arXiv:1505.02318v1 [cs.DM] 9 May 2015Parametersforminimal unsatisability:SmarandacheprimitivenumbersandfullclausesOliverKullmann1andXishunZhao21ComputerScienceDepartment,SwanseaUniversity2InstituteofLogicandCognition, SunYat-senUniversityAbstract. Weestablishanewbridgebetweenpropositional logicandelementarynumbertheory. Afull clausecontainsall variables, andwestudytheminminimally unsatisable clause-sets (MU); suchclausesare strongstructural anchors, whencombinedwithother restrictions.Countingthemaximal numberof full clausesforagivendeciency k,weobtainacloseconnectiontotheso-calledSmarandacheprimitivenumberS2(k),thesmallestnsuchthat2kdividesn!.Thedeciencyk1ofanMUisthedierencebetweenthenumberofclauses and the number of variables. We also consider the subclass UHITof MU given by unsatisable hitting clause-sets (every two clauses clash).We study the four fundamental quantities FCH, FCM, VDH, VDM :N N, denedasthemaximumnumberof full clausesinUHITresp. MU,resp. the maximal minimal number of occurrences of a variable (thevariabledegree) inUHITresp. MU, independencyonthedeciency.Wehavetherelations FCH(k) FCM(k) VDM(k) andFCH(k) VDH(k) VDM(k),togetherwithVDM(k) nM(k) k +1 +log2(k),usingthenon-MersennenumbersnM(k)asestablishedin[21].WeshowthelowerboundS2(k)FCH(k); indeedweconjecturethisto be exact. The proof rests ontwo methods: Applyingsubsumptionresolutionanditsinverse, andanalysingcertainrecursions, combininganapplication-specicrecursionwitharecursionfromtheeldofmeta-Fibonacci sequences. TheS2-lowerboundtogetherwiththenM-upper-boundyieldsagoodhandleonthefourquantities, whichwedeterminefor1 k 13.1 IntroductionLong clauses oftenoccur inpractical instances; we studythe most extremecase,theoccurrences of fullclauses,clausesof maximalpossiblelength,inmin-imalunsatisableclause-sets(F MU).Themainparameteristhedeciency(F) = c(F) n(F) 1,thenumberofclausesminusthenumberofvariables.WedenotebyFCM(k)themaximalpossiblenumberoffullclauses inF MUwith(F) =k(short: F MU=k). From[21, Theorem15] followstheup-perboundFCM(k)nM(k)forthenon-Mersennenumbers nM(k)N, withThis researchwas partiallysupportedbyNSFCGrand61272059, NSSFCGrant13&ZD186andMOEGrant11JJD7200020.2k +log2(k +1) nM(k) k +1 +log2(k)([21, Corollary 10]). Untilnow nogeneral lower bound on FCM(k) was known, and we establish S2(k) FCM(k).Here S2(k),as introduced in [28],is thesmallest n N0such that 2kdivides n!,and various number-theoretical results on S2and the generalisation Spfor primenumberspareknown.Actuallyweshowastronger lowerbound,namelywedonotconsiderallF MU=k,butonlythoseFwhicharehitting(F UHIT ),that is, where every two clauses clash, yielding FCH(k) with FCH(k) FCM(k),and we show S2 FCH. The elements of UHITare known in the DNF languageas orthogonal or disjoint tautological DNF,and when considering arbitrarybooleanfunctions, thenalsodisjointsumsof products(DSOP)ordisjointcuberepresentations areused;see[25,Section4.4]or[4,Chapter7].ThecentralunderlyingresearchquestionistheprogrammeofclassicationofMUinthedeciency, thatis, thecharacterisationofthelayersMU=kfork N. A special case of the general classication is the classication of UHIT=k.Theearliest source [1]showed (in modern notation) (F) 1 for F MU,andcharacterised the special case SMU=1 MU=1, where SMU