Space in Weak Propositional Proof Systems PDF
Preview Space in Weak Propositional Proof Systems
Ilario Bonacina Space in Weak Propositional Proof Systems Space in Weak Propositional Proof Systems Ilario Bonacina Space in Weak Propositional Proof Systems 123 Ilario Bonacina Departament deCiències dela Computació Universitat PolitècnicadeCatalunya Barcelona Spain ISBN978-3-319-73452-1 ISBN978-3-319-73453-8 (eBook) https://doi.org/10.1007/978-3-319-73453-8 LibraryofCongressControlNumber:2017963852 ©SpringerInternationalPublishingAG,partofSpringerNature2017 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. 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Printedonacid-freepaper ThisSpringerimprintispublishedbytheregisteredcompanySpringerInternationalPublishingAGpart ofSpringerNature Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland tomyfamily Preface Inthisbookyou,thereader,aregoingtoseesomeresultsonthespacecomplexityof somepropositionalproofsystems.ThisbookisarevisedversionofmyPhDthesis1 andindeeditisnotintendedtobeasurveyofalltheresultsknownonthespace complexityofpropositionalproofsystems.Itwillratherbealongwalktouching sometopicsinproofcomplexity,mostlyaboutspaceofcourse,butnotexclusively. Hopefullythiscouldbeusedtooasaratherreader-friendlyexpositionofsomegame theoreticmethodsusedinproofcomplexity.Thisisindeedanundergroundtheme connectingmostoftheresultsweshow.Ofcoursetherewillbesomesurvey(-ish) partsbutmainlythefocuswillbeonthenewgametheoretictechniquesandtheir applicationtotheanalysisofthespacecomplexityofpropositionalproofsystems. That is the results arising from my PhD thesis [Bon15] and some earlier works + [BG13,BGT14,BT15,BG15,BBG 17,BT16a,BT16b,BGT16,Bon16]. Thisisaworkaboutproofcomplexity,solet’sstartbyintroducingitinformally. Proofcomplexityisaresearchareathatstudiestheconceptofcomplexityfromthe pointofviewoflogic.Inparticular,inproofcomplexityweareinterestedinquestions suchas:“howdifficultisittoproveatheorem?”Or,moreprecisely,givenaformal system,weareinterestedinmeasuringthecomplexityofatheorem,thatisanswering questionssuchas“whatistheshortestproofofthetheoreminagivenformalsystem?” Thismirrorsquestionsincomputationalcomplexityabout,forexample,thenumber ofstepsthataTuringmachineneedstocomputeagivenfunction f;orthesizeof circuitsneededtocompute f.2 Inthisbookweinvestigatethespacecomplexityofpropositionalproofsystems, sowhatisthespaceofaproof?Wecouldstatethisquestionpictoriallyas“what isthesmallestblackboardateacherneedstopresenttheproofofatheoremtoa class of students?”3 As before, this notion is analogous to the space complexity 1Thisrevisedversionisduetothefactthatmythesiswasawarded“BestItalianPhDThesisin TheoreticalComputerScience”fortheyear2016bytheItalianchapteroftheEuropeanAssociation forTheoreticalComputerScience(EATCS). 2Ontheotherhand,wecouldalsomeasurethecomplexityofatheoremasthestrengthofatheory neededtoprovethetheorem.Thisalsohasacounterpartincomputationalcomplexity,itislinked withquestionsaboutthesmallestcomplexityclasstowhichagivenfunctionbelongs. 3Wesupposeherethatthestudentscanunderstandjustproofswrittenontheblackboardinsome given formal system and they do not have any additional memory except the minimal one to vii viii Preface in the context of uniform computations, measured, for example, as the size of a working-tapeneededbyaTuringmachinetocomputeagivenfunction. Propositionalproofcomplexity,thatisthecomplexityofpropositionalproofs, plays a role in the context of feasible proofs as important as the role of Boolean circuitsinthecontextofefficientcomputations.Althoughtheoriginalmotivationsto studythecomplexityofpropositionalproofscamefromproof-theoreticalquestions aboutfirst-ordertheories,itturnsoutthat,essentially,thecomplexityofpropositional proofs deals with the following question: “what can be proved by a prover with boundedcomputationalabilities?”Forexample,ifitscomputationalabilitiesarelim- itedtosmallcircuitsfromsomecircuitclass.Hence,propositionalproofcomplexity mirrorsnon-uniformcomputationalcomplexityandindeedthereisaveryproductive cross-fertilizationoftechniquesbetweenthetwofields.Ourunderstandingofpropo- sitionalproofsystemsis,unfortunately,similartothegeneralsituationincomplexity theory.Inbothfieldswecanprovelowerboundsinveryspecialcasesandindeed thereareseveralmajoropenproblemsthatareverybasic,waymorebasicthanthe