Generating clause sequences of a CNF formula Theoretical Computer Science

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Theoretical Computer Science

www.elsevier.com/locate/tcs

Generating clause sequences of a CNF formula

Kristóf Bérczi

^a

^,∗ , Endre Boros

^b

, Ondˇrej ˇCepek

^c

, Khaled Elbassioni

^d

, Petr Kuˇcera

^c

, Kazuhisa Makino

^e

aMTA-ELTEEgerváryResearchGroup,DepartmentofOperationsResearch,EötvösLorándUniversity,Budapest,Hungary bMSISDepartmentandRUTCOR,RutgersUniversity,NJ,USA

cCharlesUniversity,FacultyofMathematicsandPhysics,DepartmentofTheoreticalComputerScienceandMathematicalLogic,Praha,Czech Republic

dEECSDepartment,KhalifaUniversityofScienceandTechnology,AbuDhabi,UnitedArabEmirates eResearchInstituteforMathematicalSciences(RIMS),KyotoUniversity,Kyoto,Japan

a rt i c l e i nf o a b s t ra c t

Articlehistory:

Received16February2020

Receivedinrevisedform23November2020 Accepted7December2020

Availableonline14December2020 CommunicatedbyL.M.Kirousis

Keywords:

CNFformulas Clausesequences Enumeration Generation

Given aCNFformula withclauses C1,. . . ,Cm and variables V= {^x¹,. . . ,xn}^,^a^truth assignmenta:^V→ {⁰,1} ^ofleadstoaclausesequence

σ

(a)=(C1(a),. . . ,Cm(a))∈ {⁰,1}^m^where^Ci(a)=^{1 if}^clause^Cievaluatesto1 underassignmenta,otherwiseCi(a)=^0.

The setofallpossible clausesequencescarriesalotofinformationontheformula,e.g.

SAT, MAX-SATandMIN-SATcan beencodedintermsofﬁndingaclausesequencewith extremalproperties.

WeconsideraproblemposedatDagstuhlSeminar19211“EnumerationinDataManage- ment”(2019)about thegenerationofallpossible clausesequencesofagivenCNFwith bounded dimension. We prove that the problem can be solved in incrementalpolyno- mialtime. Wefurthergiveanalgorithmwithpolynomialdelayfortheclassoftractable CNFformulas.Wealsoconsiderthegenerationofmaximalandminimalclausesequences, andshowthatgeneratingmaximalclausesequencesisNP-hard,whileminimalclausese- quencescanbegeneratedwithpolynomialdelay.

©²⁰²⁰^TheÂuthor(s).^Published^byÊlsevier^B.V.^Thisîsânôpenâccessârticleûnder^the CCBYlicense(http://creativecommons.org/licenses/by/4.0/).

1. Introduction

Theconceptofwell-designedpatterntreeswasintroduced byLetelieretal.[1] asaconvenientgraphicrepresentationof conjunctive queries extendedbytheoptionaloperator.Thenodesofsucha treecorrespond tothequeries,whilethetree itselfrepresentstheoptionalextensions.Well-designedpatterntreeshavebeenstudiedfromacomplexitypointofviewin severalaspects.One ofthemostinteresting problemsinthecontext ofquerylanguagesisthegenerationproblem,that is, generatingthesolutionsoneaftertheotherwithoutrepetition.

Previouswork The generation problemwas studied forFirst-Order andConjunctive Queries [2–5] andfor well-designed patterntrees[1].Recently,Krölletal.[6] initiatedasystematicstudyofthecomplexity ofthegenerationproblemofwell- designed pattern trees. Theyidentiﬁed several tractable and intractable cases of the problem both from a classical and fromaparameterizedcomplexitypointofview.Oneclassofpatterntreeshoweverremainedunclassiﬁed.ForaclassC of

*

Correspondingauthor.

E-mailaddresses:berkri@cs.elte.hu(K. Bérczi),endre.boros@rutgers.edu(E. Boros),cepek@ktiml.mff.cuni.cz(O. ˇCepek),khaled.elbassioni@ku.ac.ae (K. Elbassioni),kucerap@ktiml.mff.cuni.cz(P. Kuˇcera),makino@kurims.kyoto.ac.jp(K. Makino).

https://doi.org/10.1016/j.tcs.2020.12.021

0304-3975/©²⁰²⁰^TheÂuthor(s).^Published^byÊlsevier^B.V.^Thisîsânôpenâccessârticleûnder^the^CC^BY^license (http://creativecommons.org/licenses/by/4.0/).

