Allegories: decidability and graph homomorphisms

13 Pages·2017·0.91 MB·English

Allegories: decidability and graph homomorphisms
Damien Pous, Valeria Vignudelli
To cite this version:
Damien Pous, Valeria Vignudelli. Allegories: decidability and graph homomorphisms. Logic in Com-
puter Science, Jul 2018, Oxford, United Kingdom. ￿10.1145/3209108.3209172￿. ￿hal-01703906v2￿
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Allegories: decidability and graph homomorphisms∗
DamienPous ValeriaVignudelli
UnivLyon,CNRS,ENSdeLyon,UCBLyon1,LIP UnivLyon,CNRS,ENSdeLyon,UCBLyon1,LIP
Lyon,France Lyon,France
Abstract A key result about allegories is that an inequationu ≥ v is
AllegorieswereintroducedbyFreydandScedrov;theyformafrag- universallyvalidforbinaryrelations(i.e.,itholdsforallinstantia-
mentofTarski’scalculusofrelations.Weshowthattheirequational tionofitsvariableswithbinaryrelations)ifandonlyifthereisa
theoryisdecidablebycharacterisingitintermsofaspecificclass graphhomomorphismfromg(u)tog(v).Forinstance,theinequa-
ofgraphhomomorphisms. tiona·b ∩a·c◦ ≥a·(b ∩c◦)∩d canbeprovedbyexhibitingthe
Weactuallydosoforanextensionofallegorieswhichweprove followinghomomorphism:
tobeconservative:allegorieswithtop.Thismakesitpossibleto
a b
exploitacorrespondencebetweentermsandK4-freegraphs,for a
whichisomorphismswereknowntobefinitelyaxiomatisable. c
Keywords Allegories,Algebra,Graphs,Treewidth,Minors,De- a b
cidability,Homomorphisms c
d
1 Introduction
ThischaracterisationwassketchedbyFreydandScedrov[12,page
In the nineties, Freyd and Scedrov proposed the notion of alle- 208]andlaterprovedbyAndrékaandBredikhin[2,Theorem1].
gory [12],anaxiomatisationofcategorieswithsomeadditional It actually appeared earlier under a different and more general
structurepresentinthecategoryofrelations(withsetsasobjects form, to prove that the database problem of conjunctive queries
andbinaryrelationsasmorphisms).Allegoriesariseinregularcate- containmentisdecidable[5,Lemma13].
gories[18,ChapterA3];theywerealsoappliedtocircuitdesign[4]. Thischaracterisationhoweveronlyappliestorepresentablealle-
Weshowinthispaperthattheirequationaltheoryisdecidable. gories,thoseallegoriesthatareisomorphictoanalgebraofconcrete
Forgettingthecategoricalstructure,allegoriesformafiniteand binaryrelations.Indeed,theequationaltheoryofallegoriesisin-
purelyequationalaxiomatisationofthepositivecalculusofrela- completewithrespecttothesemodels.FreydandScedrovgivea
tions[3].Theirsyntaxisthefollowing: counter-example[12,p.210]:therearehomomorphismsthatcor-
u,v ::=a |u·v |u ∩v |u◦ |1 respondtoinequationsthatarenotderivablefromtheaxiomsof
allegories.Incompletenessalsofollowsfromageneralnegative
Letterarangesoverasetofvariables.Thefirstthreeoperations resultbyAndrékaandMikulás[3]:anyfinitefirst-orderaxiomati-
intuitivelydenoterelationalcomposition(·),intersection(∩)and sationmustbeincompletewhentheconsideredfragmentcontains
converse(_◦).Theconstant1correspondstotheidentityrelation. atleasttheoperationsofcomposition,intersectionandconverse.
Wecanassociatetoeachtermualabelled,directedgraphg(u) Whenlookingatcounter-examplestocompleteness,onecansee
withtwodesignatedverticesforinputandoutput.Inthisconstruc-
thattheproblemsalwaysarisefromhomomorphismsthatequate
tion,a isadirectededgelabelledwitha,·isseriescomposition morethantwoverticesatatime.Infact,FreydandScedrovsuggest
ofgraphs,obtainedbymergingtheoutputofthefirstgraphand
that“theequationschosenasthedefinitionofallegoryhappento
theinputofthesecondone,∩isparallelcomposition,merging bepreciselythosethataccountforallcontainmentsobtainableby
theinputsandtheoutputsofthetwographsrespectively,(_◦)ex- identifyingtheverticestwoatatime”[12,p.210].
changesinputandoutput,and1isthegraphwithnoedgeanda
Weobtaindecidabilitybyprovingthisclaim,whichhappensto
singlevertex.Forinstance,thegraphsofthetermsa·(b ∩c◦)∩d bemoredifficultthanexpected.Anattemptatexhibitingaproof
and1∩a·barethefollowingones: wasproposedinGutierrez’sdoctoraldissertation[16].However,
a b hisproofbuildsonalemmawhichhappenstobefalseandcannot
a befixed[1,13].Gutierrezalsoclaimeddecidabilityearlier,inashort
c b abstract[15]whereheproposesanalternativecharacterisation,but
d proofsarenotavailableandsomeofhisassertionsseemhighlynon-
∗Thispaperisthefullversionof[20](Proc.LiCS’18),withappendices.Thiswork trivialtoprove.(SeeAppendixE).Tothebestofourknowledge,
hasbeensupportedbytheEuropeanResearchCouncil(ERC)undertheEuropean theproblemisthuscurrentlyconsideredasopen.
Union’sHorizon2020programme(CoVeCe,grantagreementNo678157),andbythe
LABEXMILYON(ANR-10-LABX-0070)ofUniversitédeLyon,withintheprogram Thekeydifficultyisthatsomegraphscannotberepresentedbya
"Investissementsd’Avenir"(ANR-11-IDEX-0007)operatedbytheFrenchNational term.Thus,eventhougheveryhomomorphismcanbedecomposed
ResearchAgency(ANR).
intoasequenceofhomomorphismsequatingatmosttwoverticesat
LICS’18,July9–12,2018,Oxford,UnitedKingdom atime,thereisnoguaranteethattheintermediategraphsappearing
©2018Copyrightheldbytheowner/author(s).Publicationrightslicensedtothe insuchadecompositionarethegraphsofsometerms.Considerfor
AssociationforComputingMachinery.
instancethegraphsinFigure1.Therearehomomorphismsfrom
Thisistheauthor’sversionofthework.Itispostedhereforyourpersonaluse.Not
forredistribution.ThedefinitiveVersionofRecordwaspublishedinLICS’18:LICS theoutergraphstotheinnerone,obtainedbymergingvertices
’18:33rdAnnualACM/IEEESymposiumonLogicinComputerScience,July9–12,2018, depictedwiththesamesymbol.Whilethosethreegraphsaregraphs
Oxford,UnitedKingdom,https://doi.org/10.1145/3209108.3209172.
ofterms,weshallseethatonlythefirstonecanbedecomposedinto
1
LICS’18,July9–12,2018,Oxford,UnitedKingdom DamienPousandValeriaVignudelli
1≜ G·H ≜ G H
⇀ ↼̸
G
⊤≜ G ∩H ≜
H
Figure1.Validandinvalidhomomorphisms.(Theedgesofthe dom(G)≜ G G◦≜ G
graphinthemiddleshouldbelabelledwithdifferentlettersand
orientedarbitrarily;theorientationandlabellingoftheedgesof a≜ a
theoutergraphsarethendeterminedbythetwohomomorphisms.)
