassignmentisnotsufficient,sincetheanswermustcaptureresponsestodifferentmovesbytheopponent.
Whatisdesiredisastrategy. Inaddition,astrategycanpotentiallyverifythataQBFistrue. Differentmethods
ofrecordingstrategiesanddifferentwaysofencodingthemhavebeendeveloped. Unfortunately,thesemethods
areusuallyeitherpronetocombinatorialexplosions,arecomputationallyhardtoverify,orboth.
Thus, the bias in CNF has also affected certificates: while there was a uniform and easy-to-verify method
for certifying FALSE QBF formulas, there was no such method for TRUE formulas. Conversely, there have
been methods to generate strategies for TRUE QBF formulas (winning strategies for the existential player), but
nocounterpartfor FALSE QBFs(winningstrategiesfortheuniversal). Inthisthesis, weshowhowbyutilizing
the non-CNF structure it is easy to obtain Q-resolution proofs for TRUE formulas. Moreover, we show that Q-
resolution proofs encode all the necessary information about the winning strategy, and thus can themselves be
viewed as a strategy representation. We present an algorithm which, given a Q-resolution proof, interactively
extractsastrategyforthewinningplayer.
So,bybeingabletoconstructQ-resolutionproofsforbothTRUE andFALSE QBFformulas,andbeingableto
extractstrategiesfromtheseproofs,weareabletohaveuniformandeasy-to-verifycertificatesforbothpolarities,
aswellascomputestrategiesforboththeexistentialandtheuniversalplayers.
InChapter2wewillpresentsomeoftheformalbackground. Then,inChapter3wepresentanon-CNFQBF
solverusedasthebasisforourapproach,whichtakesthefirststepstowardsusingnon-CNFinformationforbetter
reasoningfortheuniversalplayer. InsteadofCNF,itusesalogicalcircuitastheinternalrepresentationforthe
formula. InChapter4wepresentthecoreofourapproach. Weshowhowthecircuit-basedQBFsolvercanbe
modifiedtoachievecompletedualitybetweentheexistentialandtheuniversalplayerandbeabletoreasonequally
efficientlyforboth. Wealsoincludeafewsmallerimprovementsandsimplificationtechniquesthatallowusto
further speed up the solver. In Chapter 5 we present experimental results. We show the merit of our approach
CHAPTER1. INTRODUCTION 5
by demonstrating how the addition of dual propagation improves our circuit-based solver. We also compare
ourapproachwithothernon-CNFapproaches,andwithCNF-basedapproachestowhichwefeedtheconverted
versionsoftheformulas.
InChapter6wepresentourapproachforcertificategeneration. Weshowhowwecanobtaincertificatesand
strategiesforboth TRUE and FALSE formulas. Weexperimentallydemonstratethefeasibilityofourapproach,
showingthattheoverheadofproducingthecertificateisextremelysmall,andthatourcertificatescanbeverified
bythird-partyverifiers.
InChapter7wereconcileourapproachwithexistingCNF-basedsolvers. Indeed,byusinganon-CNFrepre-
sentation,oursolvermissesoutontheefficientCNF-baseddatastructuresandtechniquesthathavebeendevel-
oped. We show that the most common CNF-based solvers have indeed all the necessary mechanisms for using
dual propagation. We show how these mechanisms can be correctly initialized to obtain the full power of dual
propagation while retaining the benefits of CNF-specific techniques and data structures. We also include some
theoreticalresultsdescribingtheconditionsthatmakeinitializationsound. WeexperimentallyshowthattheCNF
techniques do offer advantages, and that the resulting solver is more efficient than the non-CNF versions. We
then take this approach a step further, and show how dual propagation can be used in cases when the non-CNF
representation is unavailable. Instead of using the complete representation of the problem, we can still benefit
from partial structural information that can be obtained by various heuristic methods from the CNF. Of course,
theweakerthereconstruction,thelessusefuldualpropagationwillbe. However,ourapproachisabletobenefit
frompartialinformationwhilecausingnooverheadifthereconstructionwaspoor. Weshowthefeasibilityofthis
approach.
InChapter8weconcludethethesisbysummarizingourcontributionsanddiscussingfuturework.