well-knownquestionP=? NP.Thesituationissimilarinthesensethatwecanprove super-polynomial lower bounds on the length of proofs only for restricted proof systems.Indeed,byaresultof[CR79],provingsuper-polynomiallowerboundson thelengthofproofsforeverypropositionalproofsystemisequivalenttoshowing thatNP(cid:2)=coNP,whichinturnisoneoftheopenandveryimportantproblemsin computationalcomplexity.Propositionalproofcomplexityisimportantalsofrom thepracticalpointofview.Theimplementationsofstate-of-the-artSATalgorithms ultimately rely on rather simple propositional proof systems. Hence the study of thosesystemshelpsinclarifyingthelimitationsofsuchalgorithmsthatareessential invariousaspectsofcomputerscience,cf.[Nor15]. Wewillfocusonthespacecomplexityoftwoparticularproofsystems:resolution, awellstudiedproofsystemthatisatthecoreofstate-of-the-artSAT-solvers;and polynomialcalculus,aproofsystemthatusespolynomialstorefutepropositional formulasthatarecontradictions.Wewillshowsomegenericcombinatorialtechniques to prove space lower bounds in both those systems and then we will apply those techniquestoshowconcretespacelowerboundsforrefutationsofseveralparticular (unsatisfiable) propositional formulas. Since the very first exponential size lower bound for resolution size in [Hak85], game theoretic methods and combinatorial characterisationsofhardnessmeasureshavealonghistoryinproofcomplexity.This bookcouldbeseenasthelatestcontributiontothistopic. For resolution the new techniques we introduce allowed for the first time to obtain—inaquiteeasywayactually—lowerboundsforthespaceofproofswhenthe spaceismeasuredasthetotalnumberofvariablestobekeptinmemory(totalspace). Forpolynomialcalculusthetechniquesweintroduce—whichismoreinvolvedthan thoseforresolution—allowustoaddressspacelowerboundswhenthespacetakes into account the number of distinct monomials to be kept in memory (monomial space).Notablythosetechniquesallowustoprove,amongotherresults,thatalmost understandthecontentoftheblackboard.Moreovertheteacherhastowritewithfontsofafixed size. Preface ix allk-CNFformulasarehardwithrespecttototalspaceinresolutionandmonomial spaceinpolynomialcalculus.Thatisweproveasymptoticallyoptimallowerbounds forthemonomialspace(andtotalspaceinresolution)forrandomk-CNFformulas innvariablesandalinearnumberofclauses.Thiswasanopenproblemmentioned for the first time in [BS01, ABRW02] and since then reported many times in the literature. BookStructureAfteranintroductiontopropositionalproofcomplexity(Chap.1), this work consists of 3 parts. Each chapter ends with a section containing open questionsandaHistorysectioncollectingsomefactsaboutthemaintheoremsof thechapterandhowtheyfitinthepreviousliterature. In Part I there are two chapters on resolution: one containing results already knownintheliteraturebeforethiswork(Chap.2)andonejustfocusedonspacein resolution(Chap.3).Morepreciselyonthecombinatorialtechniquestoprovetotal spacelowerbounds.Thenwemovetopolynomialcalculusanditsspacecomplexity (Chap.4).Thefocuswillbenowonthecombinatorialtechniquetoprovemonomial spacelowerbounds. InPartIIwecollectthemainapplicationsofthetechniqueswebuiltpreviously. Firstthereisashortchapterabouttheproofcomplexityandspacecomplexityofthe pigeonprinciples(PHPmanditsvariations),cf.Chap.5.Thenthereisaninterlude n onsomenewtypeofgames,thecovergames,definedonbipartitegraphs(Chap.6). Thischapterisessentiallyindependentfromtherestofthebookanditcollectssome resultsongraphtheory.Themotivationbehindthischapterthoughisthattheresults initwillbeneededinChap.7toprovethespacelowerboundforrandomk-CNF formulasandothergraph-basedpropositionalformulas. Inthelastpart,PartIII,weanalysethesizeofresolutionproofsinconnection withtheStrongExponentialTimeHypothesis(SETH)incomplexitytheory.More preciselyweprovestrongsizelowerboundsforarestrictedversionofresolutionwe callδ-regularresolution.Althoughnotdirectlyrelatedtospace,theresultsweshow hererelyonsomecombinatorialcharacterisationsandgamesanalogoustotheone usedtoprovespacelowerbounds. AcknowledgmentsFirstofall,IwanttothankmyadvisorNicolaGalesiatSapienza UniversityofRome.Hecultivatedmyinterestinproofcomplexitysupportingthe goodideasandshootingdownthe(many)buggyones.Thisbooksimplywouldn’t haveexistedwithouthim. Iamtrulygratefultothemanypeoplefromtheproofcomplexitycommunitythat Ihadtheprivilegetomeet.Myviewofproofcomplexityandtheoreticalcomputer sciencewasshapedbyallofthediscussionswehad.InparticularIwanttothank (theorderisnotrelevanthere):JakobNordström,MassimoLauria,MladenMikša, MarcVinyals,SusannaFigueiredodeRezende,NavidTalebanfard,TonyHuynh, Paul Wollan, Pavel Pudlák, Jan Krajícˇek, Neil Thapen, Olaf Beyersdorff, Rahul Santhanam,AlbertAtserias,JacoboTorán,andYuvalFilmus. Stockholm, IlarioBonacina May31,2017 Contents 1 Introduction................................................... 