Figure2.Graphoperations.
asequencewhereallintermediategraphsaregraphsofterms(see
AppendixAformoredetails),andthuscorrespondstoaninequality
Definition1. AgraphisatupleG =⟨V,E,s,t,l,ι,o⟩,whereV is
provablefromallegoriesaxioms.
afinitesetofvertices,Eisafinitesetofedges,s,t :E→V aremaps
Weproceedintwosteps. indicatingthesourceandtargetofeachedge,l :E →Aisamap
1. Firstwesolvetheproblemforallegorieswithtop,thatis, indicatingthelabelofeachedge,andι,o ∈V arethedesignated
allegoriesextendedwithaneutralelementforintersection, vertices,respectivelycalledinputandoutput.
whosegraphisthedisconnectgraphwithnoedgesandtwo
vertices,inputandoutput.Doingsogivesusmoreflexibility: Definition2. AhomomorphismfromG =⟨V,E,s,t,l,ι,o⟩toG′=
therearemoregraphsthatcanberepresentedbyaterm(for ⟨V′,E′,s′,t′,l′,ι′,o′⟩isapairh=⟨f,д⟩offunctionsf :V →V′
instance,thedisconnectedones),andthereisaclearcharac- andд:E→E′thatrespectthevariouscomponents:s′◦д= f ◦s,
terisationoftheclassofgraphsofterms:theyareprecisely t′◦д= f ◦t,l =l′◦д,ι′= f(ι),ando′= f(o).
thegraphsoftreewidthatmosttwo,orequivalently,the Asurjective(resp.injective)homomorphismisahomomorphism
graphsexcludingK4 asaminor.Thismovealsomakesit whosetwocomponentsaresurjective(resp.injective)functions.A
possibletoexploitarecentaxiomatisationofisomorphisms (graph)isomorphismisasurjectiveandinjectivehomomorphism
onsuchgraphs[7]:weshowthatthecorrespondingaxioms whosetwocomponentsarebijectivefunctions.WewriteG ≃G′
arederivableinallegorieswithtop(Proposition19),andwe whenthereexistsanisomorphismbetweengraphsGandG′.
canthenreasonmoduloisomorphisms.Thislatterpossibility
Weconsiderthefollowingsignaturesfortermsandalgebras:
iscrucialinmostofourproofs.
2. Thenweprovethatallegorieswithtopareaconservative Σ=(cid:8)·2,∩2,_◦1,10(cid:9) Σ⊤=Σ∪{⊤0} Σdom=Σ∪{dom1}
extensionofallegories:everyequationoverthesignatureof
Weusuallyomitthe·symbolandweassignprioritiessothatthe
allegoriesthatholdsinallallegorieswithtopactuallyholds
term(a·(b◦))∩ccanbewrittenjustasab◦∩c.
inallallegories.Wedosousingmodel-theoreticmeans,by
Graphsformalgebrasforthosesignaturesbyconsideringthe
showinghowtoembedanygivenallegoryintoanallegory
operationsdepictedinFigure2,whereinputsandoutputsarerepre-
withtop(Proposition44).Wesolveinpassingaproblem
sentedbyunlabelledingoingandoutgoingarrows.Theoperations
thatwasleftopenin[7]:wegiveafiniteaxiomatisationof
composition (·) and intersection (∩) respectively correspond to
isomorphismsforconnectedK4-freegraphs. seriesandparallelcomposition,converse(_◦)justexchangesinput
Outlineandcontributions Wefirstrecallthecorrespondence andoutput,anddomain(dom(_))relocatestheoutputtotheinput.
betweentermsandK4-freegraphs[7](Section2)andsetuptools Byinterpretingalettera∈AasthegraphafromFigure2,one
toextracttermsfromgraphs(Section3).Thenwedefineallegories canthusassociateagraphg(u)toeverytermovertheconsidered
withtopandweprovelawsthatarerequiredinthesequel(Sec- signaturesandwithvariablesinA.
tion4).Section5isdevotedtoourmaincontribution:therewe Observethatintersectingagraphwith1amountstomerging
characterisetheinequationaltheoryofallegoriesintermsofse- itsinputanditsoutput.Asaconsequence,thedomainoperation
quencesofappropriatehomomorphisms(Theorem16).Thischarac- isderivableinthesignatureΣ⊤thankstotheisomorphismbelow.
terisationleadstodecidability(Section6)andtoanotionofnormal Intuitively,relocatingtheoutputtotheinputcanbeimplemented
form(Section7).Wefinallyproceedwiththeconservativityre- byfirstdisconnectingtheoutput(bymultiplicationwith⊤onthe
sults(Section8),whichmakeitpossibletoliftourcharacterisation right),andthenmergingitwiththeinput(byintersectionwith1).
anddecidabilityprooftopureallegories,andtoprovideafinite
dom(G)≃1∩G⊤
axiomatisationofisomorphismsforconnectedK4-freegraphs.
Theconservativityresultsandtheequationalproofsneededin Accordingly,wewillusethefollowingshorthandwhenworking
thepaperhavebeenformallyverifiedusingtheCoqproofassistant. withΣ⊤-terms:dom(u)≜1∩u⊤.
Thedevelopmentcanbedownloadedandbrowsedonline[21]. Therearegraphswhicharenotthegraphofanyterm.Forin-
stance,thisisthecaseforthefollowinggraphs,whatevertheori-
2 Termsandgraphs entationandlabellingoftheiredges.
We let a,b... range over the letters of a fixed alphabet A. We
(1)
considerlabelleddirectedgraphswithtwodesignatedvertices.We
justcallthemgraphsinthesequel.Notethatweallowmultiple
Wenowrecallsomestandardgraphtheorynotions,tostatethe
edgesbetweentwovertices,aswellasself-loops.
characterisationofthegraphsofΣ⊤-termsfrom[7].
2
Allegories:decidabilityandgraphhomomorphisms LICS’18,July9–12,2018,Oxford,UnitedKingdom
Asimplegraphisanunlabelledundirectedgraphwithatmost
oneedgebetweentwoverticesandwithoutself-loops.Weusestan- 1∩a(bd∩e)(c∩c′)◦ dom(a(bd∩e)∩c∩c′) b d
wdaerddennootatetiobnyaknjdatperomteinntoilaolgeydfgreombegtwraepehnthtweooryve[r9t]i.cIenspkaartnidcujl;aar, 11∩∩((aa((bbdd∩∩ee))∩∩cc)′c)c′◦◦ dom(a∩(c∩c′)(d◦b◦∩e◦)) a e c,c′
kj-pathisa(possiblytrivial)pathwhoseendsarekandj;G+kjis
thesimplegraphobtainedfromGbyaddingtheedgekjifkandj
werenotalreadyadjacent. Figure3.Differenttermsdenotingthesameconnectedgraph.
Definition3. AminorofasimplegraphGisasimplegraphob-
tainedfromG byasequenceofthefollowingoperations:delete Thegraphsof1,a,a◦,ab∩c,1∩a,and1∩ab anddom(a)areall
anedgeoravertex,contractanedge(i.e.,deleteitandmergeits prime.Thegraphsofab,a(b∩c),1∩a∩bc anddom((1∩a)(b∩c◦))
endpoints).AsimplegraphisH-freeifH isnotoneofitsminors. arenot,thelattertwobeingthegraphof(1∩a)(1∩bc).Aprimeis
eitherapetaloraneye.Petalscanbecharacterisedasfollows.