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conjunctivequeries,awell-designedpatterntreeT isgloballyinCifforeverysubtreeTofT thecorrespondingconjunctive queryisalsoinC^.^The^treewidth^of^aconjunctivequeryisthetreewidthofitsGaifman-graph[7].In[6],thecomplexityof thegeneration problemfortheclassofwell-designedpatterntreesfalling globally intheclass ofqueriesoftreewidth at mostkandhavingc-semi-boundedinterfacewasleftopen(see [6,Table1onpage16]).

AttheDagstuhlSeminar19211“EnumerationinDataManagement”,Kröllproposedanopenproblemonthegeneration of clause sequences of CNF formulas [8, Problem 4.7]. The problem is motivated by the fact that it can be reduced to the above mentioned unsolved caseof patterntrees, thus anybound on the generation complexity would be helpful in understandingthegeneralproblem.Agenerationalgorithmoutputstheobjectsinquestiononebyonewithoutrepetition.We callitapolynomialdelayprocedureifthecomputingtimebetweenanytwoconsecutiveoutputsisboundedbyapolynomial oftheinputsize.Wecallitincrementallypolynomial,ifforanyktheﬁrstkobjectscanbegeneratedinpolynomialtime in theinputsizeandk.Finally,itiscalledtotalpolynomial ifall N objectsaregeneratedinpolynomialtime intheinputsize andN.

Theproblemstudiedinthispapercanbeformalizedasfollows.LetV = {^x1

, . . . ,

xn}^be^a^set^ofⁿ^Boolean^variables.^We denoteby ¯xj=1−xj thenegation ofvariable xj.Variables andtheir negations together are calledliterals.A clauseisan elementarydisjunctionofliteralsandaCNFisaconjunctionofclauses.Weviewclausesalsoassubsetsoftheliterals,and CNFsassetsofclauses.GivenaCNF

=^C1∧ · · · ∧^Cmandanassignmenta:^V → {⁰

,

1}^,^thecorrespondingbinarysequence

σ

(

a

)

=

(

C1

(

a

), . . . ,

Cm

(

a

))

iscalled a signature¹ of

,that is, Ci

(

a

)

=^{1 if} ^clause ^Ci evaluates to 1 under assignment a, andCi

(

a

)

=0 otherwise.Inparticular, thismeansthat

issatisﬁableifandonlyifthereexistssome assignmenta with

σ

(

a

)

=

(

1

, . . . ,

1

)

.Moreover,MAX-SATandMIN-SATcanbeencodedbyaskingforasignaturewiththelargestandsmallest sumofelements,respectively.

As an example,consider theCNF formula

=^C1∧^C2∧^C3∧^C4,where C₁=^x1∨ ¯^x3,C₂= ¯^x2, C₃=^x1∨^x2∨^x3 and C₄=^x2∨ ¯^x3. Then assignment a₁= {^x1→¹

,

x₂→¹

,

x₃→¹} ^leads ^to ^signature

σ

(

a₁

)

=

(

1

,

0

,

1

,

1

)

, while assignment a2= {^x1→⁰

,

x2→⁰

,

x3→¹}^leads^to^signature

σ

(

a2

)

=

(

0

,

1

,

1

,

0

)

.Itiseasytoseethat

hassixdifferentsignatures.In general,ifthenumberofsignaturesis

(

2ⁿ

)

,thengeneratingthemintotalpolynomialtimeisnotdiﬃcult.However,their numbermaybeo

(

2ⁿ

)

,presentingapotentialchallengeforgeneration.

Given a CNF

=^C1∧ · · · ∧^Cm, the number of literals in clause Ci is denoted by |^Ci|^. ^We ^denote ^by ^dim

()

= max_i₌₁_,...,_m|^Ci|^,^where |^Ci| ^denotes^the ^number^of^literals^of ^Ci.Wecall

a d-CNF ifdim

()

≤^d.^The^number^of^clauses andthenumberofliteralsappearingin

are denotedby ||^and^,respectively.The occurrenceofavariableina CNF isthenumberofclausesinvolvingthatvariableoritsnegation.Vectorsarewrittenusingboldfontsthroughout,e.g. x.The problemaskedin[8] isford-CNFformulaswheredisaﬁxedpositiveinteger,butwe alsoconsiderthesameproblemfor generalCNFs.

MotivatedbyMAX-SATandMIN-SAT,wealsoconsidermaximalandminimalsignatures.AsignatureofaCNF

iscalled maximal(resp.minimal)ifaninclusionwisemaximal(resp.minimal)subsetoftheclausestakesvalue1.

