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C L , R S ,
LAUSE EARNING ESOLUTION PACE
& P
EBBLING
by
Philipp Hertel
AThesisSubmittedinConformityWiththeRequirements
FortheDegreeofDoctorofPhilosophy
GraduateDepartmentofComputerScience
UniversityofToronto
Copyright(cid:13)c 2008byPhilippHertel
Abstract
ClauseLearning,ResolutionSpace,&Pebbling
PhilippHertel
DoctorofPhilosophy
GraduateDepartmentofComputerScience
UniversityofToronto
2008
Currently,themosteffectivecompleteSATsolversarebasedontheDPLLalgorithmaugmentedbyClause
Learning. These solvers can handle many real-world problems from application areas like verification,
diagnosis,planning,anddesign. ClauseLearningworksbystoringpreviouslycomputed,intermediatere-
sultsandthenreusingthemtoprunethefuturesearchtree. WithoutClauseLearning,however,DPLLloses
mostofitseffectivenessonrealworldproblems. Recentlytherehasbeensomeworkonobtainingadeeper
understanding of the technique of Clause Learning. In this thesis, we contribute to the understanding of
ClauseLearning,andtheResolutionproofsystemthatunderliesit,inanumberofways.
WecharacterizeClauseLearningasanew,intuitiveResolutionrefinementwhichwecallCL. Wethen
showthatCLproofscaneffectivelyp-simulategeneralResolution. Furthermore,thisresultholdsevenfor
themorerestrictiveclassof greedy, unitpropagating CLproofs, whichmoreaccuratelycharacterizeClause
Learningasitisusedinpractice.ThisresultissurprisingandindicatesthatClauseLearningissignificantly
morepowerfulthanwaspreviouslyknown.
Since Clause Learning makes use of previously derived clauses, it motivates the study of Resolution
space. WecontributetothisareaofstudybyprovingthatdeterminingthevariablespaceofaResolution
derivationisPSPACE-complete. Thereductionalsoyieldsasurprisingexponentialsize/spacetrade-offfor
Resolutioninwhichanincreaseofjust3unitsofvariablespaceresultsinanexponentialdecreaseinproof-
size. ThisresultrunscountertotheintuitionsofmanyintheSAT-solvingcommunitywhohavegenerally
believedthatproof-sizeshoulddecreasesmoothlyasavailablespaceincreases.
In order to prove these Resolution results, we need to make use of some intuition regarding the rela-
tionshipbetweenBlack-WhitePebblingandResolution. Infact,determiningthecomplexityofResolution
variable space required us to first prove that Black-White Pebbling is PSPACE-complete. The complex-
ityoftheBlack-WhitePebblingGamehasremainedanopenproblemfor30yearsandresistednumerous
attemptstosolveit. Itssolutionistheprimarycontributionofthisthesis.
ii
Acknowledgements
To start with, I would like to thank my supervisor Toni Pitassi. I have learned far more from Toni than I
probablyknowhowtoarticulate. TonitaughtmeagreatdealjustbybeingToni. Byfollowingtheexample
ofherdedicationtoherworkandherfamily, Ihavebecomeamuchbetterresearcherandamoremature
person in general. Her fun and free-spirited nature have also made working with her an incredibly cool
andenjoyableexperience.
IwasveryluckytohaveanincrediblePhDcommitteeandIwouldliketothankitsmembers,starting
withAlasdairUrquhart. Despitebeingnotallerthanme, Alasdairisanintellectualgiantamongusmere
mortals. Alasdair knows everything about everything and it has been an honour and a treat learning
fromhim. InmanywaysFahiemBacchus’spresenceonmycommitteewasveryreassuringsincehewas
always calm and collected even in the face of the mind-numbing complexity of clause learning. Fahiem
isalsoagreatwriterandIhavelearnedalotaboutbeingpersuasivefromhim. ProfessorCookisatruly
inspirationalfigurewhohasmadesuchfundamentalcontributionstocomputersciencethatworkingwith
him makes one believe that the hardest problems might just be solvable. I would also like to thank my
externalexaminerNickPippenger. ReadingNick’ssurveypaperonPebblinghelpedmyresearchagreat
dealanditwasarealpleasuremeetinghim. ThoughCharlieRackoffwasonlyaddedtomycommitteefor
myfinaldefense,IfeellikeIhavealottothankhimfor.CharlieisprobablythemostoriginalthinkerIhave
ever met and he has really taught me the value of asking the right questions, which is arguably the most
importantlessonanyoneinresearchcouldeverbetaught.