RobertsonandSeymour’sgraphminortheorem[22],statesthat
(simple)graphsarewell-quasi-orderedbytheminorrelation.Asa Lemma7. AtestGisapetalifeither
consequence,theclassesofgraphsofboundedtreewidth[9],which • G ≃1,orG ≃1∩aforsomelettera,or
areclosedundertakingminors,canbecharacterisedbyfinitesets • G has no self-loop on its input, is connected, and remains
ofexcludedminors.Twosimpleandstandardinstancesarethe connectedwhenremovingtheinput.
followingones:thegraphsoftreewidthatmostone(theforests)
Asexpected,everyconnectedgraphcanbedecomposedasa
arepreciselythoseexcludingthecyclewiththreevertices(C3); seriescompositionofprimes.Thiscantypicallybedepictedasfol-
thoseoftreewidthatmosttwoarethoseexcludingthecomplete
lows,whereeyesaregreenandpetalsareyellow.Thefourdepicted
graphwithfourvertices(K4)[10]. verticesarecalledcheckpoints:theymustbevisitedbyany(undi-
rected)pathfromtheinputtotheoutput.Propercheckpointsare
(C3) (K4) thosedifferentfrominputandoutput.
Definition4. TheskeletonofagraphGisthesimplegraphSob-
tainedfromGbyforgettinginput,output,labelling,edgedirections,
edgemultiplicities,andself-loops.ThestrongskeletonofGisS+ιo
Thisdecompositionisnotuniquehowever:therecanbesuperfluous
ifι(cid:44)o,andSotherwise.
occurrencesof1,andtheorderinwhichcontiguouspetalsappear
Asanexample,K4 isthestrongskeletonofallinstancesofthe doesnotmatter.IfthestartinggraphbelongstoTW2,thensodo
graphsin(1). itsprimecomponents;thisallowsonetoproceedrecursively.
Thefollowingpropositionmakesitpossibletodecomposenon-
Proposition5([7,Corollary26]). LetGbeagraph.Thefollowing
trivialeyes.Thisisaconsequenceof[7,Proposition21(i)].
areequivalent.
1. ThereexistsaΣ⊤-termusuchthatG ≃g(u). Proposition8. LetG ∈TW2beaneye.EitherGconsistsofasingle
2. ThestrongskeletonofGhastreewidthatmosttwo. edge,orthereareconnectedgraphsG1,G2 ∈TW2s.t.G ≃G1∩G2.
3. ThestrongskeletonofGisK4-free. WewriteG[k;j]forthegraphGwithinputandoutputrespectively
Inthesequel,wewriteTW2forthesetofgraphssatisfyingthese settokandj.AsillustratedinFigure3,therecanbeseveralways
conditions.Theresultsfrom[7]alsoentailthatProposition5adapts ofextractingatermfromatest.Weshallmostlyusethefollowing
toconnectedgraphsjustbyrestrictingtoΣdom-terms.(Σ-terms observation,toresorttothecasewhereinputandoutputdiffer:
alonearenotenough,considerforinstancethegraphdom(a).) Observation9. LetGbeatestandletkbeavertexofG.Wehave
G ≃dom(G[ι;k]).
3 Parsinggraphs
Toextractatermusingthisobservation,onemusthowevermake
Manydifferenttermscandenotethesamegraph.Firstbecauseof
associativity,commutativity,andneutralelements.Butalso,and surethatG[ι;k]belongstoTW2.WhenGisalreadyknowntobe
moreimportantly,becauseofgraphswhoseinputandoutputare inTW2,onecanusefork anyneighbouroftheinput:G[ι;k]is
equal.ConsiderforinstancethegraphinFigure3,whichisthe necessarilyinTW2insuchacase,sinceitsstrongskeletonisthe
graphofthefivetermsgivenontheleft.Notethatthetermsinthe sameasthatofG.Thisishowthetwotermsinthesecondcolumn
ofFigure3areextracted.Otheroptionsareoftenpossible,consider
secondcolumndonotexistinthesyntaxofpureallegories(the
forinstancethefollowinggraph:
signatureΣ):weneedeitherthedomainoperationoritsencoding
through the constant ⊤. The ability to write such terms in the a b
syntaxofallegorieswithtopiscrucialinSection5.
Wenowproveafewresultsthatallowustoextracttermsfrom Choosingtheneighbouroftheinputyieldsdom(adom(b)),while
agivengraphinTW2.Wefirstfocusonconnectedgraphs. choosingtheothervertexyieldsdom(ab).Conversely,someoptions
areforbidden:thetopmostvertexinthegraphofFigure3cannot
Definition6(Primes,tests,petals,eyes). AgraphGisprimeifit
isconnectedandforallgraphsG1,G2,G ≃G1·G2entailsG1≃1 beDchisocsoennn,ethcteesdtrgornagphsskeclaentobneopfatrhseedreassuflotilnlogwgsr:aphbeingK4.
orG2≃1.Agraphisatestifitsinputisequaltoitsoutput.Apetal
isaprimetest.Aneyeisaprimewithdistinctinputandoutput. Proposition10. LetG ∈TW2beadisconnectedgraph.
3
LICS’18,July9–12,2018,Oxford,UnitedKingdom DamienPousandValeriaVignudelli
u ∩u =u (A0) u v
v u
u ∩(v ∩w)=(u ∩v)∩w (A1) u w u
w
u ∩v =v ∩u (A2)
u ∩⊤=u (A3) u v v u
w w
u·(v·w)=(u·v)·w (A4)
u·1=u (A5) Figure5.SurjectivehomomorphismsforAxioms(AD)and(AM).
u◦◦=u (A6)
(u ∩v)◦=u◦∩v◦ (A7)
(u·v)◦=v◦·u◦ (A8) Axiom(AD),semi-distributivity,isequivalenttomonotonicity
uv ∩uw ≥u(v ∩w) (AD) ofcompositionontheright,andthusalsoontheleftbyduality,so
(v ∩wu◦)u ≥vu ∩w (AM) thatalloperationsaremonotoneintheend.
Axiom(AM)iscalledmodularidentity.Thisistheonlyunusual
axiom.Itgeneralisesthenotionofmodularityforlattices[8].Its
Figure4.Axiomsofallegorieswithtop(All⊤). dualisthefollowinglaw:
u(v ∩u◦w)≥uv ∩w (AM’)
1. IfGhasaconnectedcomponentH whichcontainsneitherthe
inputnortheoutput,thenthegraphG′obtainedbyremoving Asymmetricalconsequenceofmodularidentityisthefollowing
HfromGbelongstoTW2andforeveryvertexkinH,wehave inequation,knownasDedekindlaw:
H[k;k]∈TW2andG ≃G′∩⊤H[k;k]⊤.
2. Otherwise,GhasexactlytwoconnectedcomponentsG′and (v ∩wu◦)(u ∩v◦w)≥vu ∩w (DD)
G′′respectivelycontainingtheinputandtheoutput,wehave
G′[ι;ι],G′′[o;o]∈TW2andG ≃G′[ι;ι]⊤G′′[o;o]. Theaxiomsofallegoriesaresoundwithrespecttobinaryrela-
tions.Asaconsequenceofthecharacterisationofrepresentable
Proof. ItsufficestoshowthatthecomputedgraphsareinTW2.This allegoriesmentionedintheIntroduction,wehave:
followsfromtheobservationthattheirstrongskeletonisalwaysa
subgraphofthestrongskeletonofG,andthefactthattheclassof Proposition13. If⊢All⊤u ≥vthenthereexistsahomomorphism
K4-freegraphsisclosedundertakingsubgraphs. □ fromg(u)tog(v).