MUcontainsthoseFMUsuchthatnoliteralscanbeaddedtoanyclauseswithoutde-stroyingunsatisability.Later[5] characterisedMU=1viamatrices,whiletheintuitivecharacterisation viabinarytreeswasgivenin[16,AppendixB],wherealsoSMU=1=UHIT=1hasbeenshown. Intheformof S-matrices, theclass MU=1had been characterised earlier in [13,11], going back to a conjectureonQualitativeEconomics([7]), andwheretheconnectionstothiseldof ma-trixanalysis,calledQualitativeMatrixAnalysis(QMA),whererstrevealedin[18] (see[15, Subsection11.12.1] and[23, Subsection1.6.4] for overviews).SMU=2and partially MU=2were characterised in [14],with further informa-tion on MU=2 in [22]. [6] showed that all layers MU=k are poly-time decidable.AkeyelementfortheseinvestigationsintothestructureofMUisthemin-var-degreevd(F) := minvvar(F)|{C F: v var(C)}|,theminimalvariable-degree of F,and itsmaximum VDM(k)over allF MU=k.Indeedthekey tothe characterisation of MU=1 in [5] as well as in [13] was the proof of VDM(1) =2. Therstgeneral upperbound kN: VDM(k) 2kwasshownin[16,Lemma C.2]. Now in [21], mentioned above, we actually showed the upper boundVDM(k)nM(k). Usingfc(F)forthenumberof full clausesinF, obviouslyfc(F) vd(F) holds, andFCM(k) is the maximumof fc(F) over all F MU=k,thusFCM(k) VDM(k).ForthevariationVDH(k)VDM(k), whichonlyconsidershittingclause-sets, we conjecture VDH(k) = VDM(k) for all k 1. Furthermore we conjectureFCM(k) nM(k)1, and thus the quantities nM(k), VDM(k), VDH(k), FCM(k)are believed to have at most a distance of 1 to each other. On the other hand weconjecture FCH(k) = S2, where S2(k) oscillates between the linear function k+1and thequasi-linear functionk +1 +log2(k).Altogether thefourfundamen-tal quantities FCH, FCM, VDH, VDM are fascinating and important structuralparameters, whosestudycontinues toreveal newand surprising aspects of MUandUHIT .3It is also possible to go beyond MU: in [23, Section 9] it isshown that whenconsideringthemaximumofvd(F)overallF LEAN=k MU=k,thesetofallleanclause-sets,thatthennM(k)istheprecisemaximumforallk 1.Lean clause-sets were introduced in [17] as the clause-sets where it is not possibletosatisfysomeclauseswhilenottouchingtheotherclauses, andindeedwerealready introduced earlier, as non-weakly satisable formulas (matrices) in theeldofQMAby[12].Furthermoreitisshownin[23,Section10],thatthereisapolytimeautarkyreduction, removingsomeclauseswhichcanbesatisedwithout touching the other clauses, which establishes for arbitrary clause-sets Ftheupperboundvd(F)nM((F)); aninterestingopenquestionhereistondthewitnessingautarkyinpolynomialtime.Backtothemainresultofthispaper,theproofofS2 FCHisnon-trivial.Indeed the proof is relatively easy for a function S2(k) dened by an appropriaterecursion, motivated by employing full subsumption extension C C{v}, C{v}inanoptimalway.ThenthemainauxiliaryresultisS2=S2.Forthatweuse another function, namely a2(k) as considered in [24] in a more general form,whilea2wasintroducedwithasmallmodicationin[3]. Theseconsiderationsbelongtotheeldofmeta-Fibonaccisequences,wherespecialnestedrecursionsarestudied,initiatedby[8,Page145].Viaacombinatorialargumentwederivesuchanestedrecursionfromthecourse-of-valuerecursionforS2, whichyieldsS2=2a2. Wealsoshow2a2=S2(thisequalitywasconjecturedontheOEIS[27]),andweobtainS2= S2.Overview. Themainresults of this paper areas follows. Theorem18provesS2=2a2. Theorem31showsameta-Fibonacci