It was Charles Morgan who gave me the most encouragement to go to graduate school. I would like
to thank him for his mentorship and his unerring moral compass. If everyone could know someone like
Charlesduringtheirdevelopmentalyears,theworldwouldnotbesuchascrewedupplace.
I would also like to thank NSERC for supporting me financially throughout graduate school. I don’t
reallyknowwhoisresponsibleformeoverthere, butwhoeveryouare, thankyouformakingitpossible
formetohavearoofovermyheadandfoodinmytummy.
AllofmyfriendsattheUniversityofTorontoalsodeservelotsofthanks.Firstandforemost,Iwouldlike
to acknowledge Stratis Ioannidis. Stratis, you were a great roommate and continue to be a terrific friend.
EmmanuelleandIhadawonderfultimetravelingtoGreecetovisityouandyourfamily. Ihopethatyou
andGrace(whoIwouldalsoliketoacknowledge!) remainashappyasever. IwouldliketothankMarc
Tedder for being human and allowing me to waste hours of his time discussing a huge range of issues.
I would also like to thank my friend Nazanin Mirmohammadi. Your honesty, intelligence, and curiosity
inspiremyown. IwouldalsoliketothankAlanSkelley,KleoniIoannidou,andMarkBravermanforbeing
greatfriends. Iamsurewewillcontinuetobeintouchformanyyearstocome.
Of course, I would never have been able to do this without my family. I would like to thank my wife
Emmanuelle,whoseloveandsupportmadeeverythingpossible. Thankyouforyoursenseofhumourand
allofyourpatienceandunderstanding. IwouldalsoliketothankmybrotherAlexanderwhohasalways
been a great older brother and role-model to me. Last, but certainly not least, I would like to thank my
Mother. Thejourneyofmyeducationstartedalongtimeagowithyou,andhasbeenpunctuatedalongthe
waywithmanyofyoursacrifices. Forthat,andformanyotherthings,Iwillbeeternallygrateful.
iii
Contents
1 Introduction 1
1.1 Summary&Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Definitions&PreviousResults 6
2.1 GeneralLogicDefinitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 DAG/CircuitDefinitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 ProofComplexityDefinitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3.1 p-Simulation&ExponentialSeparation . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3.2 Effectivep-Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3.3 Automatizability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 ResolutionDefinitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4.1 ResolutionProof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
ClassicResolutionProof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
ResolutionProofsasDAGs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Configuration-StyleResolutionProof . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4.2 ResolutionRefinements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4.3 ResolutionComplexityMeasures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.5 PreviousResolutionResults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.5.1 ResolutionHierarchy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.5.2 ResolutionSpaceResults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.5.3 RelationshipsBetweenComplexityMeasures. . . . . . . . . . . . . . . . . . . . . . . . 17
2.5.4 PreviousAutomatizabilityResults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
PositiveResults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
NegativeResults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.6 PebblingGameDefinitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.6.1 BlackPebbling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.6.2 Black-WhitePebbling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.6.3 Frugality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.7 PreviousPebblingResults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