Remark11. Theoccurrencesoflettersinatermareinonetoone Axioms(A1-A8)actuallyallcorrespondtographisomorphisms.
correspondencewiththeedgesofitsgraph.Asaconsequence,ifC[] Idempotency(A0)correspondstoaninjectivehomomorphismfrom
isatermcontextsothatC[a]isatermwithadesignatedoccurrence righttoleft,andtoasurjectivehomomorphismfromlefttoright;
ofthelettera,thengivenatermu,thegraphofC[u]isobtained Axioms (AD) and (AM) correspond to the surjective homomor-
fromthegraphofC[a]byreplacingtheedgecorrespondingtothe phismsdepictedinFigure5.Notethatthosethreefamiliesofsur-
selectedoccurrenceofawiththegraphofu. jectivehomomorphismsmayequatearbitrarilymanyvertices:the
graphofumighthavemanyverticesinadditiontoinputandoutput.
4 Allegories (ThisissuewillbeaddressedinSection5.4.)
AsaconsequenceofProposition13,wehave⊢ 1≥uifand
Definition12. AnallegorywithtopisaΣ⊤-algebrasatisfyingthe onlyifg(u)isatest.Wethuscalltests thetermAslsl⊤atisfyingthis
axioms in Figure 4, where an inequation of the formu ≥ v is
condition.Equivalently,atestisatermthatisprovablyequalto
ashorthandfortheequationu ∩ v = v.Given Σ⊤-termswith sometermoftheshape1 ∩v.FollowingnotationsfromKleene
variablesinA,wewrite⊢ u =vwhenthisequationisderivable
All⊤ algebrawithtests(KAT)[19],weletα,β rangeovertests.
from the axioms in Figure 4 (equivalently, when it holds in all
allegorieswithtop,forallinterpretationofthevariables).Similarly Recallthatdom(u)≜1∩u⊤isaderivedoperationinallegories
withtop.Suchanoperationisnotdefinableinpureallegories,but
forinequations.
AnallegoryisaΣ-algebrasatisfyingthoseaxiomsbut(A3).We wecandefineasimilaroperationbysettingdom′(u) ≜ 1 ∩uu◦.
Bothdomainoperationsaretestsbydefinition,butthegraphsof
define⊢ u =vaccordingly.
All dom(u)anddom′(u)arenotisomorphic;theyaredepictedbelow.
Axioms(A0-A8)capturethemostnaturalpropertiesoftheoper-
ators:intersectionisidempotent,associative,commutative,andhas
u u u
⊤asaneutralelement(sothatthederivedrelation≥isapartial
orderwith⊤asmaximumelementandintersectionasmeet);com-
Theseoperationsarehoweverinterchangeableinallegorieswith
positionisassociativeandhasneutralelement1;converseisan
involutionthatreversescompositions.Theyentail1◦=1,⊤◦=⊤, top:wewillprovetheequationbelowafterProposition14.
monotonicityofintersectionandconverse,andanotionofduality:
everystatementwhichholdsuniversallyalsoholdswhenreversing ⊢All⊤ dom(u)=dom′(u) (2)
allcompositions.Weusesuchlawsfreelyinthesequel.
4
Allegories:decidabilityandgraphhomomorphisms LICS’18,July9–12,2018,Oxford,UnitedKingdom
Proposition14. Thefollowinglawsarederivableinallegories.
1∩1=1 (A9)
dom′(u ∩v)=1∩uv◦ (3) dom(u ∩v)=1∩u·v◦ (A10)
α(v ∩w)=αv ∩w (4) u·⊤=dom(u)·⊤ (A11)
αβ =α ∩β =βα (5) (1∩u)·v =(1∩u)·⊤∩v (A12)
α =α◦=αα =dom′(α) (6)
dom′(uv)=dom′(udom′(v)) (7) Figure6.Axiomsfor2p-algebras(with(A1-A8)).
Proof. Weprove(3)bydoubleinclusion.1∩uv◦ ≥ dom′(u ∩v)
followsbymonotonicityfromu ≥u ∩v andv◦ ≥u◦ ∩v◦.For TheclassTW2isclosedundersubgraphs,soG (cid:44)→HandH ∈TW2
theotherdirection,weuseDedekindlaw(DD)withw =1: impliesG ∈ TW2.SincewerequireG andH tobeK4-freewhen
1∩(u ∩v)(u◦∩v◦)≥1∩(uv◦∩1)=1∩uv◦ G(No(cid:7)teHt,hwatewahlseonhha:vGet→hatHGis⇀asHurajencdtiHve∈hoTmWom2iomrpphliiessmGeq∈uaTtWin2g.
ForEquation(4),wehave exactlytwovertices,thenneitherG ∈TW2impliesH ∈TW2nor
theconverse.SeeAppendixB.)
α(v ∩w)≥α(v ∩α◦w) (1≥α◦)
Relation⇀isapreorderand⇌isanequivalencerelation.The
≥αv ∩w (by(AM’)) remainderofthissectionisdevotedtoshowingthattherelation
≥αv ∩αw (1≥α) ⇌issoundandcompletew.r.t.provabilityinAll⊤:
≥α(v ∩w) (by(AD)) Theorem16. Wehave⊢ u =vifandonlyifд(u)⇌д(v).
All⊤
Itsufficestoprovethefirstequationin(5),whichfollowsfrom Westartwiththebackwardimplication,forwhichitsufficesto
idempotencyand(4):αβ =α(1∩β)=α1∩β =α ∩β. show thatд(u) (cid:44)→ д(v) entails ⊢ u ≥ v (Section 5.2) and
Forthefirstequationin(6),wehave д(u) д(v)entails⊢ u ≥v(SecAtilol⊤n5.3).Forbothimplications,
All⊤
(1∩u)◦=1∩1u◦=dom′(1∩u) (by(3)) acru(cid:7)cialpreliminarystepconsistsindealingwithisomorphisms
(Section5.1).Thenweprovetheforwardimplication(Section5.4).
=dom′(u ∩1)=1∩u1◦=1∩u (by(3))
5.1 Isomorphisms
Theotherequationsin(6)followusing(5).Weget(7)asfollows.
dom′(udom′(v))≥dom′(udom′(u◦uv ∩v)) Definition17([7,Section3]). A2p-algebraisaΣ⊤-algebrasatis-
fyingtheaxioms(A1)-(A8)fromFigure4andtheaxioms(A9)-(A12)
=dom′(u(1∩u◦uvv◦)) (by(3)) inFigure6.Wewrite⊢2p u =vwhentwotermsuandvarecon-
≥dom′(u ∩uvv◦) (by(AM)) gruentmodulothoseaxioms,orequivalently,whentheequation
=1∩uvv◦u◦=dom′(uv) (by(3)) holdsinall2p-algebras.
≥1∩u(1∩vv◦)u◦ Notethatidempotency(A0)isnotincludedintheaxiomsof2p-
=1∩u(1∩vv◦)(1∩vv◦)◦u◦ (by(6)) algebra.TW2isthefree2p-algebra;inparticular,wehave
=dom′(udom′(v)) □ Theorem18([7,Corollary34]). Wehave⊢2pu =viffg(u)≃g(v).
Wenowobservethateveryallegorywithtopisa2p-algebra.