recursionforS2, whereS2isintroducedbyarecursiondirectlyrelatedtoourapplication.Theorem33thenprovesS2=S2. After thesenumber-theoreticpreparations, weconsider sub-sumptionresolutionandits inversion(extension); Theorem38combines sub-sumptionextensionandtherecursionmachinery,andshowsS2FCH.Intheremainder of thepaper, this fundamental result is applied. Theorem 39 proves atight upperbound on S2,whileTheorem 42 considers thecases wherethelowerboundvia S2and theupperbound vianMcoincides.FinallyinTheorem 47 wedeterminethefourfundamentalquantitiesfor1 k 13(seeTable1).2 PreliminariesWeuseZforthesetof integers, N0:={nZ: n0}, N:=N0\ {0}, andnallyR Zisthesetofrealnumbers.Formapsf, g: XRwewritef gif x X: f(x) g(x).OnthesetLIT of literalswehavecomplementationxLIT x, withx=xandx=x. WeassumeZ \ {0}LIT , withz=zforzZ \ {0}.Variables VALIT with NVAare special literals, andthe underlyingvariableof aliteral isgivenbyvar: LIT VA, suchthatforvVAholdsvar(v) =var(v) =v, while for xLIT \ VAholds x=var(x). For asetL LIT wedeneL:= {x :xL}.AclauseisanitesetCofliteralswith4C C= (Cisclash-free).Aclause-setisanitesetofclauses,thesetofallclause-setsisdenotedbyCLS.For aclause Cwe dene var(C) :={var(x) : xC} VA, andfor aclause-set F we dene var(F) :=

CF var(C) VA. We use the measuren(F):=|var(F)| N0andc(F):=|F| N0, whilethedeciencyis(F):=c(F) n(F) Z.Theset of satisableclause-sets isdenoted by SAT CLS,which isthesetof clause-sets Fsuch that there is a clause C which intersects all clauses of F, i.e.,with D F: CD = ; the unsatisable clause-sets are USAT := CLS\SAT .ThesetMUUSAT of minimallyunsatisableclause-setsis theset ofFUSAT , suchthatforFFholdsFSAT . Theunsatisablehittingclause-setsaregivenbyUHIT := {FUSAT| C, DF, C=D:C D=}. It is easy to see that UHIT MUholds, and that for all F UHIT holds

CF 2|C|=1. Whileall denitionsaregiveninthispaper, forsomemorebackground see[15].2.1 FullclausesAfull clauseforF CLSissomeC Fwithvar(C) = var(F)(equivalently,|C| = n(F)),and thenumberof fullclausesis counted byfc : CLSN0,whichcan be dened as fc(F) := c(FA(var(F))), and where A(V ) UHITfor somenite V VA is the set of all clauses Cwith var(C) = V . Standardised versionsof theA(V )areAn:=A({1, . . . , n})fornN0. Thefollowingobservationiscontainedintheproofof[31,UtterlyTrivialObservation]:Lemma1. ForF UHIT ,F= {}, thenumber fc(F)offullclausesiseven.Proof. Letn:=n(F). Wehave

CF2n|C|=2n, andthus

CF 2n|C|iseven(dueton > 0).Since

CF,|C|=n2n|C|iseven,theassertion follows. 2.2 Thefourfundamental quantitiesForF CLSwedenethevar-degreeasvdF(v) := c({C F: v var(C)}) N0forvVA, whileincaseofvar(F)=(i.e., F / {, {}})wedenethemin-var-degreevd(F) := minvvar(F) vdF(v) N.Denition2. Fork NletFCH(k) Nbethemaximal fc(F)forF UHIT=k;FCM(k) Nbethemaximal fc(F)forF MU=k;VDH(k) Nbethemaximal vd(F)forF UHIT=k;VDM(k) Nbethemaximal vd(F)forF MU=k.Fork = 1thecaseF= {}isexcludedinthelasttwodenitions.By[21,Lemma9,Corollary 10,Theorem15]:5Theorem3([21]). VDM(k)nM(k)=k + log2(k + 1 + log2(k + 1))k + 1 +log2(k)forall k N.Here nM : N N is the enumeration of natural numbers excluding the Mersennenumbers 2n 1for nN; thelist of initial values is 2, 4, 5, 6, 8, 9, 10, 11, 12,13, 14, 16, 17(http://oeis.org/A062289). In[23, Theorem14.4] it is shownthat VDM(6) =8=nM(6) 1, extendingthistoanimprovedupper boundVDMnM1([23, Theorem14.6], where