Ourresults Wegiveapolynomialdelay algorithmfortheclassoftractableCNFformulasinSection 2.Section 3discusses CNFswithboundeddimension.Fortheclassofformulaswithboundeddimensionandvariableoccurrences,wegiveanin- crementalpolynomialalgorithminSection3.1.InSection3.2,weshowthatG S

()

canbesolvedinincrementalpolynomial timeforformulaswithaboundeddimension,thusansweringtheopenproblemposedbyKröll.Thegenerationofmaximal andminimalsignaturesisconsideredinSection4.Finally,weconcludethepaperinSection5,wherea‘reversed’variantof theproblemisproposedasanopenquestion.

2. TractableCNFs

Let

and

betwoCNFs.Iftheclausesof

arealsoclausesin

,then

iscalledasub-CNF of

,denotedby

⊆

. Wecalla familyofCNFstractableifforanyCNF

inthisfamilythesatisﬁability ofanysub-CNFof

canbedecided in polynomialtimeevenafterﬁxinganysubsetofthevariablesatarbitraryvalues.Forexample,theclassesof2-CNFsorHorn CNFsaretractable.

Theorem1.If

belongstoatractablefamilyandhasm clauses,thenitssignaturescanbegeneratedwithpolynomialdelay.

1 WepreferthetermsignatureoverthetermclausesequenceproposedbyKröll,sinceitisabinarystring,notasequenceofclauses.Thereforeweuse thetermsignatureintherestofthepaper.

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Proof. The idea is to apply the so-called‘ﬂashlight’ approach in the signature space, using SAT as a ‘ﬂashlight’ [9]. Let

=m

i=¹C_i.We aregoing tobuilda binarytreeinwhichthepaths fromthe roottothe verticesofthetreecorrespond tobinary valuesofinitialsegments ofthesetofclauses,that is,C1

, . . . ,

Ck forsome 1≤^k≤^m.^The^treeîs^built^layer^by layer,eachnewlayercorrespondingtoanewclausebeingaddedtotheexaminedprefix.Thereexistsasignaturewiththis prefixifandonlyiftheCNFformedbytheclauses settovalue oneinthissequenceissatisfiableevenafteralltheforced fixingofvariablesthatappearinclauseswhosevalueiszero(notethataclausehasvalue 0 ifandonlyifalltheliteralsin itare0). IfsuchaCNFisnotsatisfiable,we backtrackanddonot explorethesubtreerootedatthisvertexasthereexists no signaturewiththisprefix.IftheCNF issatisfiable,we continuebuildingthe correspondingsubtree whichinthiscase isguaranteedtocontainatleastonesignature.Thealgorithmwillnotbacktrackabovethisvertexbeforeoutputting all(at least one)signaturesin thissubtree.It isnot difficult toverify thatafter atmost2m calls to SAT we canoutput a new signaturenotgeneratedbefore.Afteroutputtingthelastsignature,theprocedureterminatesafteratmostmSATcalls.

Bytheabove,thesignaturesof

canbegeneratedwithadelayof O

(

m

)

SAT-calls.As

belongstoatractable family, thisimpliesapolynomialdelayalgorithm,concludingtheproofofthetheorem.

Remark2.LetusremarkthatthefamilyofmonotoneCNFsistractable,butforthiscasethereisamoreeﬃcientpolynomial delaygenerationofthesignatures.Indeed,inthiscasewecanviewaclauseasasubsetofthevariables.Consequently,the set of zeros in a signature corresponds to a union ofclauses. It is easy to see that such unions can be generated with O

(

nm

)

delay andO

(

n

)

averagedelay,wherenisthenumberofvariables,andm= ||^is^the^number^of^clauses, ^see^[10, Proposition11] and[11,Theorem15].

NotethatinthiscaseTheorem1guaranteesonlyan O

(

^m

)

delay,becauseeverySATcallrequiresO

( )

time.

3. CNFswithboundeddimension 3.1. Boundedvariableoccurrence

GivenaCNF

,we denoteby H=

(,

E

)

theconﬂictgraph of

.Theverticesof H are theclauses of

andedges areexactlytheconﬂictingpairsofclauses,i.e.,pairs

(

Ci

,

Cj

)

forwhichthereexistsaliteralu∈^Ci suchthatu¯∈^Cj.