iv
3 ThePSPACE-CompletenessofBlack-WhitePebbling 29
3.1 OverviewofMainResult . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2 Lingas: BlackPebblingMonotoneCircuitsisPSPACE-Complete . . . . . . . . . . . . . . . . . 31
3.3 ReductionFromQBFψtoBlack-WhitePebblingDAGG . . . . . . . . . . . . . . . . . . . . . 37
3.4 ProofofCorrectness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.4.1 ForwardDirection/UpperBound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.4.2 ReverseDirection/LowerBound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.5 ExponentialTime/SpaceTrade-OffforBlack-WhitePebbling . . . . . . . . . . . . . . . . . . . 85
3.6 PSPACE-CompletenessoftheSymmetricBlack-WhitePebblingGame . . . . . . . . . . . . . 85
3.6.1 DefinitionofSymmetricBlack-WhitePebbling . . . . . . . . . . . . . . . . . . . . . . . 85
3.6.2 ModificationtotheDefinitionofFrugality . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.6.3 ModificationtotheBlack-WhitePebblingReduction . . . . . . . . . . . . . . . . . . . 86
3.6.4 ProofThatSymmetricBlack-WhitePebblingisPSPACE-Complete . . . . . . . . . . . 86
4 ResolutionSpaceResults 89
4.1 ProofOverview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.2 ReductionFromQBFψToCNFFormulaPeb(G(cid:48)) . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.2.1 ConstructionofG(cid:48) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.2.2 ConstructionofPeb(G(cid:48)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.2.3 OtherPreliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.3 ProofofCorrectness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.3.1 ForwardDirection/UpperBound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.3.2 ReverseDirection/LowerBound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.3.3 Trade-off . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5 TheProofComplexityofClauseLearning 129
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.2 DPLL&RegularTreeResolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
5.2.1 TheDPLLProcedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
5.2.2 DPLLProducesTreeResolutionProofs. . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
5.2.3 TheClauseLearningProcedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
5.3 ClauseLearningProofs(CL) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
5.4 Greedyvs. Non-greedyClauseLearningProofs . . . . . . . . . . . . . . . . . . . . . . . . . . 138
5.5 ThePowerofClauseLearningProofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
5.5.1 RelationToPreviousWork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
5.5.2 ClauseLearningEffectivelyp-SimulatesResolution . . . . . . . . . . . . . . . . . . . . 142
5.5.3 ImplicationsoftheEffectivep-simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 151
5.5.4 ResolutionHierarchy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
v
6 RelatedOpenProblems 154
6.1 ClauseLearningProblems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
6.2 PebblingProblems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
6.3 ResolutionSpaceProblems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
Bibliography 157
vi
List of Figures
2.1 AportionoftheResolutionProofComplexityHierarchybeforetheinclusionofClauseLearn-
ing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.1 ThemonotonecircuitG producedforψ = ∀x ∃x ∀x (x ∨x¯ ∨x )∧(x¯ ∨x ∨x )∧(x¯ ∨
