Relationaland AlgebraicMethods inComputerScience
17thInternationalConference,RAMiCS2018
Groningen,TheNetherlands,October29 – November1,2018
Proceedings
Editors JulesDesharnais
Université Laval
Québec,QC
Canada
WalterGuttmann
UniversityofCanterbury
Christchurch
NewZealand
StefJoosten
OpenUniversiteit
Heerlen
TheNetherlands
ISSN0302-9743ISSN1611-3349(electronic)
LectureNotesinComputerScience
ISBN978-3-030-02148-1ISBN978-3-030-02149-8(eBook) https://doi.org/10.1007/978-3-030-02149-8
LibraryofCongressControlNumber:2015949476
LNCSSublibrary:SL1 – TheoreticalComputerScienceandGeneralIssues
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Preface
Thisvolumecontainstheproceedingsofthe17thInternationalConferenceonRelationalandAlgebraicMethodsinComputerScience(RAMiCS2018),whichwasheld inGroningen,TheNetherlands,fromOctober29toNovember1,2018.
Theplantoinitiatethisseriesofconferenceswasputinplaceduringthe38th BanachSemesteronAlgebraicMethodsinLogicandTheirComputerScience ApplicationinWarsaw,Poland,inSeptemberandOctober1991.The firstnumbered occurrencewasaDagstuhlseminaronRelationalMethodsinComputerScience (RelMiCS1),heldinGermanyin1994.Fromthenonanduntil2009,atintervalsof aboutoneyearandahalf,therewasaRelMiCSconference.Startingin2003,RelMiCS conferenceswereheldjointlywithApplicationsofKleeneAlgebras(AKA)conferences.AtRelMiCS11/AKA6inDoha,Qatar,itwasdecidedtousethecurrentname fortheseries,thatis,RelationalandAlgebraicMethodsinComputerScience (RAMiCS).
RecurrenttopicsofRAMiCSconferencesincludesemiring-andlattice-based structuressuchasrelationalgebrasandKleenealgebras,theirconnectionswithprogramlogicsandotherlogics,theiruseintheoriesofcomputing,theirformalization withtheoremprovers,andtheirapplicationtomodelingandreasoningaboutcomputingsystemsandprocesses.
Intotal,30papersweresubmittedtoRAMiCS2018andtheProgramCommittee selected21ofthemforpresentationattheconference.Intheseproceedingstheselected papersaregroupedunderthreeheadings: “TheoreticalFoundations,”“Reasoning AboutComputationsandPrograms,” and “ApplicationsandTools.” Eachsubmission wasevaluatedbyatleastfourindependentreviewers,andfurtherdiscussedelectronicallyduringtwoweeks.ThechairsareverygratefultoallProgramCommittee membersandtotheexternalreviewersfortheirtimeandexpertise.
RolandBackhouse,ManuelBodirsky,andPhilippaGardnerkindlyacceptedour invitationtopresenttheirresearchattheconference.TheabstractsofRolandBackhouse’s talk, “TheImportanceofFactorisationinAlgorithmDesign,” andPhilippaGardner ’stalk, “ScalableReasoningAboutConcurrentPrograms,” areincludedintheproceedings.Sois thefullpaperrelatedtoManuelBodirsky’stalk, “FiniteRelationAlgebraswithNormal Representations.” Thisconferencefeaturedatutorialonmotivatingstudentsforrelation algebrapresentedbyoneofus(StefJoosten).Theabstractofthistutorialisalsoincluded inthisvolume.Anexperimentalsessionofthisconferencewasameetingwithsoftware engineerswhoworkinpractice,mostofwhomhavelittlecontactwiththetheories discussedinourconference.Thismeetingwasplannedasawaytofosterapplicationsof relationalandalgebraicmethodsincomputerscience.
WethanktheRAMiCSSteeringCommitteefortheirsupportandtheOpenUniversiteitNederlandfororganizingthisconference.Wegratefullyacknowledgethe financial supportofOrdinaNederlandBVfortheirsponsorshipandtheirhelpwithorganizingthis
conference.WealsothanktheGroningenCongresBureauandtheRijksuniversiteit GroningenforthewarmwelcomewereceivedintheAcademiegebouwandthecityitself. AspecialthankyouisduetoSebastiaanJ.C.Joosten,authorinpreviouseditionsandthis timeaveryactive,talentedpublicitychair.
WealsoappreciatetheexcellentfacilitiesofferedbytheEasyChairconference administrationsystem,andAlfredHofmannandAnnaKramer ’shelpinpublishingthis volumewithSpringer.Finally,weowemuchtoallauthorsandparticipantsfortheir supportofthisRAMiCSconference.