Let S⊆

be a maximal independent set of H, andlet L

(

S

)

=

Ci∈SC_i denote the setof literals appearing in the clauses of S.Wedeﬁne apartial assignmentaS:^L

(

S

)

→ {⁰

,

1}^by^setting^all ^literals^of^L

(

S

)

tozero(andhencethecom- plementaryliteralsaresetto 1).The signatureassociatedtoS isthendeﬁnedas

σ

(

S

)

:=

σ

(

aS

)

=

(

y1

, . . . ,

ym

)

∈ {⁰

,

1}^m^. The coordinatesof

σ

(

S

)

are well-definedas y_i=^{0 if}ândônlyîf^Ci∈^S ^forⁱ=¹

, ...,

m.We willdismissthesubscript

whenever theCNF inquestion isclear fromthecontext. Notethat fordifferentmaximal independentsets S=^S ^of ^H wehave

σ (

S

)

=

σ (

S

)

.ItisworthmentioningthatallmaximalindependentsetsofH canbegeneratedwithpolynomial delay[12–14],whichishenceagoodstartforCNFsignaturegeneration.

Assumethat

hasboundeddimension,i.e.,foraconstantdwehave|^Ci|≤^d^for^allⁱ=¹

, ...,

m.LetusdeﬁneXj= {^Ci∈

|^xj∈^Ci or^x¯j∈^Ci}^.^We^say^that

isof

ω

-boundedoccurrenceif|Xj|≤

ω

for j=¹

, ...,

nand

ω

isaﬁxedconstant.

Theorem3.If

hasboundeddimensionandoccurrence,thenitssignaturescanbegeneratedinincrementalpolynomialtime.

Proof. An inducedmatching of an undirected graph isa matching which forms an induced subgraph,that is,no two of its edges are joined by an edge ofthe graph.A maximal induced matching canbe found by a simple greedyapproach:

repeatedlyselectanedgee,addittothesolution,anddeletetheend-verticesoftheedgetogetherwiththesetofvertices adjacenttoatleastoneofthem.

Let M⊆Ê ^be â ^maximal înduced ^matchingⁱⁿ ^H.Let usdenoteby

μ

thenumber ofedges in M andby N the 2

μ

verticesincidenttoedgesinM.Notethat H hasatleast2μmaximalindependentsets(bychoosingarbitrarilyoneofthe endpointsfromeachedgeofM andthengreedilyextendingthissettoamaximalindependentset)andhenceatleastthis manysignaturescanbegeneratedwithpolynomialdelay,asexplainedabove.WedenotebyW⊆

thesetofclausesthat haveedgesin H connectingthemtosomeoftheclausesinN,formally

W

= {

^C

∈ | ∃

^C

∈

^N

: (

C

,

C

) ∈

^E

}

NotethatN⊆^W^.^Moreover,^let^us^denote^U=

\^W^.Îtîsêasy^to^see^that Û îsânindependentsetinH,sinceanyedge withbothendpointsinU couldbeusedtoenlargeM,thuscontradictingitsmaximality.

Assumethat|^Ci|≤^d^for^allⁱ=¹

, ...,

m,and|Xj|≤

ω

forall j=¹

, ...,

n,wheredand

ω

areﬁxedconstants.Observethat withtheseassumptions allclauses inN containtogether atmost2

μ

dliteralsandsowe have|^W|≤²

μ

d

ω

.Let K be the setofvariablesinvolvedinclausesofW andletusdenoten= |^K|^.^Now^we^haveⁿ≤^d|^W|^,^implying

n

≤

²

μ

d²

ω .

Finally,letusdenotebyLthe(possiblyempty)setofvariablesthatappearonlyinclausesofU.Clearly,allvariablesinLare monotonein

(somevariablesappearonlypositivelywhilesomeothersappearonlynegatively)sinceU isanindependent setinH.Thusallliteralsin

thatcorrespondtovariablesinLcanbesimultaneouslyassignedzero.

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Theprocedurethatgeneratesallsignaturesof

worksinthreesteps.Intheﬁrststep,allliteralsinLaresetto0 and theresultingCNFinnvariablesfromK isdenoted

.Thenwegeneratewithpolynomialdelaythemaximalindependent sets S,

=¹

, ...,

kofH (see[12–14]),andthecorrespondingsignatures

σ (

S

)

,

=¹

, ...,

kof

,wherek≥²^μ^(note^that asignatureof

inthiscaseextendsuniquelytoasignatureof

byaddingzerosforclausesthatconsistonlyofvariables fromL).