3 2 1 3 2 1 3 2 1 3
x ∨x¯ )∧(x ∨x¯ ∨x¯ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2 1 3 2 1
3.2 AnExampleofaLiteralWidgetfromLingas’sConstruction . . . . . . . . . . . . . . . . . . . 33
3.3 TheExistentialWidgetFromLingas’sConstruction. . . . . . . . . . . . . . . . . . . . . . . . . 34
3.4 TheUniversalWidgetFromLingas’sConstruction. . . . . . . . . . . . . . . . . . . . . . . . . 34
3.5 Aliteralwidgetforvariablex (left). Aliteralwidgetforvariablex inthetruestate(center).
i i
Aliteralwidgetforvariablex inthefalsestate(right). . . . . . . . . . . . . . . . . . . . . . . 38
i
3.6 Aclausewidgetforclausez =(l1∨l2∨l3)(left). Theconnectionofz toG (center). And
j j j j m 0
a4-slide({v1,v2,v3,v4},{u1,u2,u3,u4})(right). . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.7 LegendexplainingthecomponentsofFigures3.8and3.9.. . . . . . . . . . . . . . . . . . . . . 40
3.8 AuniversalwidgetfortheBlack-WhitePebblingresult. . . . . . . . . . . . . . . . . . . . . . . 41
3.9 AnexistentialwidgetfortheBlack-WhitePebblingresult.. . . . . . . . . . . . . . . . . . . . . 42
3.10 AnexampleofGforψ =∀x ∃x ∀x (x ∨x¯ ∨x )∧(x ∨x ∨x¯ )∧(x¯ ∨x ∨x¯ )∧(x¯ ∨x¯ ∨x ) 44
3 2 1 1 2 3 1 2 3 1 2 3 1 2 3
3.11 Subintervalsof[tα,tω]duringwhichnodesoftheuniversalwidgetmustbeemptyormust
containablackpebble.Intervalsduringwhichanodemustcontainablackpebbleareshown
as a thick blue line, while intervals during which a node must be empty are shown as a
thinnerredline. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.12 Subintervalsof[tα,tω]duringwhichnodesoftheexistentialwidgetmustbeemptyormust
containablackpebbleincase2a. Intervalsduringwhichanodemustcontainablackpebble
areshownasathickblueline,whileintervalsduringwhichanodemustbeemptyareshown
asathinnerredline. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.13 Subintervalsof[tα,tω]duringwhichnodesoftheexistentialwidgetmustbeemptyormust
containablackpebbleincase2b. Intervalsduringwhichanodemustcontainablackpebble
areshownasathickblueline,whileintervalsduringwhichanodemustbeemptyareshown
asathinnerredline. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.14 ModificationmadetoG toformG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
sym
vii
4.1 ClauseWidgetforz =(x ∨x ∨x )inG(cid:48). (Alledgesaredirectedfromlowernodestoward
j 1 2 3
highernodes.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.1 AT-RESrefutationofF =(x¯ ∨x¯ )∧(x¯ )∧(x¯ ∨x¯ ∨x¯ )∧(x ∨x¯ )∧(x ∨x¯ ∨x¯ )∧(x ∨
1 2 5 1 2 5 1 2 2 3 4 2
x ∨x )∧(x ∨x ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
4 5 2 3
5.2 ACLrefutationofF = (x¯ ∨x¯ )∧(x¯ )∧(x¯ ∨x¯ ∨x¯ )∧(x ∨x¯ )∧(x ∨x¯ ∨x¯ )∧(x ∨
1 2 5 1 2 5 1 2 2 3 4 2
x ∨x )∧(x ∨x ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
4 5 2 3
5.3 General form of the CL refutation. The refutation is drawn as a tree for clarity’s sake, but
therewillbecross-edgespointingfromsomesub-proofsbacktosomeearliersub-proofs. . . 143
5.4 Subproofi.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
5.5 General form of the gCL refutation. The refutation is drawn as a tree for clarity’s sake, but
therearecrossedgespointingfromsub-proofsbacktoearliersub-proofsandfromthederiva-
tionof∅backtosub-proofs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
5.6 Subproofi.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.7 ThestructureofP(p,i),attachedtothebackboneofsub-proofi. . . . . . . . . . . . . . . . . . 148
5.8 TheResolutionProofComplexityHierarchyincludingClauseLearning. . . . . . . . . . . . . 153
viii
Chapter 1
Introduction
1.1 Summary & Motivation
Thisthesiscoversthreemaintopics: ClauseLearning, ResolutionSpace, andBlack-WhitePebbling. At
first glance, clause learning and black-white pebbling seem to be unrelated topics, but they are linked by
their mutual connections to the study of Resolution space measures. In this chapter we introduce each
ideaandgivesomeintuitionastowhytheyareinterestingandimportanttopicsofstudy. Wealsobriefly
describe the novel contributions made in this thesis. We shall use [Sip96] and [CK01] as our standard
references for complexity theory and proof complexity, and begin our introduction by discussing black-
whitepebbling.
Thecomplexityoftheblack-whitepebblinggamehasremainedanopenproblemforalmost30years.In
thisthesiswesettlethisopenproblembyshowingthattheblack-whitepebblinggameisPSPACE-complete.
WethenusesimilarideasinamorecomplicatedreductiontoprovethePSPACE-completenessofResolution
space. Thereductionalsoyieldsasurprisingexponentialtime/spacespeedupforResolutioninwhichan
increaseof3unitsofspaceresultsinanexponentialdecreaseinproof-size.