1 1.1 PropositionalProofSystems ................................. 1 1.1.1 FregeSystems....................................... 3 1.1.2 AlgebraicProofSystems.............................. 5 1.1.3 GeometricProofSystems ............................. 8 1.1.4 ConnectionwithSAT-Solvers.......................... 8 1.2 SpaceofProofs ............................................ 10 1.3 SummaryofResults ........................................ 11 PartI GeneralResultsandTechniques 2 Resolution..................................................... 15 2.1 SubsystemsofResolution ................................... 15 2.2 Size...................................................... 16 2.3 WidthandAsymmetricWidth ................................ 20 2.3.1 CombinatorialCharacterizations ....................... 23 2.3.2 The“Equivalence”ofWidthandAsymmetricWidth....... 26 2.4 OpenProblems ............................................ 28 3 SpaceinResolution ............................................ 29 3.1 SemanticResolution ........................................ 30 3.2 ClauseSpace .............................................. 31 3.2.1 SemanticClauseSpace ............................... 33 3.3 TotalSpace................................................ 34 3.3.1 SemanticTotalSpace................................. 36 3.4 SpaceandDepth ........................................... 38 3.5 OpenProblems ............................................ 39 4 SpaceinPolynomialCalculus.................................... 41 4.1 FromCNFFormulastoPolynomials .......................... 42 4.1.1 CompleteTreeFormulasCT .......................... 44 n xi xii Contents 4.2 SizeandDegree............................................ 44 4.3 Space .................................................... 45 4.4 SemanticPolynomialCalculus ............................... 47 4.5 MonomialSpaceLowerBounds .............................. 48 4.5.1 CT RequiresLargeMonomialSpace ................... 51 n 4.5.2 ProofsofTheorem4.2andTheorem4.3 ................ 52 4.6 OpenProblems ............................................ 56 PartII Applications 5 PigeonholePrinciples........................................... 61 5.1 The(Standard)PigeonholePrinciples.......................... 61 5.2 TheBit-PigeonholePrinciple................................. 65 5.3 TheXORPigeonholePrinciple ............................... 67 5.4 OpenProblems ............................................ 69 6 Interlude:CoverGames ........................................ 71 6.1 C-Matchings............................................... 71 6.2 SomeHall’sTheorems ...................................... 72 6.2.1 AHall’sTheoremforVW-Matchings ................... 73 6.3 CoverGames .............................................. 75 6.4 RandomBipartiteGraphs.................................... 77 6.5 WinningStrategiesforCoverGames .......................... 79 6.5.1 AWinningStrategyfortheGameonV-Matchings ........ 79 6.5.2 AWinningStrategyfortheGameonVW-Matchings ...... 83 7 SomeGraph-BasedFormulas ................................... 89 7.1 FromCoverGamestoSpaceLowerBounds .................... 89 7.2 Randomk-CNFFormulas ................................... 92 7.3 PigeonholePrinciplesoverGraphs ............................ 95 7.4 TseitinFormulas ........................................... 97 7.5 OpenProblems ............................................100 PartIII APostlude 8 StrongSizeLowerBoundsfor(aSubsystemof)Resolution .........105 8.1 SETHandProofComplexity.................................105 8.2 GameCharacterizationsofWidthandSize .....................108 8.3 AStrongWidthLowerBound................................113 8.4 OpenProblems ............................................ 117 References.........................................................119 Index .............................................................129
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