Inthesecond stepwegeneratealltheremainingsignaturesstemming fromassignments whereall literalsin Lare set tozero.Wetryall2ⁿ binaryassignments tothevariablesinK andcheckforeachofthemwhetheranewsignaturewas generated.Note that k≥²^μ≥

(

2ⁿ

)

¹^/^2d²^ω using theinequality n≤²

μ

d²

ω

,which inturnimplies 2ⁿ≤^k^2d²^ω^.^Hence ^this step canbe doneinan incremental polynomialtime, inparticularin O

(

mnk^2d²ω

)

time sincegeneratinga signaturefora givenassignmenttakesO

(

mn

)

time.

Forthethirdstepletusassumethatk≥^k^distinct^signatures^were^generatedⁱⁿ^the^firstând^second^step.^By^switching the assignmentto literalsin L,we mayget newsignatures,resultingfromchanging some ofthe zerosin asignature to one. Forany partial assignmentto the variables in K we obtain a monotone CNF on variablesin L, andhencethis isa set-uniongenerationproblemthatcanbesolvedwithpolynomialdelayasobservedintheprevioussection.Sinceitsuffices toconsideronlythosepartialassignmentstothevariablesin K whichproducedasignatureinthefirsttwosteps,wemay getinthiswaythesamesignaturemultipletimes,butnomorethanktimes,andthusatthissteptheadditionalsignatures arealsogeneratedinincrementalpolynomialtime.

3.2. Unboundedoccurrence

Intheprevioussection,weconsideredCNFswithboundeddimensionandoccurrence.Therunningtimeofthealgorithm providedby Theorem3dependsexponentially on

ω

,henceitisnotsuitableforhandlingthegeneralcase. Inthepresent section,amoregeneralprocedureisgivenbasedonadifferentapproach.

ForaCNF

,wedenoteby G=

(,

E

)

thesocalleddualgraphof

[15].TheverticesofG aretheclausesof

and edgesareexactlythepairsofclauses

(

C_i

,

C_j

)

forwhichthereexistsavariablethatoccursinbothC_iandC_j(complemented ornot).If S⊆

isanindependentsetofG,thentheclausesofShavepairwisedisjointsetsofvariablesinvolved.

Theorem4.ThereexistsanalgorithmAthatgeneratesthesignaturesofaCNF

consistingofm clausesinn binaryvariablesin O

(

dm²nk

(

^d₂

))

totaltime,whered=^dim

()

andk isthenumberofsignatures.

Proof. Weprovetheclaimbyinductionond.Ford≤^{2 the}^claim^follows^by^Theorem^1.

Assume now thatwe already proved theclaim forall d

<

d, andlet usconsidera CNF

=^C1∧^C2∧ · · · ∧^Cm with dim

()

=^d.^Let^us^associate^to

itsdualgraphGasdeﬁnedabove.LetS⊆

beamaximalindependentsetofG.Such a setcan be obtainedby asimple greedyprocedure inpolynomial time inthe sizeof

. Notethat clauses in S involve pairwisedisjointsetsofvariables,duetothefactthat S isanindependentsetofG.Thus,wecanchoosealiteral uC∈^C foreach clause C∈^S,^set âllôther ^literalsⁱⁿ ^C ^to^zero,^set âllôther ^variables ^not ôccurringⁱⁿ^clauses ôf ^S ^to^zero,ând makeallpossibletruthassignmenttotheliteralsuC,C∈^S.^This^way^weôbtain^k0=²^|^S^| ^different^binary ^signaturesôf

. Notethatwecanoutputthesek0 signatureswithpolynomialdelay.

The totalnumberofvariablesinvolvedinclauses of S isn≤^d|^S|^.^Hence ^we^can ^assignⁱⁿ^all ^possible^ways^values^to thesevariables,andproduce2ⁿ≤^k^d₀ subproblems

j, j=¹

, ...,

2ⁿ intheremainingvariablesinO

(

mn2ⁿ

)

=^O

(

mnk^d₀

)

time whichispolynomialintheinputsizeandk0,sincedisaﬁxedconstant.Notethateachofthesesubproblemsisobtained from

by ﬁxingthen variables atabinary assignment,andsucha substitutioncan bedone in O

(

mn

)

time. Notealso, that each ofthese residualproblems isofdimension atmostd−^1. Îndeed,êach ôf^the ^clauses ^not ⁱⁿ ^S ^shares ât^least onevariablewiththeclausesofS,since SisamaximalindependentsetofG,andnowthatsharedvariableisfixedata binaryvalue.