October2018JulesDesharnais
WalterGuttmann
StefJoosten
Organization
OrganizingCommittee
ConferenceChair
StefJoostenOpenUniversiteit,TheNetherlands
PCChairs
JulesDesharnaisUniversité Laval,Canada
WalterGuttmannUniversityofCanterbury,NewZealand
PublicityChair
SebastiaanJoostenUniversiteitTwente,TheNetherlands
ProgramCommittee
LucaAcetoReykjavíkUniversity,Iceland GranSassoScienceInstitute,Italy
RudolfBerghammerChristian-Albrechts-UniversitätzuKiel,Germany
JulesDesharnaisUniversité Laval,Canada
UliFahrenberg ÉcolePolytechnique,France
HitoshiFurusawaKagoshimaUniversity,Japan
WalterGuttmannUniversityofCanterbury,NewZealand
RobinHirschUniversityCollegeLondon,UK
PeterHöfnerData61,CSIRO,Australia
MarcelJacksonLaTrobeUniversity,Australia
Jean-BaptisteJeanninUniversityofMichigan,USA
PeterJipsenChapmanUniversity,USA
StefJoostenOpenUniversiteit,TheNetherlands
WolframKahlMcMasterUniversity,Canada
BarbaraKönigUniversitätDuisburg-Essen,Germany
DexterKozenCornellUniversity,USA
AgiKuruczKing’sCollegeLondon,UK
TadeuszLitakFriedrich-AlexanderUniversitätErlangen-Nürnberg, Germany
RogerMadduxIowaStateUniversity,USA
AnnabelleMcIverMacquarieUniversity,Australia
SzabolcsMikulásBirkbeck,UniversityofLondon,UK
AliMiliNewJerseyInstituteofTechnology,USA
BernhardMöllerUniversitätAugsburg,Germany
José N.OliveiraUniversidadedoMinho,Portugal
AlessandraPalmigianoTechnischeUniversiteitDelft,TheNetherlands
DamienPousCNRS,France
MehrnooshSadrzadehQueenMaryUniversityofLondon,UK
JohnStellUniversityofLeeds,UK
GeorgStruthUniversityofSheffi eld,UK
MichaelWinterBrockUniversity,Canada
SteeringCommittee
RudolfBerghammerChristian-Albrechts-UniversitätzuKiel,Germany
JulesDesharnaisUniversité Laval,Canada
AliJaouaQatarUniversity,Qatar
PeterJipsenChapmanUniversity,USA
BernhardMöllerUniversitätAugsburg,Germany
José N.OliveiraUniversidadedoMinho,Portugal
EwaOrłowskaNationalInstituteofTelecommunications,Poland
GuntherSchmidtUniversitätderBundeswehrMünchen,Germany
MichaelWinter(Chair)BrockUniversity,Canada
AdditionalReviewers
MusaAl-Hassy
BenjaminCabrera
ChristianDoczkal
JérémyDubut
SabineFrittella
IliasGarnier
Sponsors
RolandGlück
GiuseppeGreco
LucyHam
SebastiaanJoosten
FeiLiang
KonstantinosMamouras
OrdinaNederlandBV(MainSponsor)
GroningenCongresBureau
LocalOrganizers
ChrisjaMurisOpenUniversiteit
HeleenBakkerOpenUniversiteit
HillebrandMeijerOrdina
ErickKosterOrdina
AstridPatzOrdina
MargotSpeeGroningenCongresBureau
KellyScholtensGroningenCongresBureau
KokiNishizawa
FilipSieczkowski
JorgeSousaPinto
NorihiroTsumagari
ThorstenWißmann
AbstractsofInvitedTalks
Tutorial:RelationAlgebraintheClassroom withAmpersand
StefJoosten
OpenUniversityoftheNetherlands stef.joosten@ou.nl
Thistutorialexploresawaytomotivatestudentsforrelationalgebrasbyapplyingitin softwareengineering.Participantswillgethands-onexperiencewithAmpersand[1], whichisacompilerthattransformsaspeci ficationinrelationalgebrainworking software.Theywillbedirectedtothedocumentationsite[2]formorepreciseinformationaboutthelanguageandtools.
Thistutorialstartswithamotivation:WhymightworkingwithAmpersandmotivatestudentsforrelationalgebra?Thentheparticipantswillcreateaninformation systemonline,justlikestudentsdointhecourseRuleBasedDesign[3].Forthis purposeweaskparticipantstobringtheirlaptops.Thepresentationproceedsby pointingoutwhichlearningpointsarerelevantinthetutorial.It finalizesbygivingan overviewintheavailablematerialsandtools.Allmaterialsarefreelyavailableinopen source,soparticipantscantakeittotheirownclassrooms.
Professorswhowanttousethesematerialsarecordiallyinvitedtopartakeinthe furtherdevelopment.
Background
Ampersandwasoriginallyintendedasameanstospecifyrequirements[4]in heterogeneousrelationalgebra[5].Thetoolsetevolvedintoatoolforstudents[6].The Ampersandtoolsethasbeenusedsince2013[7]attheOpenUniversityintwocourses andnumerousresearchassignments.
ThenovelfeatureofAmpersandisthatatheoryinrelationalgebraisbeingusedas adatabaseprogram.Itspeci fiespersistence(i.e.thedatabase)anduserinterfaces.This isachievedbyusingoneinterpretationofthealgebra:arelationisinterpretedasa fi nite setofpairs.
Theusergetsaprogramminglanguagethatisdeclarative,stronglytyped,and easilysubjectedtoproofsofcorrectness[8].Thebenefitsarefastdevelopment(because theAmpersandcompilergeneratesworkingsoftware),maintainability(becausesoftwareiseasilydividedintoindependentchunks),andadaptability(becausegenerating anapplicationanddeployingitisautomated).