Theabovelawsalsoholdinallegorieswithtop.Equation(2)follows
byinstantiatingvwith⊤in(3). Proposition19. If⊢2pu =vthen⊢All⊤u =v.
Proof. It suffices to prove axioms (A9)-(A12) from Figure 6. Ax-
5 Graphtheoreticalcharacterisation
iom(A9)isatrivialinstanceofidempotency.Axiom(A10)follows
WedefineaclassofhomomorphismsonTW2graphs,thatwewill from(3)and(2),andAxiom(A12)from(4).ForAxiom(A11)we
provetocharacteriseinequationsinAll⊤.Suchhomomorphisms, haveu⊤≥1∩u⊤andby(AM):
denotedby⇀,arethosethatcanbedecomposedasasequence
ofhomomorphismswhosesourceandtargetarebothinTW2and u⊤=1⊤∩u⊤≤(1∩u⊤⊤◦)⊤=dom(u)⊤ □
equateatmosttwovertices.Thisisthecaseforthefirsthomo-
Corollary20. Ifg(u)≃g(v)then⊢ u =v.
morphisminFigure1(onemustmergetheblackcirclesfirst,see All⊤
AppendixA),butnotforthesecondone:merginganytwover- Thisresultisfundamentalforthefollowingproofs:itallowsusto
ticeswiththesameshapeinthegraphontherightyieldsagraph reasonuptoisomorphisms,andtofreelychoosethewaywewant
containingK4asaminor.) toreadagivengraph.Recallforinstancethefivetermsdenoting
WeletR∗denotethereflexive-transitiveclosureofarelationR. thesamegraphinFigure3;thankstotheabovecorollary,weknow
thatthosefivetermsareprovablyequalinallegorieswithtop,so
Definition15. Definethefollowingrelationsongraphs: thatwecanfreelyreplaceonebytheother.
• G H ifG,H ∈ TW2 andthereisasurjectivehomomor-
phi(cid:7)smh:G →H suchthathcollapsesatmosttwovertices; 5.2 Injectivehomomorphisms
• G (cid:44)→H ifthereisaninjectivehomomorphismh:G →H; WriteG (cid:44)→v Hifthereexistsaninjectivehomomorphismwhichis
• G ⇀H ifG ( ∪(cid:44)→)∗H; bijectiveonedgesandaddsexactlyonevertex(i.e.,suchthatthere
• G ⇌H when(cid:7)G ⇀H andH ⇀G. isexactlyonevertexthatisnotinitsrange),andG (cid:44)→e H ifthere
5
LICS’18,July9–12,2018,Oxford,UnitedKingdom DamienPousandValeriaVignudelli
existsaninjectivehomomorphismwhichisbijectiveonvertices k ∈g(w)andk′∈g(α).Byisomorphism,wecanassume
andaddsexactlyoneedge(idem).Wehave thatk′istheinputofg(α).Theng(u)≃g(C[1∩⊤α⊤]),
(cid:44)→=(cid:44)→∗(cid:44)→∗ where1correspondstovertexk,andg(v)≃g(C[1∩α]).
v e Theresultfollowsfrom⊢ ⊤α⊤≥α.
Itthussufficestoshowthat(cid:44)→v and(cid:44)→e yieldproofsinallegories b. g(u)hasexactlytwoconnAelcl⊤tedcomponentsrespectively
withtop.Weexploittheresultaboutisomorphismstodoso. containingtheinputandtheoutput,andthereareαandβ
Proposition21. Ifg(u)(cid:44)→v g(v)then⊢All⊤u ≥v. scuocnhcltuhdaetgb(yu)in≃dugc(αti⊤onβ)i.fLtihkeecinoltlhaepsperdevvioeurtsicceassek,,wke′caarne
Proof. Observethatg(u)(cid:44)→v g(v)entailsg(u ∩⊤⊤)≃g(v),and eitherbothing(α)orbothing(β).Supposek ∈g(α)and
thus⊢All⊤u ∩⊤⊤=vbyCorollary20.Wefinallyget k′ ∈ g(β).Byhypothesis,collapsingk andk′givesusa
All⊤⊢u =u ∩⊤≥u ∩⊤⊤=v . □ gorfagp(αh)ga(vn)dikn′TisWc2o.nSnineccteevdetrotetxhekoisuctpountnoefctge(dβ)to,wtheedienrpivuet
Proposition22. Ifg(u)(cid:44)→e g(v)then⊢All⊤u ≥v. thatgraphsg(α)[ι,k]andg(β)[k′,o]areinTW2.Hence,
thereexistw,x s.t.g(α)[ι,k] ≃ g(w),g(β)[k′,o] ≃ g(x).
Proof. Supposetheaddededgeislabelledbya,andwritev =C[a] Wehaveg(u)≃g(w⊤x)andg(v)≃g(wx),andtheresult
byselectingthecorrespondingoccurrenceofainv(Remark11). followsby⊢ w⊤x ≥wx.
Wehaveд(u)≃д(C[⊤]),so⊢All⊤u =C[⊤]byCorollary20.Weget 2. g(u) is a connAelclt⊤ed test. Since g(u) has at least two ver-
All⊤⊢u =C[⊤]≥C[a]=vbymonotonicityofalloperations. □ tices,thereissomevertexk adjacenttotheinput,butdif-
NotethatinadditiontoCorollary20,wearemakingacrucialuse ferentfromit.Thenthereissometermwsuchthatg(u)≃
ofthepresenceof⊤inthesyntaxintheabovetwoproofs.We g(dom(w))andg(u)[ι;k]≃g(w).Theexistenceofahomo-
couldgetridofitwhenworkingwithconnectedgraphs,butthis morphismfromg(u)tog(v)impliesthatg(v)isatestas
requires convoluted arguments (for instance, we can no longer well.Moreover,bythedefinitionofhomomorphism,h(k)is
handleverticesandedgesseparatelyandinanarbitraryorder). eitheradjacenttotheinputing(v)ortheinputitself(incase
theverticesidentifiedbythehomomorphismareexactlyk
5.3 Surjectivehomomorphisms andtheinputofg(v)).Inbothcases,thereisatermx such
Likeaboveforinjectivehomomorphisms,fortwographsG,H ∈ thatg(v) ≃ g(dom(x))andg(v)[ι;h(k)] ≃ g(x).Thefunc-
TW2 writeG v H if there exists a surjective homomorphism tionhisstillasurjectivehomomorphismfromg(w)tog(x)
whichisbijecti(cid:7)veonedgesandequatesexactlytwovertices,and collapsingtwovertices,sincetheonlydifferencebetween
G e Hifthereexistsasurjectivehomomorphismwhichisbijective g(dom(w))andg(w)isthattheoutputhasbeenrelocatedto
on(cid:7)verticesandequatesexactlytwoedges.Wehave k,andanalogouslytheonlydifferencebetweeng(dom(x))
= = ∗ andg(x)isthattheoutputhasbeenrelocatedtoh(k).The
v e graphg(w)hasthesamenumberofverticesandedgesas
(cid:7) (cid:7) (cid:7)
(Where v=isthereflexiveclosureof v.)Wenowshowthat v g(u),buthasinputdifferentfromoutput,sowecanapply
and e y(cid:7)ieldproofsinallegorieswithto(cid:7)p.Thisiseasyforthelat(cid:7)ter, theinductivehypothesistog(w)andderive⊢All⊤ w ≥ x.
butt(cid:7)heformerrelationrequiresamuchdeeperanalysis. Hence,⊢ dom(w)≥dom(x).