Weapply algorithmAtoeach oftheresidualsub-CNFs

j, j=¹

, ...,

2ⁿ,onebyone. Thiswaywe producesignatures thatextendthepatternon Sdeﬁnedbyx^j∈ {⁰

,

1}ⁿ^,^for^all ^j=¹

, ...,

2ⁿ onebyone.Twooftheseextendedsignaturesmay coincide,butonlyifthey areextensionsoftwodifferentx^j-s, sincefora ﬁxedx^j algorithmAgeneratespairwisedistinct extensions.Thus, we mayproducethesamesignature no morethan 2ⁿ times.Since 2ⁿ=^O

(

k^d₀

)

,we canshow that this procedureworksintotalpolynomialtime.

Toseethisletusintroducesomeadditionalnotation.Wedenoteby Xj⊆^Y= {⁰

,

1}ⁿ^, ^j=¹

, ...,

2^|^S^|thenonemptysetsof (partial)assignmentsthatproducethesamesignatureontheclausesof S.Notethatthe Xj-spartitionthesetY ofpartial truth assignments.For x∈^Y^,^let ^us^denote^by

(

x

)

the residualCNF,andby k

(

x

)

the numberofsignaturesof

(

x

)

.We denotebyg

()

therunningtimeoftheabovedescribedrecursivealgorithmonCNF

andletG

(

m

,

n

,

d

,

k

)

bethemaxima of g

()

overallCNFswithatmostmclausesonnvariableshavingdim

()

≤^d^and^having^at^most^ksignatures.

The totalcomputational timeinthe ﬁrstphase oftheabove procedurethat endswithproducinga listof2ⁿ residual CNFs,eachofdim≤^d−^{1 is}^bounded^by ^O

(

m²n

)

+^O

(

mnk0

)

+^O

(

mnk^d₀

)

≤^{K m}²^nk^d₀ ^forâ^suitable^constant ^K ^that^does^not depend onm,n,andk0.Thefirst termontheleft handsideisthetime tobuild G andtofindamaximal independent set S.Thesecondtermisthetimeweneedtogeneratethek₀ initialsignatures.Thethirdtermisthetimetogeneratethe 2ⁿ≤^k^d₀subproblems.

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For x∈^Xj andx∈^Xj with j=^j ^the ^CNFs

(

x

)

and

(

x

)

cannot sharesignatures,since thosemust alreadydiffer on S bythedeﬁnitionofthesets X_j for j=¹

, ...,

k₀.However, forx

,

x∈^Xj CNFs

(

x

)

and

(

x

)

mayshare(some,even many)signatures.Discountingtheonesignature wealreadyproducedwiththegiven0-1 values on S,we canstillexpect k_j different signaturesproduced by algorithmA whenwe use itforCNFs

(

x

)

,x∈^Xj, wheremax_x_∈_X_j[^k

(

x

)

−¹]≤^kj≤

x∈^Xj[^k

(

x

)

−¹]^.^Thus,ⁱⁿ^total^we^get^k = ^k0+^k1+ · · · +^k2^|^S^| differentsignaturesfor

.ThetotalrunningtimeonCNFs

(

x

)

,x∈^Xjcanbeboundedby

x∈^Xj

g

((

x

)) ≤ |

^Xj

|

^G

(

m

,

n

,

d

−

¹

,

k_j

).

Thus,forthetotalrunningtimeofalgorithmAon

weget

g

() ≤

^G

(

m

,

n

,

d

,

k

) ≤

^{K m}²^nk^d0

+

k0

j=1

|

^Xj

|

^G

(

m

,

n

,

d

−

¹

,

kj

)

≤

^{K m}²^nk^d₀

+

^k^d₀^G

(

m

,

n

,

d

−

¹

,

k

),

whereforthelastinequalityweusedkj≤^k^for^all ^j=¹

, ...,

k0,implyingG

(

m

,

n

,

d−¹

,

kj

)

≤^G

(

m

,

n

,

d−¹

,

k

)

,whichallows thisquantity tobe factoredout ofthe sum,that canbe then upperboundedby k0

j=¹|^Xj|=²ⁿ≤^k^d₀^.^Using ^this^we^can showbyinductionondthatG

(

m

,

n

,

d

,

k

)

≤^Ldm²^nk

(

^d₂

)

_for_some_constant_L_(we_will_choose _L≥^K⁾^which^will^complete^the proofofourclaim.Now

G

(

m

,

n

,

d

,

k

) ≤

^{K m}²^nk^d₀

+

^k^d₀^G

(

m

,

n

,

d

−

¹

,

k

)

≤

^{K m}²^nk^d₀

+

^k^d₀^L

(

d

−

¹

)

m²nk

(

^d⁻₂¹

)

≤

^Lm²^nk^d

+

^k^d^L

(

d

−

¹

)

m²nk

(

^d⁻₂¹

)

≤

Lm²nk^d

+

L

(

d

−

1

)

m²nk

(

^d⁻₂¹

)

+^d

≤

Ldm²nk

(

^d₂

).