The Black-White Pebbling Game was introduced by Cook and Sethi in 1976 [CS76] in an attempt to
separateP fromNL. TheBlack-WhitePebblingGamereceivedconsiderableattentionthroughoutthenext
decade due to its numerous applications including VLSI design, compilers, and algebraic complexity. In
1983, determining its complexity was rated as “An Open Problem of the Month” in David Johnson’s NP-
Completeness Column [Joh83]. An excellent survey of pebbling results from this period can be found in
Pippenger [Pip80]. Recently, there has been a resurgence of interest in pebbling games due to their links
withpropositionalproofcomplexity[BS02,ET01,Nor06,HU07,Her08].
TheBlack-WhitePebblingGamewasprecededbytheBlackPebblingGame,whichhasalsobeenwidely
studied [Pip80]. Let G = (V,E) be a DAG with one distinguished output node, s. In the Black Pebbling
Game,aplayertriestoplaceapebbleonswhileminimizingthenumberofpebblesplacedsimultaneously
on G. The game is split up into distinct steps, each of which takes the player from one pebbling configu-
rationtothenext. Initially,thegraphcontainsnopebblesandeachsubsequentconfigurationfollowsfrom
thepreviousbyoneofthefollowingrules:
1
CHAPTER1: INTRODUCTION 2
• Atanypointtheplayermayplaceablackpebbleonanysourcenodev.
• Atanypointtheplayermayremoveablackpebblefromanynodev.
• For any node v, if all of v’s predecessors have pebbles on them, then the player may place a black
pebbleonv,ormayslideablackpebblefromapredecessoruofvtov.
The Black Pebbling Game models deterministic space-bounded computation. Each node represents a
resultandtheplacementofablackpebbleonanoderepresentsthedeterministiccomputationoftheresult
from previously computed results. A sequence of moves made by the player is called a pebbling strategy.
If a strategy manages to pebble s using no more than k pebbles, then that strategy is called a k-pebbling
strategy.
TheBlack-WhitePebblingGameisamorepowerfulextensionoftheBlackPebblingGamethatincludes
whitepebbles,whichbehaveinadualmannertotheoriginalblackpebbles. Asbefore,theplayerattempts
to place a black pebble on s while minimizing the number of pebbles placed simultaneously on G at any
time. However,unliketheBlackPebblingGame,theBlack-WhitePebblingGamedoesnotenduntilevery
nodeotherthansisempty. Sotheplayermustremoveanyoutstandingpebblesonceshasbeenreached.
TheBlack-WhitePebblingGameextendstheBlackPebblingGamewiththeadditionofthefollowingrules:
• Atanypointtheplayermayremoveawhitepebblefromanysourcenodev.
• Atanypointtheplayermayplaceawhitepebbleonanynodev.
• Foranynodev withawhitepebbleonit,theplayermayslidethepebbletoanemptypredecessoru
ofvifallofv’sotherpredecessorsarepebbled,ortheplayermayremovethewhitepebbleifallofv’s
predecessorsarepebbled.
• Thegameendswhenscontainsablackpebbleandeveryothernodeisempty.
As before, the placement of each black pebble is meant to model the derivation of a deterministically
computed result, while the placement of each white pebble is meant to model a non-deterministic guess
whoseverificationrequiresallofitsantecedentstobederived.
In1978,LingasshowedthatageneralizationoftheBlackPebblingGame,playedonmonotonecircuits
instead of DAGs, is PSPACE-complete [Lin78]. This was a surprising result since the PSPACE-complete
games of the time involved two players and it was clear how the alternation between them led to each
game’shighcomplexity. In1980,Gilbert,Lengauer,andTarjanelaboratedonthebasicstructureofLingas’s
constructiontoprovethePSPACE-completenessoftheBlackPebblingGameonDAGs[GLT80]. Themain
difficultyinmovingfrommonotonecircuitstothemorerestrictedclassofDAGsisthecreationofanOR
widgetusingonlytheglobalboundonthenumberofpermissiblepebblesandnodeswhichactlikeAND
gates.
While the above results settle the complexity of black pebbling, determining the complexity of black-
whitepebblinghasresistednumerousattempts. Incontrasttoblackpebbles,whitepebblesallowamuch
richer choice of strategies since they can be placed anywhere on the graph regardless of previous pebble