All⊤
3. g(u)isaconnectedgraphwithinputdifferentfromoutput,
Proposition23. Ifg(u)(cid:7)e g(v)then⊢All⊤u ≥v. andisnotprime,i.e.,thereareu1,u2bothnotequivalentto1
Proof. Theonlydifferencebetweeng(u)andg(v)isthatthereare suchthatg(u)=g(u1)·g(u2).Letk,k′betheverticesmerged
twoparalleledgeswiththesamelabelaing(u)thatarereplaced bythehomomorphism.Ifeitherk,k′arebothing(u1)or
byasingleedgeaing(v).Letv′bethetermobtainedfromv by theyarebothing(u2)(possiblyincludingthecasewhenone
replacingbya∩atheoccurrenceofacorrespondingtothissingle ofk,k′isthecheckpointbetweeng(u1)andg(u2)ing(u)),
edgeing(v).Wehave⊢ v =v′byidempotency.Nowobserve wecanapplytheinductivehypothesis.Otherwise,k isin
thatg(u)≃g(v′),sothaAtl⊢l⊤ u =v′byCorollary20. □ g(u1)andk′ ising(u2),andneitherofthemisthecheck-
All⊤ pointbetweeng(u1)andg(u2).AsdiscussedinSection3,
Proposition24. Ifg(u) v g(v)then⊢All⊤u ≥v. g(u)canbedecomposedasasequenceofprimecomponents
(cid:7)
Proof. Lethbethesurjectivehomomorphismfromg(u)tog(v)col- g(u1)·····g(un),whereeachcomponentiseitherapetalor
lapsingexactlytwovertices.Weprovethestatementbyinduction aneye.W.l.o.g.,wecanassumethatk isinthefirstprime
on|g(u)|,where|G|isthelexicographicproductof: component,andthatk′isinthelastprimecomponent(mod-
ulothepresenceofpetalsequivalentto1).Ifitwerenotthe
• thenumberofedgesandverticesofG, case,theinductivehypothesiscouldbeappliedasabove.We
• 1ifGisatest,and0otherwise. considerdifferentcasesdependingonwhetherk,k′areina
Weproceedbycasesonthestructureofg(u). petalorinaneye.
1. g(u)isadisconnectedgraph.ThenbyProposition10we a. Bothkandk′areinpetals.Sincekandk′canbemerged,
havetwocases: andsincek′isconnectedtotheoutput,theoutputofthe
a. g(u)hasaconnectedcomponentwhichcontainsneither firstpetalcanberelocatedtok.Analogously,theinput
theinputnortheoutput.Hence,therearewandα such ofthelastpetalcanberelocatedtok′.Thenthereare
thatg(u) ≃ g(w ∩⊤α⊤).Letk,k′bethetwocollapsed w,x,zs.t.g(u)≃g(w⊤x ∩z),asrepresentedbelow,with
vertices.Ifk,k′ areeitherbothing(w)orbothing(α), ktheoutputofg(w)andk′theinputofg(x),andg(v)≃
wederivetheresultbytheinductivehypothesis.Suppose g(wx ∩z).Weconcludeby⊢ ⊤≥1.
All⊤
6
Allegories:decidabilityandgraphhomomorphisms LICS’18,July9–12,2018,Oxford,UnitedKingdom
• • andbycollapsingkandk′wewouldhavethatg(v)
hasK4asaminor.
b. Bothkandk′areinaneye.Then(thestrongskeletonof) • k′isstrictlyinaneyeofg(w)havingasoutputor
g(u)hasthefollowinggraphasaminor(theloweredge inputapropercheckpointofg(w).Theng(u)has
asaminoroneofthefollowinggraphs:
beingtheoneaddedbythedefinitionofstrongskeleton).
k k′ k′ k′
k k
WethusgetK4bycollapsingkandk′,whichcontradicts and,asinthepreviouscase,bycollapsingkandk′
thefactthatg(v)isthegraphofaterm. weobtainK4.
c. kisinapetalandk′isinaeye.Weconsidertwocases: 4. g(u)isaneyenotreducedtoanedge.Theng(u)≃g(u1)∩
i. Supposethepetalandtheeyearecontiguous,i.e.,that g(u2)withbothg(u1)andg(u2)containingatleastoneedge.
therearenoprimecomponentsbetweenthem(modulo Ifk,k′areinthesameparallelcomponentg(ui),thenwe
thepresenceof1).Thenthepetalisnotequivalentto canapplytheinductivehypothesis.Otherwise,supposethat
1,otherwisekandk′wouldbeinthesamecomponent. theyarerespectivelying(u1)andg(u2).Wecanassumethat
Wehaveg(u)≃g(αw ∩x),withk ∈g(α)andg(w ∩x) bothg(u1)andg(u2)haveatleastonepropercheckpoint,
theeyecontainingk′,withk′strictlyinsideg(w)and sinceotherwisetherewouldbeawaytodecomposeg(u)
g(x)containingatleastoneedge(thisdecomposition intoparallelcomponentssuchthatk,k′areinthesameone.
ofaneyealwaysexistsbyProposition8). Wehavethreecases:
•
•
• •
Thenwehaveg(v)≃g(z∩x)withg(αw) v g(z)and •
weconcludebytheinductivehypothesis. (cid:7)
ii. Supposethepetalandtheeyearenotcontiguous.If • k,k′ are respectively proper checkpoints of g(u1) and
thereareonlypetals(ofwhichatleastonenotequiv- g(u2).Thentherearew,w′,x,x′s.t.g(u)≃g(ww′∩xx′)
alentto1)betweenthem,wecanmovethembyiso- andg(v)≃g((w ∩x)(w′∩x′)),andweuseAxiom(AD).
morphismbeforethepetalcontainingk,andapplythe • kisstrictlyinthepetalofapropercheckpointofg(u)(or
inductivehypothesis.Hence,supposethereisatleast symmetricallyfork′).Theng(u)hasasaminor
oneeyebetweenthefirstandlastprimecomponentof k
g(u).Wegetdifferentcasesdependingonwherekand
k′arerespectivelylocatedinthepetalandintheeye. k′
A. Ifk doesnotcoincidewiththeinput,theng(u)has andbycollapsingkandk′weobtainK4.
asaminor • kisstrictlyinaneyeofg(u1)havingasoutputorinputa
k k′ propercheckpointofg(u1)(orsymmetricallyfork′).Then
g(u)hasasaminoroneofthefollowinggraphs
andbycollapsingkandk′wewouldhavethatg(v)
k k
hasK4asaminor,whichisacontradiction.
B. kistheinputandtheeyecontainingk′hastwopar- k′ k′
allelcomponentsw,z,fork′∈g(w)andg(w)withat and,asabove,bycollapsingkandk′weobtainK4.
leastonepropercheckpoint.Wehavethreecases: 5. Itremainstoconsiderthecasewheng(u)isaneyereduced
• toanedge.Theng(u)≃g(a),g(v)≃g(a∩1)andtheresult
• followsby⊢All⊤u ≥u ∩1. □
CombiningCorollary20andPropositions21,22,23,and24,we
• k′ is a proper checkpoint of g(w). Then g(u) ≃ obtainthattherelation⇀issoundforallegorieswithtop:
g(x(w1w2∩z))withk theinputofx andwithk′ Theorem25. Ifg(u)⇀g(v)then⊢ u ≥v.
theoutputofw1,forg(w)≃g(w1w2).Bycollapsing All⊤
k,k′weobtaing(v)≃g((x ∩w1◦)z∩w2)andwe 5.4 Completeness
deriveinAll⊤: FortheconverseofTheorem25,weintroduceanintermediatein-
equationalpresentationofallegorieswithtop.Thisideawasalready
x(w1w2∩z)≥(x ∩w1◦)((x ∩w1◦)◦w2∩z) sketchedin[12];wegivedetailshereforthesakeofcompleteness.