Remark5.Since2-CNFsaretractable,therunningtimeofthealgorithmcanbeslightlyimprovedbystoppingtherecursion whend=^2,^as^pointed^out^by^Strozecki^[16].

Corollary6.Thealgorithmconstructedintheaboveproofworksinincrementalpolynomialtime.

Proof. Usingtheabovetheorem,wecanprovethisclaimbyinductiononthedimensiond.Whend=^1,^the^claim^is^trivially true.

Considernowthegeneralcase,asintheproofoftheabovetheorem.Asweremarkedthere,producingtheﬁrstk₀=²^|^S^| signaturesinfactcanbedonewithpolynomialdelay.AfterthiswestartprocessingtheCNFs

(

x

)

forx∈^Xj, j=¹

, ...,

k0. Notethatthesignaturesproducedfrom

(

x

)

,x∈^Xjand

(

x

)

,x∈^Xjarealldifferentif j=^j^.^Note^also^that^dim

((

x

))

≤ d−^{1 for}^all^x∈^Xj, j=¹

, ...,

k₀,andthuswecanassumebyinductionthattheirsignaturescanbeproducedinincremental polynomial time in thesize of

(

x

)

,whichis boundedby the size of

.Thus, if X_j= {^x1

, ...,

x}^,^then ^we ^can ^produce k

(

x1

)

newsignaturesinincremental polynomialtime,infactregardlesshowmanyweproduced previously(includingthe k0 wehavefromtheﬁrstphase). Letusdenotebyq

(

m

,

n

,

k

(

x1

))

thepolynomialboundingthetotaltimeprocessing

(

x1

)

.If k

(

x₂

) >

k

(

x₁

)

,thenmaybetheﬁrstk

(

x₁

)

signaturesproducedfrom

(

x₂

)

coincidewiththeoneswealreadygeneratedfrom

(

x1

)

,butstillafteratmostq

(

m

,

n

,

k

(

x1

))

timewegetanewsignature.Intheworstcase,wehavekj=^k

(

x1

)

≥^k

(

xi

)

forall xi∈^Xj,i=^1,ⁱⁿ^which^case^processing

(

xi

)

,i=²

, ...,

maynotproduceanynewsignatures.Since

≤^k^d0,thismeansthat thelargestgapbetweentheoutputofthelastsignatureof

(

x1

)

andnextnewsignatureisnotmorethank^d₀q

(

m

,

n

,

k

(

x1

))

, atamomentwhenwehavealreadyproducedk≥^k0+^k

(

x₁

)

signatures.Thusthislargesttimegapbetweentwooutputsis stillboundedbyapolynomialofm,n,andthenumberofsignaturesk≥^k0+^k

(

x1

)

producedsofar.Asbothn andmare boundedbytheinputsize

^,^the^corollary^follows.

4. Generatingmaximalandminimalsignatures

GenerationofmaximalsignaturesisdiﬃcultasitincludesSATasaspecialcase.

Theorem7.UnlessP=NP,themaximalsignaturescannotbegeneratedintotalpolynomialtime.

Proof. Letusconsider aCNF

,and observethat its unique maximal signature isthe all-onevector ifandonly if

is satisﬁable.Assumebycontradictionthatwehaveatotalpolynomialtimealgorithmforgeneratingallmaximal signatures,

(6)

anddenote byt

(

k

, )

thepolynomial bound forits terminationwhen theinput hassize k andoutput involves exactly

signatures.Letusrunthisalgorithmfort

(||||,

¹

)

time,whichispolynomialintheinputsizeforinput

.Ifthisalgorithm outputs the all-onevector then

is satisﬁable, andotherwise itis not. Hence itwould decidethe satisﬁability of

in polynomialtime.AsSATisNP-complete[17],thetheoremfollows.

Itturnsoutthatminimalsignaturescanbegeneratedeﬃciently.

Theorem8.MinimalsignaturescanbegeneratedwithpolynomialdelayforarbitraryCNFformulas.