References
1.Joosten,S.:RAP3(2009–2018). rap.cs.ou.nl/RAP3
2.Joosten,S.,Hageraats,E.:DocumentationofAmpersand(2016–2018). ampersandtarski. gitbook.io/documentation/
3.Rutledge,L.,Wetering,R.v.d.,Joosten,S.:CourseIM0103RuleBasedDesignforCS(2009–2018). www.ou.nl/-/IM0403_Rule-Based-Design
4.Dijkman,R.M.,FerreiraPires,L.,Joosten,S.M.M.:Calculatingwithconcepts:atechniquefor thedevelopmentofbusinessprocesssupport.In:Evans,A.(ed.)ProceedingsoftheUML 2001WorkshoponPracticalUML-BasedRigorousDevelopmentMethods:Counteringor IntegratingtheeXstremists,LectureNotesinInformatics,vol.7.FIZ,Karlsruhe(2001)
5.vanderWoude,J.,Joosten,S.:Relationalheterogeneityrelaxedbysubtyping.In:deSwart,H. (ed.)RAMICS2011.LNCS,vol.6663,pp.347–361.Springer,Heidelberg(2011)
6.Michels,G.,Joosten,S.,vanderWoude,J.,Joosten,S.:Ampersand:applyingrelationalgebra inpractice.In:Proceedingsofthe12thConferenceonRelationalandAlgebraicMethodsin ComputerScienceRAMICS2011.LNCS,vol.6663,pp.280–293.Springer-Verlag,Berlin (2011)
7.Michels,G.,Joosten,S.:ProgressivedevelopmentandteachingwithRAP.In:Proceedings ofthe3rdComputerScienceEducationResearchConferenceonComputerScienceEducation Research,CSERC2013,pp.3:33–3:43.OpenUniversity,Heerlen,TheNetherlands(2013)
8.Joosten,S.:Relationalgebraasprogramminglanguageusingtheampersandcompiler.J.Log. AlgebraicMethodsProgram. 100,113–129(2018)
Contents
InvitedPaper
FiniteRelationAlgebraswithNormalRepresentations.................3 ManuelBodirsky
TheoreticalFoundations
C-Dioidsand l-ContinuousChomsky-Algebras.....................21 HansLeiß andMarkHopkins
CoequalizersandTensorProductsforContinuousIdempotentSemirings....37 MarkHopkinsandHansLeiß
Distances,NormsandErrorPropagationinIdempotentSemirings.........53 RolandGlück
T-NormBasedOperationsinArrowCategories.....................70 MichaelWinter
DecidabilityofEquationalTheoriesforSubsignaturesofRelationAlgebra...87 RobinHirsch
CompositionofDifferent-TypeRelationsviatheKleisliCategory fortheContinuationMonad...................................97 KokiNishizawaandNorihiroTsumagari
AxiomatizingDiscreteSpatialRelations...........................113 GiuliaSindoni,KatsuhikoSano,andJohnG.Stell
AModalandRelevanceLogicforQualitativeSpatialReasoning.........131 PranabKumarGhoshandMichaelWinter
OntheStructureofGeneralizedEffectAlgebrasandSeparationAlgebras...148 SarahAlexander,PeterJipsen,andNadiyaUpegui
CountingFiniteLinearlyOrderedInvolutiveBisemilattices.............166 StefanoBonzio,MichelePraBaldi,andDiegoValota
MIX H-AutonomousQuantalesandtheContinuousWeakOrder.........184 MariaJoãoGouveiaandLuigiSantocanale
ReasoningAboutComputationsandPrograms
CalculationalVerificationofReactiveProgramswithReactiveRelations andKleeneAlgebra.........................................205
SimonFoster,KangfengYe,AnaCavalcanti,andJimWoodcock
VerifyingHybridSystemswithModalKleeneAlgebra................225
JonathanJuliánHuertayMuniveandGeorgStruth
AlgebraicDerivationofUntilRulesandApplicationtoTimerVerification...244
JessicaErtel,RolandGlück,andBernhardMöller
FalseFailure:CreatingFailureModelsforSeparationLogic.............263
CallumBannisterandPeterHöfner
TowardsanAnalysisofDynamicGossipinNetKAT ..................280
MalvinGattingerandJanaWagemaker
CoalgebraicToolsforRandomness-ConservingProtocols...............298
DexterKozenandMatveySoloviev
ApplicationsandTools
AlgebraicSolutionofWeightedMinimaxSingle-FacilityConstrained LocationProblems.........................................317
NikolaiKrivulin
ASetSolverforFiniteSetRelationAlgebra.......................333
MaximilianoCristiá andGianfrancoRossi
OntheComputationalComplexityofNon-dictatorialAggregation........350
LefterisKirousis,PhokionG.Kolaitis,andJohnLivieratos
CalculationalRelation-AlgebraicProofsintheTeaching Tool
WolframKahl
InvitedPaper
FiniteRelationAlgebraswithNormal Representations
ManuelBodirsky(B)
Institutf¨urAlgebra,TUDresden,01062Dresden,Germany manuel.bodirsky@tu-dresden.de
Abstract. Oneofthetraditionalapplicationsofrelationalgebrasis toprovideasettingforinfinite-domainconstraintsatisfactionproblems. Complexityclassificationforthesecomputationalproblemshasbeenone ofthemajoropenresearchchallengesofthisapplicationfield.Thepast decadehasbroughtsignificantprogressonthetheoryofconstraintsatisfaction,bothoverfiniteandinfinitedomains.Thisprogresshasbeen achievedindependentlyfromtherelationalgebraapproach.Thepresent articletranslatestherecentfindingsintothetraditionalrelationalgebra setting,andpointsoutaseriesofopenproblemsattheinterfacebetween modeltheoryandthetheoryofrelationalgebras.
1Introduction
Oneofthefundamentalcomputationalproblemsforarelationalgebra A is the networksatisfactionproblemfor A,whichistodetermineforagiven Anetwork N whetheritissatisfiableinsomerepresentationof A (fordefinitions, seeSects. 2 and 3).RobinHirschnamedin1995the ReallyBigComplexity Problem(RBCP) forrelationalgebras,whichisto ‘clearlymapoutwhich(finite) relationalgebrasaretractableandwhichareintractable’ [Hir96].Forexample, forthePointAlgebrathenetworksatisfactionproblemisinPandforAllen’s IntervalAlgebraitisNP-hard.Oneofthestandardmethodstoshowthatthe networksatisfactionproblemforafiniterelationalgebraisinPisviaestablishing localconsistency.Thequestionwhetherthenetworksatisfactionproblemfor A canbesolvedbylocalconsistencymethodsisanotherquestionthathasbeen studiedintensivelyforfiniterelationalgebras A (see[BJ17]forasurveyonthe secondquestion).
If A hasafullyuniversalsquarerepresentation(wefollowtheterminologyof Hirsch[Hir96])thenthenetworksatisfactionproblemfor A canbeformulated asaconstraintsatisfactionproblem(CSP)foracountablyinfinitestructure. Thecomplexityofconstraintsatisfactionisaresearchdirectionthathasseen quitesomeprogressinthepastyears.The dichotomyconjecture ofFederand
ManuelBodirsky—TheauthorhasreceivedfundingfromtheEuropeanResearch CouncilundertheEuropeanCommunity’sSeventhFrameworkProgramme (FP7/2007-2013GrantAgreementno.257039,CSP-Infinity).
c SpringerNatureSwitzerlandAG2018
J.Desharnaisetal.(Eds.):RAMiCS2018,LNCS11194,pp.3–17,2018. https://doi.org/10.1007/978-3-030-02149-8 1
Vardifrom1993statesthateveryCSPforafinitestructureisinPorNPhard;the tractabilityconjecture [BKJ05]isastrongerconjecturethatpredicts preciselywhichCSPsareinPandwhichareNP-hard.Twoindependentproofs oftheseconjecturesappearedin2017[Bul17, Zhu17],basedonconceptsand toolsfromuniversalalgebra.AnearlierresultofBartoandKozik[BK09]gives anexactcharacterisationofthosefinite-domainCSPsthatcanbesolvedbylocal consistencymethods.