≥(x ∩w1◦)z∩w2 (by(AM’)) TheinequationaltheoryAll⊤≥isgeneratedbytheaxioms(A1)-
(A8)inFigure4,whereu =visaderivedoperatordefinedasu ≤v
andv ≤u,andtheaxiomsinFigure7.Atomicidempotencyisanax-
• k′isstrictlyinapetalofg(w)withinputaproper
iomschemeparameterisedbylettersa∈A.Thepointofthissystem
checkpointofg(w).Theng(u)hasasaminor
isthatitsaxiomsallcorrespondtosimplegraphhomomorphisms,
k′ thatcaneasilybeseentobelongtotherelation⇀:
k
Lemma26. Forallaxiomsu ≥vofAll≥,wehaveg(u)⇀g(v).
⊤
7
LICS’18,July9–12,2018,Oxford,UnitedKingdom DamienPousandValeriaVignudelli
6 Decidability
u ≥u ∩v (leftinequality)
Wenowshowthattherelation⇀isdecidable,andthatthereisa
u ≥v ∩u (rightinequality)
notionofnormalformforallegorieswithtop.Thekeyobservation
1∩1≥1 (1idempotency) isthatsurjectivehomomorphismscanalwaysbeappliedfirst.
a∩a ≥a (atomicidempotency,foralla∈A)
Lemma31. Wehavethefollowinginclusion:(cid:44)→ ⊆ (cid:44)→.
u·v ∩x·w ≥(u ∩x)·(v ∩w) (separatedsemi-distributivity) (cid:7) (cid:7)
(u ∩w·x◦)·v ≥u·(v ∩x)∩w (separatedmodularity) Proof. Leti :G (cid:44)→G′andh:G′ H.Thegraphhi(G)isK4-free,
beingasubgraphofH,andthefu(cid:7)nctionд:G →hi(G),definedas
hi,isasurjectivehomomorphismcollapsingatmosttwovertices.
Figure7.Inequationalpresentationofallegorieswithtop(All⊤≥). Indeed,functionд istriviallysurjective,andeitherthevertices
collapsedbyharebothini(G),inwhichcaseдcollapsesthem,orat
leastoneofthecollapsedverticesisnotini(G),whichimpliesthat
Proof. Axioms(A1)-(A8)fromFigure4aswellasidempotencyfor1 дisanisomorphism.Therefore,wehaveG hi(G).Sincehi(G)
correspondtographisomorphisms.Inequalityaxiomscorrespond injectsinH,weconcludethatG (cid:44)→H. (cid:7) □
to injective homomorphisms, unlessv is a test andu is not. In (cid:7)
suchacase,e.g.,forleftinequality,wehaveg(u) v g(u)∩1(cid:44)→ Asaconsequence,weobtainthefollowingcharacterisation,which
g(u ∩v).Atomicidempotencycorrespondsto e.(cid:7)Separatedsemi- givesdecidability:
distributivityandseparatedmodularitycorresp(cid:7)ondeitherto or
to≃.E.g.,forseparatedsemi-distributivitywehave≃ifboth(cid:7)uvand Proposition32. WehaveG ⇀H iffG ∗(cid:44)→H.
(cid:7)
xaretestsorbothvandwaretests,andwehave v otherwise. □ Corollary33. Therelation⇀isdecidable.
(cid:7)
Notethatunrestrictedidempotency,semi-distributivityandmodu- Proof. ItsufficestodecidewhetherG ∗(cid:44)→H.Wehavethatthe
laridentity(Axioms(A0),(AD)and(AM)inFigure4)couldnotbe sets{G′ |G ∗G′}and{H′ |H′(cid:44)→H(cid:7)}arefiniteandcomputable
handledinsuchawaysincetheycorrespondtohomomorphisms (uptoisomo(cid:7)rphism);itsufficestotestwhethertheyintersect. □
potentiallyequatingmanyverticesandedges.Thattheyarenever-
thelesscapturedbytherelation⇀isobtainedonlyaposteriori. Onecanactuallygetanon-deterministicpolynomialalgorithm:
Byfurthershowingthat⇀is‘closedundercontexts’,wededuce guessasequenceG0,...,Gn ofgraphsobtainedfromG =G0by
thatthesystemAll≥issoundfor⇀. mergingtwoverticesatatime,checkthatthesegraphsbelong
⊤ toTW2,computethegraphH′obtainedfromGn bymergingall
Lemma27. Forallterm-contextsC,wehave paralleledgeswiththesamelabel(sothatGn e∗ H′)andcheck
1. ifg(u) g(v)theng(C[u]) g(C[v]); thatH′(cid:44)→H.Thelattertestcanbedoneinpoly(cid:7)nomialtimeonce
2. ifg(u)(cid:44)(cid:7)→g(v)theng(C[u])(cid:7)(cid:44)→g(C[v]). Gn andthusH′areknowntohaveboundedtreewidth[6,11,14].
Corollary34. TheequationaltheoryofAll⊤isinNP.
Proof. ByinductiononC.Forthefirstitem,whenC =[·]∩w,if
themergedverticesaretheinputandoutputofuandifwisatest 7 Normalforms
thenwehaveg(C[u])≃g(C[v]),whichisaspecialcaseof . □
Whenstudyinghomomorphismequivalenceongraphs,oneoften
(cid:7)
usesthenotionofcore,thosegraphswhereeveryendomorphismis
Theorem28. If⊢All≥ u ≥vtheng(u)⇀g(v). anisomorphism.Everygraphhasacore,whichisaminimalgraph
⊤
initsequivalenceclassmodulohomomorphismequivalence[17].
ItremainstoshowthattheinequationalpresentationAll⊤≥iscom- Onedefinesasimilarnotionhereforallegories,usingourrestricted
pleteforallegorieswithtop.Westartbyprovingthat(unrestricted)
formofhomomorphismequivalence(⇌).
idempotencyisderivableinAll⊤≥: ThenormalformofagraphG,writtennf(G)isagraphwhichis
minimalw.r.t.thenumberofverticesandedgesinitsequivalence
Lemma29. Foralltermsu,wehave⊢All⊤≥ u ∩u =u. classmodulo⇌.Normalformsareuniqueuptoisomorphism:
Proof. Weprove⊢All≥ u ∩u ≥ubyinductiononu.(Theconverse Proposition35. WehaveG ⇌H iff nf(G)≃nf(H).
⊤
inequalitytriviallyholdsbytheinequalityaxioms).Thebasecases
Definethefollowing(computable)relation:
aregivenbytheaxiomsofAll≥.Forseriescompositionwehave
⊤
⊢All≥ uv ∩uv ≥(u ∩u)(v ∩v)byseparatedsemi-distributivity, G (cid:123)H ≜ G ∗H (cid:44)→G
and⊤thenweconcludebytheinductivehypothesis. □ (cid:7)
Proposition36. Therelation (cid:123) isaconfluentandterminating
Theorem30. If⊢All⊤u ≥vthen⊢All⊤≥ u ≥v preorderandforallgraphsGwehaveG (cid:123)nf(G).