Proof. Weclaimthatthereisaone-to-onecorrespondencebetweenminimalsignaturesofaCNF

andmaximalindepen- dent setsofitsconﬂict graph H.Since H canbe builtinpolynomialtime from

andmaximal independentsetsofa graphcanbegeneratedwithpolynomialdelay[12–14],thiswouldprovethetheorem.

Tosee theabove claim,assume ﬁrst thata signature

σ

= {

σ

C|^C∈

}

îsâ ^minimal ^signature ôf

.Notethat the set S= {^C∈

|

σ

C=⁰} ^is ^an independent set in H. For any C∈

with

σ

C=^{1 there} ^must êxist â ^conflict ^between ^C andsome C∈^S,^sinceôtherwise^we ^could^set

σ

C tozerowithoutforcing anyoftheclauses in S tochangetheir values, contradictingtheminimalityof

σ

.Thus S mustbeamaximalindependentset.

TheotherdirectionfollowsfromthefactthatifS isamaximalindependentsetofHandwesetalltheclausesinS to zero,thenallotherclausesof

areforcedtotakevalueoneduetotheconﬂictsbetweenS andotherverticesofH. 5. Conclusions

In thispaperweshow that all signaturesofagiven CNFwitha boundeddimensioncan be generatedinincremental polynomialtime,answeringanopenproblemposedbyKröll[8,Problem4.7].Afasterincrementalpolynomialalgorithmis providedfortheclassofformulaswhereboththedimensionandtheoccurrencearebounded.Moreover,itisalsoshown thatthesametaskcanbedonewithpolynomialdelay iftheinputCNFisfromatractableclass(inthiscasenoboundon dimensionoroccurrenceisnecessary).Finally,itisprovedthatmaximalsignaturescannotbegeneratedintotalpolynomial timeunless P=^{N P}^,^while^minimal^signatures^can^be^generated^with^polynomial^delay^for^arbitrary^CNF^formulas.

Inthiscontextitisinterestingtonote thatgivena3-CNF

withmclauses andthevector y=

(

1

,

1

, ...,

1

)

∈ {⁰

,

1}^m ^it isNP-hardtotestwhether y isasignatureof

,ornot(y isasignatureifandonlyif

issatisﬁable).Ontheotherhand, ourresultsshowthatgeneratingallsignaturesof

canbe doneinincrementalpolynomialtime.Thisisaratherunusual behaviorforagenerationproblem.Typically,ifallsolutionsofagivenproblemcanbegeneratedinincrementalpolynomial time,checkingifagivencandidateisasolutionornotiscomputationallyeasy.

An additionalproblemrelatedtoCNF signatureswasstatedattheDagstuhlSeminar19211byTurán.Givena set S⊆ {⁰

,

1}^m^,^does^there êxistâ ^CNF ^with^m ^clauses ^such ^that ^S îs êxactlyîts ^set ôfâll signatures? Ifyes,can such aCNF be computedefficiently?This‘reverse’problem(getthesignatures,outputclauses)totheproblempresentedinthispaper(get theclauses,outputsignatures)istothebestofourknowledgecompletelyopen.

Declarationofcompetinginterest

The authors declare that they haveno known competingﬁnancial interests or personal relationships that could have appearedtoinﬂuencetheworkreportedinthispaper.

Acknowledgements

We are truly grateful forYann Strozecki forhis valuable observations andsuggestions that helped usto improvethe paper.Wewouldalsoliketothanktheanonymousrefereesfortheircarefulreadingofthemanuscriptandtheirinsightful comments.

Kristóf Bérczi was supported by the János Bolyai Research Fellowship of the Hungarian Academy of Sciences and by the ÚNKP-19-4 New National ExcellenceProgram ofthe Ministry forInnovation and Technology.Ondˇrej ˇCepek andPetr Kuˇcera gratefully acknowledge a support by the Czech Science Foundation (Grant 19-19463S). Projects no. NKFI-128673 and“ApplicationDomainSpeciﬁcHighlyReliableITSolutions”havebeenimplementedwiththesupportprovidedfromthe NationalResearch,DevelopmentandInnovation Fund ofHungary,ﬁnancedunderthe FK_18andtheThematicExcellence ProgrammeTKP2020-NKA-06(NationalChallengesSubprogramme)fundingschemes,respectively.Thisworkwassupported bytheResearchInstituteforMathematicalSciences,anInternationalJointUsage/ResearchCenterlocatedinKyotoUniversity.

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