Usually,thenetworksatisfactionproblemforafiniterelationalgebra A cannotbeformulatedasaCSPforafinitestructure.However,suprisinglyoftenit canbeformulatedasaCSPforacountablyinfinite ω -categorical structure B Foranimportantsubclassof ω -categoricalstructureswehaveatractabilityconjecture,too.TheconditionthatsupposedlycharacterisescontainmentinPcan beformulatedinmanynon-triviallyequivalentways[BKO+17, BP16, BOP17] andhasbeenconfirmedinnumerousspecialcases,seeforinstancethearticles[BK08, BMPP16, KP17, BJP17, BMM18, BM16]andthereferencestherein.
Inthelightoftherecentadvancesinconstraintsatisfaction,bothoverfinite andinfinitedomains,werevisittheRBCPanddiscussthecurrentstateofthe art.Inparticular,weobservethatif A hasa normalrepresentation (again,we followtheterminologyofHirsch[Hir96]),thenthenetworksatisfactionproblem for A fallsintothescopeoftheinfinite-domaintractabilityconjecture.Wealso showthatthereisanalgorithmthatdecidesforagivenfiniterelationalgebra A withafullyuniversalsquarerepresentationwhether A hasanormalrepresentation.(Inotherwords,thereisanalgorithmthatdecidesforagiven A whether theclassofatomic A-networkshastheamalgamationproperty.)Thescopeof thetractabilityconjectureislarger,though.Wedescribeanexampleofafinite relationalgebrawhichhasan ω -categoricalfullyuniversalsquarerepresentation (andapolynomial-timetractablenetworksatisfactionproblem)whichisnot normal,butwhichdoesfallintothescopeoftheconjecture.
Whethertheinfinite-domaintractabilityconjecturemightcontributetothe resolutionoftheRBCPingeneralremainsopen;wepresentseveralquestions inSect. 7 whoseanswerwouldshedsomelightonthisquestion.Thesequestions concerntheexistenceof ω -categoricalfullyuniversalsquarerepresentationsand areofindependentinterest,andinmyviewtheyarecentraltothetheoryof representablefiniterelationalgebras.
2RelationAlgebras
A properrelationalgebra isaset B togetherwithaset R ofbinaryrelations over B suchthat
1.Id:= {(x,x) | x ∈ B }∈R; 2.If R1 and R2 arefrom R,then R1 ∨ R2 := R1 ∪ R2 ∈R; 3.1:= R∈R R ∈R; 4.0:= ∅∈R;
5.If R ∈R,then R :=1 \ R ∈R;
6.If R ∈R,then R := {(x,y ) | (y,x) ∈ R}∈R;
7.If R1 and R2 arefrom R,then R1 ◦ R2 ∈R;where R1 ◦ R2 := {(x,z ) |∃y ((x,y ) ∈ R1 ∧ (y,z ) ∈ R2 )} .
Wewanttopointoutthatinthisstandarddefinitionofproperrelationalgebrasitis not requiredthat1denotes B 2 .However,inmostexamples,1indeed denotes B 2 ;inthiscasewesaythattheproperrelationalgebrais square.The inclusion-wiseminimalnon-emptyelementsof R arecalledthe basicrelations oftheproperrelationalgebra.
Example1(ThePointAlgebra). Let B = Q bethesetofrationalnumbers,and consider R = {∅, =,<,>, ≤, ≥, = , Q2 } .
Thoserelationsformaproperrelationalgebra(withthebasicrelations <,>, =, andwhere1denotes Q2 )whichisknownunderthename pointalgebra.
The relationalgebraassociatedto (B, R)isthealgebra A withthedomain A := R andthesignature τ := {∨, , 0, 1, ◦, , Id} obtainedfrom(B, R)inthe obviousway.An abstractrelationalgebra isa τ -algebrathatsatisfiessomeof thelawsthatholdfortherespectiveoperatorsinaproperrelationalgebra.We donotneedtheprecisedefinitionofanabstractrelationalgebrainthisarticle sincewedealexclusivelywith representable relationalgebras:a representation ofanabstractrelationalgebra A isarelationalstructure B whosesignature is A;thatis,theelementsoftherelationalgebraaretherelationsymbolsof B.Eachrelationsymbol a ∈ A isassociatedtoabinaryrelation aB over B suchthatthesetofrelationsof B inducesaproperrelationalgebra,andthe map a → aB isanisomorphismwithrespecttotheoperations(andconstants) {∨, , 0, 1, ◦, , Id}.Inthiscase,wealsosaythat A isthe abstractrelation algebraof B.Anabstractrelationalgebrathathasarepresentationiscalled representable.For x,y ∈ A,wewrite x ≤ y asashortcutforthepartialorder definedby x ∨ y = y .Theminimalelementsof A \{0} withrespectto ≤ are calledthe atoms of A.Ineveryrepresentationof A,theatomsdenotethebasic relationsoftherepresentation.Wementionthatthereareabstractfiniterelation algebrasthatarenotrepresentable[Lyn50],andthatthequestionwhethera finiterelationalgebraisrepresentableisundecidable[HH01].
Example2. The(abstract)pointalgebraisarelationalgebrawith8elements and3atoms,=, <,and >,andcanbedescribedasfollows.Thevaluesofthe compositionoperatorfortheatomsofthepointalgebraareshowninthetable ofFig. 1.Notethatthistabledeterminesthefullcompositiontable.Theinverse (<) of < is >,andIddenotes=whichisitsowninverse.Thisfullydetermines therelationalgebra.Theproperrelationalgebrawithdomain Q presentedin Example 1 isarepresentationofthepointalgebra.