Proof. That (cid:123) isapreorderfollowsfrom ∗ and(cid:44)→beingpre-
Proof. Itsufficestoderivemodularityandsemi-distributivityfrom orders.Terminationfollowsfromtheobserva(cid:7)tionthatthesizeofa
theirseparatedversions;thisfollowsfromLemma29. □ graphdecreasesalong (onemustofcourseforbidtrivialsteps).
Confluenceisprovedb(cid:7)yinductiononthesumofthesizesofthe
CombiningTheorems25,28,and30wefinallyobtainTheorem16. consideredgraphs,usingProposition32(seeAppendixC). □
8
Allegories:decidabilityandgraphhomomorphisms LICS’18,July9–12,2018,Oxford,UnitedKingdom
Gutierrezprovesasimilarresultinthecontextoffinitecategories
1∩1=1 (A9)
withanepi-monofactorisationsystem[16,Chapter4.2].Onecannot
reusehisresultdirectly:theinjective(resp.surjective)homomor- dom(u ∩v)=1∩u·v◦ (A10)
phismsweuseherearenotexactlythemono(resp.epi)morphisms dom(u·v)=dom(u·dom(v)) (A13)
ofthenaturalcategoryassociatedtotherelation⇀. dom(u)·(v ∩w)=dom(u)·v ∩w (A14)
To illustrate this rewriting system, consider the four graphs
depictedbelow.Wehighlightpairsofverticesthatcanpotentially
bemergedbyrepresentingthemwiththesamesymbol,usingthe Figure8.Axiomsfor2pdom-algebras(with(A1,A2,A4-A8)).
sameconventionasinFigure1forlabelsandorientationofedges.
G4isthenormalformofthefourgraphs,andtherearethreeways Graphsforma2pdom-algebra:theaxiomsaresound.Toprove
ofreachingitfromG1:directly,orbygoingthroughG2orG3. thattheyarecompleteforgraphisomorphisms,werelyonTheo-
rem18andweprovethatevery2pdomalgebracanbeembedded
=G1 (cid:123) G2= ina2palgebra.Combinedwiththeotherresultsfrom[7],thisalso
yieldsthatconnectedK4-freegraphsformthefree2pdom-algebra.
(cid:123) (cid:123) WefixaΣdom-algebra⟨X,·,∩,_◦,dom(_),1⟩intheremainder
=G3 (cid:123) G4= ofthissection.WewriteT forthesetoftestsinX.Weconstruct
Inthisexample,everyattempttocollapsetwoverticesresultsina thefollowingΣ⊤-algebra:
graphinTW2.Thisisnotalwaysthecase,asshownbelow. Definition38. LetX bethesetX ⊎T2.Foru ∈X,wewriteufor
uasanelementofX.Forα,β ∈T,wewriteα⋄βforthepair⟨α,β⟩
asanelementofX.WeturnX intoaΣ⊤-algebrabysetting:
u·v ≜u·v u ∩v ≜u ∩v
H1 H2 H3 (α⋄β)·(γ⋄δ)≜α⋄δ (α⋄β)∩(γ⋄δ)≜αγ⋄βδ
(cid:7) (cid:7)
H3isthenormalformofthosethreegraphs.Theuniquehomomor- (α⋄β)·v ≜α⋄dom(v◦β) (α⋄β)∩v ≜αvβ
phismfromH1toH3canbefactorisedthroughthegraphH2 ∈TW2
obtainedbyfirstmergingtheverticesdepictedwithtriangles.Since u·(γ⋄δ)≜dom(uγ)⋄δ u ∩(γ⋄δ)≜γuδ
Hno3t(cid:44)H→1 (cid:123)H1,Hw2e,sdinedceucHe2Hd1oe(cid:123)snoHt3e.mWbeedalsinoHha1v.eIfHin2st(cid:123)eadHw3e, btruyt (α⋄β)◦≜β⋄α 1≜1
tocollapsefirsttheverticesdepictedwithblackcirclesinH1,we u◦≜u◦ ⊤≜1⋄1
obtainagraphthatdoesnotbelongtoTW2.
WhenX isthealgebraofconnectedgraphs,X intuitivelyrepre-
8 Conservativityarguments sentgraphswhereallverticesareconnectedeithertotheinputor
totheoutput:anelementudenotesaconnectedgraph,whilean
Wenowshowthattheresultspresentedintheprevioussections
elementα⋄β denotesthedisconnectedgraphα⊤β.
extendtopureallegories(withouttop).Wedosobyprovingthat
Whencomposingtwo‘disconnectedelements’α⋄β andγ⋄δ
All⊤isaconservativeextensionofAll,i.e.,thatforalltermsu,v inseries,wethrowawayacomponentthatshouldintuitivelybe
inthesyntaxofallegories(i.e.,forall⊤-freeterms),⊢All⊤ u =v createdandwhichisnotconnectedtotheinputortotheoutput:
ifandonlyif⊢ u =v.Consequently,theequationaltheoryof
All βγ.ThismeansthatX cannotbethefree2p-algebra:whateverthe
allegoriesisdecidableandTheorem16alsoholdsforAll.
starting2pdom-algebraX,X alwayssatisfiesthelaw⊤u⊤=⊤.
Weproveconservativitybyshowingthateveryallegoryem-
Notehoweverthatthefunctionmappinganelementu ∈X tou
bedsintoanallegorywithtop(Proposition44).Ithappensthatwe
canfactorisethisconstructiontoshowinpassinghowtohandle isaninjectiveΣ-homomorphismfromX toX.
isomorphismsofconnectedK4-freegraphs.Thisiswhatwedofirst. Lemma39. In2pdom,anelementuisatestiffdom(u)=u.
Proofsinthissectionaremostlyequationalandofteninvolve
manycases.Wepresentsketchestogiveintuitions;moredetails Proposition40. IfX isa2pdom-algebrathenX isa2p-algebra.
canbeenfoundinAppendixD;fullproofsformalisedinCoqcan Proof. WemustshowthatX satisfiesallthe2paxioms(Figure6).
bebrowsedonline[21]. Considertheassociativityofproduct(A4).Eachofthethreevari-
ablesoccurringinthisaxiomcanbeeitherinX orinT2.Theproof
8.1 IsomorphismsofconnectedgraphsinTW2 istrivialwhenallelementsareinX.Weshowtworelevantcases.
Asexplainedin[7],connectedK4-freegraphscorrespondtoterms ((α⋄β)·u)·(γ⋄δ)=(α⋄dom(u◦β))·(γ⋄δ)
overthesignatureΣdom,where⊤isnolongerpresent,anddom()
becomesaprimitiveoperation.Itwashoweverleftopenwhether =α⋄δ
isomorphismsofsuchgraphscouldbefinitelyaxiomatisedover (α⋄β)·(u·(γ⋄δ))=(α⋄β)·(dom(uγ)⋄δ)
thissyntax.Weanswerthisquestionbytheaffirmative.
((α⋄β)·u)·v =(α⋄dom(u◦β))·v
Definition37. A2pdom-algebraisaΣdom-algebrasatisfyingthe =α⋄dom(v◦dom(u◦β))
axioms(A1-A8)fromFigure4except(A3),andtheaxiomsinFig-
ure8.Wewrite⊢2pdomu =vwhentwotermsuandvarecongru- =α⋄dom(v◦u◦β) (by(A13))
entmodulothoseaxioms. (α⋄β)·(u·v)=(α⋄β)·uv □
9