Fig.1. Thecompositiontableforthebasicrelationsinthepointalgebra.
3TheNetworkSatisfactionProblem
Let A beafiniterelationalgebrawithdomain A.An A-network N =(V ; f ) consistsofafinitesetofnodes V andafunction f : V × V → A.
Anetwork N iscalled – atomic iftheimageof f onlycontainsatomsof A andif f (a,c) ≤ f (a,b) ◦ f (b,c)forall a,b,c ∈ V (1) (herewefollowagainthedefinitionsin[Hir96]);
satisfiablein B,forarepresentation B of A,ifthereexistsamap s : V → B (where B denotesthedomainof B)suchthatforall x,y ∈ V (s(x),s(y )) ∈ f (x,y )B ;
satisfiable if N issatisfiableinsomerepresentation B of A.
The (general)networksatisfactionproblemforafiniterelationalgebra A is thecomputationalproblemtodecidewhetheragiven A-networkissatisfiable. Therearefiniterelationalgebras A wherethisproblemisundecidable[Hir99]. Arepresentation B of A iscalled – fullyuniversal ifeveryatomic A-networkissatisfiablein B; – square ifitsrelationsformaproperrelationalgebrathatissquare.
Thepointalgebraisanexampleofarelationalgebrawithafullyuniversalsquare representation.Notethatif A hasafullyuniversalrepresentation,thenthe networksatisfactionproblemfor A isdecidableinNP:foragivennetwork(V,f ), simplyselectforeach x ∈ V 2 anatom a ∈ A with a ≤ f (x),replace f (x)by a, andthenexhaustivelycheckcondition(1).Alsonotethatafiniterelationalgebra hasafullyuniversalrepresentationifandonlyiftheso-calledpath-consistency proceduredecidessatisfiabilityofatomic A-networks(see,e.g.,[BJ17, HLR13]). However,notallfiniterelationalgebrashaveafullyuniversalrepresentation. Anexampleofarelationalgebrawith4atomswhichhasarepresentationwith sevenelementsbutwherepathconsistencyofatomicnetworksdoesnotimply consistency,called B9 ,hasbeengivenin[LKRL08].Arepresentationof B9 withdomain {0, 1,..., 6} isgivenbythebasicrelations {R0 ,R1 ,R2 ,R3 } where Ri = {(x,y ) | x + y = i mod7},for i ∈{0, 1, 2, 3}.Infact,everyrepresentation of B9 isisomorphictothisrepresentation.Let N bethenetwork(V,f )with V = {a,b,c,d}, f (a,b)= f (c,d)= R3 , f (a,d)= f (b,c)= R2 , f (a,c)= f (b,d)= R1 , f (i,i)= R0 forall i ∈ V ,and f (i,j )= f (j,i)forall i,j ∈ V .Then N isatomic butnotsatisfiable.
4ConstraintSatisfactionProblems
Let B beastructurewitha(finiteorinfinite)domain B andafiniterelational signature ρ.Thenthe constraintsatisfactionproblemfor B isthecomputational problemofdecidingwhetherafinite ρ-structure E homomorphicallymapsto B Notethatif B isasquarerepresentationof A,thentheinput E canbeviewed asan A-network N .Thenodesof N aretheelementsof E.Todefine f (x,y )for variables x,y ofthenetwork,let a1 ,...,ak bealistofallelements a ∈ A such that(x,y ) ∈ aE .Thendefine f (x,y )=(a1 ∧···∧ ak );if k =0,then f (x,y )=1. Observethat E hasahomomorphismto B ifandonlyif N issatisfiablein B (hereweusetheassumptionthat B isasquarerepresentation).
Conversely,when N isan A-network,thenweview N asthe A-structure E whosedomainarethenodesof N ,andwhere(x,y ) ∈ r E ifandonlyif r = f (x,y ). Again, E hasahomomorphismto B ifandonlyif N issatisfiablein B.
Proposition1. Let B beafullyuniversalsquarerepresentationofafiniterelationalgebra A.Thenthenetworksatisfactionproblemfor A equalstheconstraint satisfactionproblemfor B (uptothetranslationbetween A-networksandfinite A-structurespresentedabove).
Proof. Wehavetoshowthatanetworkissatisfiableifandonlyifithasahomomorphismto B.Clearly,if N hasahomomorphismto B thenitissatisfiablein B,andhencesatisfiable.Fortheotherdirection,supposethatthe A-network N =(V,f )issatisfiableinsomerepresentationof A.Thenthereexistsforeach x ∈ V 2 anatomic a ∈ A suchthat a ≤ f (x)andsuchthatthenetwork N obtainedfrom N byreplacing f (x)by a satisfies(1);hence, N isatomicand satisfiablein B since B isfullyuniversal.Hence, N issatisfiablein B,too.
Forgeneralinfinitestructures B asystematicunderstandingofthecomputationalcomplexityofCSP(B)isahopelessendeavour[BG08].However,if B isa first-orderreductofafinitelyboundedhomogeneousstructure (thedefinitionscan befoundbelow),thentheuniversal-algebraictractabilityconjectureforfinitedomainCSPscanbegeneralised.Thisconditionissufficientlygeneralsothatit includesfullyuniversalsquarerepresentationsofalmostalltheconcretefinite relationalgebrasstudiedintheliterature,andtheconditionalsocapturesthe classoffinite-domainCSPs.Aswewillsee,theconceptsof finiteboundedness and homogeneity areconditionsthathavealreadybeenstudiedintherelation algebraliterature.
4.1FiniteBoundedness
Let ρ bearelationalsignature,andlet F beasetof ρ-structures.ThenForb(F ) denotestheclassofallfinite ρ-structures A suchthatnostructurein F embeds into A.Fora ρ-structure B wewriteAge(B)fortheclassofallfinite ρ-structures thatembedinto B.Wesaythat B is finitelybounded if B hasafiniterelational signatureandthereexistsafinitesetoffinite τ -structures F suchthatAge(B)= Forb(F ).Asimpleexampleofafinitelyboundedstructureis(Q; <).Itiseasy
toseethattheconstraintsatisfactionproblemofafinitelyboundedstructure B isinNP.
Proposition2. Let A beafiniterelationalgebrawithafullyuniversalsquare representation B.Then B isfinitelybounded.
Proof(Proofsketch). Besidessomeboundsofsizeatmosttwothatmakesure thattheatomicrelationspartition B 2 ,itsufficestoincludeappropriatethreeelementstructuresinto F thatcanbereadofffromthecompositiontable of A
4.2Homogeneity
Arelationalstructure B is homogeneous (or ultra-homogeneous [Hod97])ifevery isomorphismbetweenfinitesubstructuresof B canbeextendedtoanautomorphismof B.Asimpleexampleofahomogeneousstructureis(Q; <).
Arepresentationofafiniterelationalgebra A iscalled normal ifitissquare, fullyuniversal,andhomogeneous[Hir96].ThefollowingisanimmediateconsequenceofPropositions 1 and 2
Corollary1. Let A beafiniterelationalgebrawithanormalrepresentation B. Thenthenetworksatisfactionproblemfor A equalstheconstraintsatisfaction problemforafinitelyboundedhomogeneousstructure.
Aversatiletooltoconstructhomogeneousstructuresfromclassesoffinite structuresis amalgamation ` alaFra¨ıss´e.Wepresentitforthespecialcaseof relationalstructures ;thisisallthatisneededhere.An embedding of A into B isanisomorphismbetween A andasubstructureof B.An amalgamation diagram isapair(B1 , B2 )where B1 , B2 are τ -structuressuchthatthereexists asubstructure A ofboth B1 and B2 suchthatallcommonelementsof B1 and B2 areelementsof A.Wesaythat(B1 , B2 )isa 2-pointamalgamationdiagram if |B1 \ A| = |B2 \ A| =1.A τ -structure C isan amalgamof (B1 , B2 ) over A iffor i =1, 2thereareembeddings fi of Bi to C suchthat f1 (a)= f2 (a)for all a ∈ A.Inthecontextofrelationalgebras A,theamalgamationpropertycan alsobeformulatedwithatomic A-networks,inwhichcaseithasbeencalledthe patchworkproperty [HLR13];westickwiththemodel-theoreticterminologyhere sinceitisolderandwell-established.
Definition1. Anisomorphism-closedclass C of τ -structureshasthe amalgamationproperty ifeveryamalgamationdiagramofstructuresin C hasanamalgamin C .Aclassoffinite τ -structuresthatcontainsatmostcountablymany non-isomorphicstructures,hastheamalgamationproperty,andisclosedunder takinginducedsubstructuresandisomorphismsiscalledan amalgamationclass.
Notethatsinceweonlylookatrelationalstructureshere(andsinceweallow structurestohaveanemptydomain),theamalgamationpropertyof C implies the jointembeddingproperty(JEP) for C ,whichsaysthatforanytwostructures B1 , B2 ∈C thereexistsastructure C ∈C thatembedsboth B1 and B2
Theorem1(Fra¨ıss´ e [Fra54, Fra86];see [Hod97]). Let C beanamalgamation class.Thenthereisahomogeneousandatmostcountable τ -structure C whose ageequals C .Thestructure C isuniqueuptoisomorphism,andcalledthe Fra¨ıss´elimit of C .
Thefollowingisawell-knownexampleofafiniterelationalgebrawhichhas afullyuniversalsquarerepresentation,butnotanormalone.
Example3. The leftlinearpointalgebra (see[Hir97, D¨un05])isarelationalgebra withfouratoms,denotedby=, <, >,and |.Hereweimaginethat‘x<y ’signifies that x is earlierintimethan y .Theideaisthatateverypointintimethepast islinearlyordered;thefuture,however,isnotyetdeterminedandmightbranch intodifferentworlds;incomparabilityoftimepoints x and y isdenotedby x|y Wemightalsothinkof x<y as x istotheleftof y ifwedrawpointsinthe plane,andthismotivatesthename leftlinearpointalgebra.Thecomposition operatoronthosefourbasicrelationsisgiveninFig. 2.Theinverse(<) of < is >,Iddenotes=,and | isitsowninverse,andtherelationalgebraisuniquely givenbythisdata.Itiswellknown(fordetails,see[Bod04])thattheleftlinear pointalgebrahasafullyuniversalsquarerepresentation.Ontheotherhand,the networksdrawninFig. 3 showthefailureofamalgamation.
Fig.2. Thecompositiontableforthebasicrelationsintheleftlinearpointalgebra.
AnalgorithmtotestwhetherafiniterelationalgebrahasanormalrepresentationcanbefoundinSect. 6
4.3TheInfinite-DomainDichotomyConjecture
Theinfinite-domaindichotomyconjectureappliestoaclasswhichislargerthan theclassofhomogeneousfinitelyboundedstructures.Tointroducethisclasswe needtheconceptof first-orderreducts. Supposethattworelationalstructures A and B havethesamedomain,that thesignatureofastructure A isasubsetofthesignatureof B,andthat RA = RB forallcommonrelationsymbols R.Thenwecall A a reductof B,and B an expansionof A.Inotherwords, A isobtainedfrom B bydroppingsome oftherelations.A first-orderreductof B isareductoftheexpansionof B byallrelationsthatarefirst-orderdefinablein B.TheCSPforafirst-order reductofafinitelyboundedhomogeneousstructureisinNP(see[Bod12]). Anexampleofastructurewhichisnothomogeneous,butareductoffinitely
Fig.3. Exampleshowingthatatomicnetworksfortheleftlinearpointalgebradonot havetheamalgamationproperty.Adirectededgefrom x to y signifies x<y ,anda dashededgebetween x and y signifies x|y
boundedhomogeneousstructureistherepresentationoftheleft-linearpoint algebra(Example 3)givenin[Bod04].
Conjecture1(Infinite-domaindichotomyconjecture). Let B beafirst-order reductofafinitelyboundedhomogeneousstructure.ThenCSP(B)iseither inPorNP-complete.
Hence,theinfinite-domaindichotomyconjectureimpliestheRBCPforfinite relationalgebraswithanormalrepresentation.InSect. 5 wewillseeamore specificconjecturethatcharacterisestheNP-completecasesandthecasesthat areinP.
5TheInfinite-DomainTractabilityConjecture
Tostatetheinfinite-domaintractabilityconjecture,weneedacoupleofconcepts thataremostnaturallyintroducedfortheclassofall ω -categoricalstructures.A theoryiscalled ω -categorical ifallitscountablyinfinitemodelsareisomorphic. Astructureiscalled ω -categorical ifitsfirst-ordertheoryis ω -categorical.Note thatfinitestructuresare ω -categoricalsincetheirfirst-ordertheoriesdonot havecountablyinfinitemodels.Homogeneousstructures B withfiniterelational signatureare ω -categorical.Thisfollowsfromaveryusefulcharacterisationof ω -categoricitygivenbyEngeler,Svenonius,andRyll-Nardzewski(Theorem 2). Thesetofallautomorphismsof B isdenotedbyAut(B).The orbit ofa k -tuple (t1 ,...,tn )underAut(B)istheset {(a(t1 ),...,a(tn )) | a ∈ Aut(B)}.Orbitsof pairs(i.e.,2-tuples)arealsocalled orbitals
Theorem2(see [Hod97]). Acountablestructure B is ω -categoricalifandonly if Aut(B) hasonlyfinitelymanyorbitsof n-tuples,forall n ≥ 1.
ThefollowingisaneasyconsequenceofTheorem 2
Proposition3. First-orderreductsof ω -categoricalstructuresare ω -categorical.
First-orderreductsofhomogeneousstructures,ontheotherhand,neednot behomogeneous.Anexampleofan ω -categoricalstructurewhichisnothomogeneousisthe ω -categoricalrepresentationoftheleftlinearpointalgebragiven in[Bod04](seeExample 3).Notethatevery ω -categoricalstructure B,andmore generallyeverystructurewithfinitelymanyorbitals,givesrisetoafiniterelationalgebra,namelytherelationalgebraassociatedtotheunionsoforbitalsof B (see[BJ17]);werefertothisrelationalgebraasthe orbitalrelationalgebra of B
Wefirstpresentaconditionthatimpliesthatan ω -categoricalstructurehas anNP-hardconstraintsatisfactionproblem(Sect. 5.1).Thetractabilityconjecturesaysthateveryreductofafinitelyboundedhomogeneousstructurethat doesnotsatisfythisconditionisNP-complete.Wethenpresentanequivalent characterisationoftheconditionduetoBartoandPinsker(Sect. 5.2),andthen yetanotherconditionduetoBarto,Oprˇsal,andPinsker,whichwaslatershown tobeequivalent(Sect. 5.3).
5.1TheOriginalFormulationoftheConjecture
Let B bean ω -categoricalstructure.Then B iscalled
–a core ifallendomorphismsof B (i.e.,homomorphismsfrom B to B)are embeddings(i.e.,areinjectiveandalsopreservethecomplementofeachrelation).
– modelcomplete ifallself-embeddingsof B areelementary,i.e.,preserveall first-orderformulas.
Clearly,if B isarepresentationofafiniterelationalgebra A,then B isacore. However,notallrepresentationsoffiniterelationalgebrasaremodelcomplete. Asimpleexampleistheorbitalrelationalgebraofthestructure(Q+ 0 ; <)where Q+ 0 denotesthenon-negativerationals:itsrepresentationwithdomain Q+ 0 has self-embeddingsthatdonotpreservetheorbital {(0, 0)}
Let τ bearelationalsignature.A τ -formulaiscalled primitivepositive if itisoftheform ∃x1 ,...,xn (ψ1 ∧···∧ ψm )where ψi isoftheform y1 = y2 oroftheform R(y1 ,...,yk )for R ∈ τ ofarity k .Thevariables y1 ,...,yk can befreeorfrom x1 ,...,xn .Clearly,primitivepositiveformulasarepreservedby homomorphisms.
Theorem3([Bod07, BHM10]). Every ω -categoricalstructureishomomorphicallyequivalenttoamodel-completecore C,whichisuniqueuptoisomorphism, andagain ω -categorical.Allorbitsof k -tuplesareprimitivepositivedefinable in C
Let B and A bestructures,let D ⊆ B n ,andlet I : D → A beasurjection. Then I iscalleda primitivepositiveinterpretation ifthepre-imageunder I of A,oftheequalityrelation=A on A,andofallrelationsof A isprimitivepositive definablein A.Inthiscasewealsosaythat B interprets A primitivelypositively. Thecompletegraphwiththreevertices(butwithoutloops)isdenotedby K3
Theorem4([Bod08]). Let B bean ω -categoricalstructure.Ifthemodelcompletecoreof B hasanexpansionbyfinitelymanyconstantssothatthe resultingstructureinterprets K3 primitivelypositively,then CSP(B) isNP-hard.
Wecannowstatetheinfinite-domaintractabilityconjecture.
Conjecture2. Let B beafirst-orderreductofafinitelyboundedhomogeneous structure.If B doesnotsatisfytheconditionfromTheorem 4 thenCSP(B)is inP.
Thisconjecturehasbeenverifiedinnumerousspecialcases(see,forinstance, thearticles[BK08, BMPP16, KP17, BJP17, BMM18, BM16]),includingtheclass offinite-structures[Bul17, Zhu17].
5.2TheTheoremofBartoandPinsker
Thetractabilityconjecturehasafundamentallydifferent,butequivalentformulation:insteadofthe non-existence ofahardness-condition,werequirethe existence ofapolymorphismsatisfyingacertainidentity;theconceptofpolymorphismsisfundamentaltotheresolutionoftheFeder-Vardiconjecturein both[Bul17]and[Zhu17].
Definition2. A polymorphism ofastructure B isahomomorphismfrom Bk to B,forsome k ∈ N.Wewrite Pol(B) forthesetofallpolymorphismsof B.
Anoperation f : B 6 → B iscalled
Siggers ifitsatisfies
f (x,y,x,z,y,z )= f (z,z,y,y,x,x)
forall x,y,z ∈ B ; – pseudo-Siggersmodulo e1 ,e2 : B → B if
e1 (f (x,y,x,z,y,z ))= e2 (f (z,z,y,y,x,x))
forall x,y,z ∈ B .
Theorem5([BP16]). Let B bean ω -categoricalmodel-completecore.Then either – B canbeexpandedbyfinitelymanyconstantssothattheresultingstructure interprets K3 primitivelypositively,or – B hasapseudo-Siggerspolymorphismmoduloendomorphismsof B.
5.3TheWonderlandConjecture
Aweakerconditionthatimpliesthatan ω -categoricalstructurehasanNP-hard CSPhasbeenpresentedin[BOP17].Forreductsofhomogeneousstructures withfinitesignature,however,thetwoconditionsareequivalent[BKO+17]. Hence,weobtainanotherdifferentbutequivalentformulationofthetractability conjecture.Theadvantageofthenewformulationisthatitdoesnotrequirethat thestructureisamodel-completecore.
Let B beacountablestructure.Amap μ :Pol(B) → Pol(A)iscalled minorpreserving ifforevery f ∈ Pol(B)ofarity k andall k -aryprojections π1 ,...,πk wehave μ(f ) ◦ (π1 ,...,πk )= μ(f ◦ (π1 ,...,πk ))where ◦ denotescomposition offunctions.ThesetPol(B)isequippedwithanaturalcompleteultrametric d (see,e.g.,[BS16]).Todefine d,supposethat B = N.For f,g ∈ Pol(B)we define d(f,g )=1if f and g havedifferentarity;otherwise,ifboth f,g have arity k ∈ N,then d(f,g ):=2 min{n∈
Theorem6(of [BOP17]). Let B be ω -categorical.Supposethat Pol(B) hasa uniformlycontinuousminor-preservingmapto Pol(K3 ).Then CSP(B) isNPcomplete.
Wementionthatthereare ω -categoricalstructureswheretheconditionfrom Theorem 6 applies,butnottheconditionfromTheorem 4
Theorem7(of [BKO+17]). If B isareductofahomogeneousstructurewith finiterelationalsignature,thentheconditionsgiveninTheorem 4 andinTheorem 6 areequivalent.
6TestingtheExistenceofNormalRepresentations
Inthissectionwepresentanalgorithmthattestswhetheragivenfiniterelation algebrahasanormalrepresentation.Thisfollowsfromamodel-theoreticresult thatseemstobefolklore,namelythattestingtheamalgamationpropertyfora classofstructuresthathastheJEPandasignatureofmaximalaritytwowhich isgivenbyafinitesetofforbiddensubstructuresisdecidable.Wearenotaware ofaproofofthisintheliterature.
Theorem8. Thereisanalgorithmthatdecidesforagivenfiniterelationalgebra A whichhasafullyuniversalsquarerepresentationwhether A alsohasa normalrepresentation.
Proof. Firstobservethattheclass C ofallatomic A-networks,viewedas Astructures,hastheJEP:if N1 and N2 areatomicnetworks,thentheyaresatisfiablein B since B isfullyuniversal,andhenceembedinto B whenviewedas structures.Since B issquarethesubstructureof B inducedbytheunionofthe imagesof N1 and N2 isanatomicnetwork,too,anditembeds N1 and N2
Let k bethenumberofatomsof A.Itclearlysufficestoshowthefollowing claim,sincetheconditiongiventherecanbeeffectivelycheckedexhaustively.
Claim. C hastheAPifandonlyifall2-pointamalgamationdiagramsof sizeatmost k +2amalgamate.
Sosupposethat D =(B1 , B2 )isanamalgamationdiagramwithoutamalgam.Let B1 beamaximalsubstructureof B1 thatcontains B1 ∩ B2 suchthat (B1 , B2 )hasanamalgam.Let B2 beamaximalsubstructureof B2 thatcontains B1 ∩ B2 suchthat(B1 , B2 )hasanamalgam.Then Bi = Bi forsome i ∈{1, 2};let C1 beasubstructureof Bi thatextends Bi byoneelement,and let C2 := B3 i .Then(C1 , C2 )isa2-pointamalgamationdiagramwithoutan amalgam.Let C0 := C1 ∩ C2 .Let C1 \ C0 = {p} and C2 \ C0 = {q }.Foreach a ∈ A thereexistsanelement ra ∈ C0 suchthatthenetwork({r,p,q },f )with f (p,q )= a, f (p,r )= f B1 (p,r ), f (r,q )= f B2 (r,q )failstheatomicityproperty(1).Let C1 bethesubstructureof C1 inducedby {p}∪{ra | a ∈ A} and A1 bethesubstructureof C2 inducedby {q }∪{ra | a ∈ A}.Thentheamalgamation diagram(C1 , C2 )hasnoamalgam,andhassizeatmost k +2.
A: Finite relation algebras with a normal representation
B: Finite relation algebras with a fully universal square representation which is a reduct of
C: Finite relation algebras with an -categorical fully universal square representation (1) (2)
D: Finite relation algebras with a fully universal square representation
E: Finite representable relation algebras
Fig.4. Subclassesoffiniterepresentablerelationalgebras.Membershipofrelation algebrasfromDtotheinnermostboxAisdecidable(Theorem 8).Example 3 separates BoxAandBoxB.Thefiniterelationalgebrafrom[Hir99]separatesBoxDandE. BoxBfallsintothescopeoftheinfinite-domaintractabilityconjecture.BoxesCand Dmightalsofallintothescopeofthisconjecture(seeProblem(1)andProblem(2)).
7ConclusionandOpenProblems
Hirsch’sReallyBigComplexityProblem(RBCP)forfiniterelationalgebras remainsreallybig.However,thenetworksatisfactionproblemofeveryfinite
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