nM1: NNcanbe denedasfollows:nM1(k) := nM(k)forkNwithk= 2nn + 1forsomen 3,whilenM1(2nn + 1) := nM(2n n + 1) 1 = 2n;seeTable1forinitialvalues.Theorem4([23]).Fork NholdsVDM(k) nM1(k) nM(k).We conclude these preparations with a special property of FCH(k) (support-ingourConjecture48thatFCH = S2),namelybyLemma1wehave:Corollary5. FCH(k)isevenforall k N.3 SomeintegersequencesWe review the Smarandache primitive numbers S2(k) and the meta-Fibonaccisequencesa2(k).WeshowinTheorem 18,thatS2= 2a2holds.3.1 SomepreparationsWedenetwogeneraloperationsaaandaPaforsequencesa. Firstthe(standard)-operator:Denition6. Fora : IR,whereI Zisclosedunderincrement,wedenea : IRbya(k) := a(k + 1) a(k).Soaismonotonicallyincreasingia0, whileaisstrictlymonotonicallyincreasingia 1.Sequenceswithexactlytwodierent-values,whereoneof these values is 0, play a special role for us, and we call them d-Delta, wheredistheothervalue:Denition7. Asequencea: N0Zis calledd-Deltafor dZ \ {0}, ifa(N0) = {a(n)}nN0= {0, d}.Whilethe-operatordeterminesthechangetothenextvalue,theplateau-operator determinessubsequencesofunchangingvalues:Denition8. For asequence a: NRwhichis non-stationary(for all ithereisj>iwithaj=ai)wedenePa: N N(theplateauoperator)byletting Pa(n)forn Nbethesize ofthe n-th (maximal)plateauofequal values(maximal intervalsof Nwhereaisconstant).SoPa(1)isthesizeoftherstplateau, Pa(2)thesizeofthesecondplateau,andsoon; i N: a(i) =a(i + 1) iPaistheconstant1-function. Forad-DeltasequenceafromPaandtheinitialvaluea1wecanreconstruct a.63.2 SmarandacheprimitivenumbersThe Smarandache Primitive Numbers were introduced in [28, Unsolved Prob-lem47]:Denition9. Fork N0letS2(k)bethesmallestn N0suchthat2kdividesn!. Usingord2(n), nN, forthemaximal mN0suchthat2mdividesn, wegetthatS2(k)fork N0isthesmallestn N0suchthatk

ni=1 ord2(i).SoS2(0)=0, andS2(N0)={0, 2}.Thefollowingiswell-knownandeasytoshow(seeSubsectionIII.1in[9]forbasicpropertiesofS2(k)):Lemma10. Thesequence S2(1), S2(2), S2(3), . . . isobtainedfromthesequence1, 2, 3, . . . of natural numbers, wheneachelement nNis repeatedord2(n)manytimes.Example11. ThenumbersS2(k)fork{1, . . . , 25}are2, 4, 4, 6, 8, 8, 8, 10, 12,12, 14, 16, 16, 16, 16, 18, 20, 20, 22, 24, 24, 24, 26, 28, 28.ThecorrespondingOEIS-entry(whichhas1asrstelement(index0), insteadof 0aswehaveit, andwhichweregard asappropriate)ishttp://oeis.org/A007843.Lemma12([30]).Fork Nholdsk + 1 S2(k) = k + O(log k).We give an independent proof for the lower bound in Lemma 40, while we sharpenthe upper bound in Theorem 39. For more number-theoretic properties of S2see[29].TounderstandtheplateausofS2,weneedtherulerfunction:Denition13. Letrun:= ord2(2n) Nforn N.TheplateausofS2aregivenbytherulerfunction: inLemma10wedeter-mined the number of repetitions of values v N as ord2(v), while for the plateausweskipzero-repetitions, whichhappenateachoddnumber, andthusfortheassociatedindexnwehaven=v2forevenv, andthenumberofrepetitionsisord2(v) = ord2(2n);weobtainLemma14. P(S2(k))kN= (run)nN.3.3 Meta-FibonaccisequencesStartedby[8, Page145], variousnestedrecursionsforintegersequenceshavebeenstudied. Oftenthe focus inthis eldof meta-Fibonacci sequencesisonchaoticbehaviour,butweconsiderhereonlyawell-behavedcase(butindetail):Denition15. In [24] the sequence a2: N0N03), has been dened recursivelyviaa2(k) = a2(k a2(k 1)) + a2(k 1 a2(k 2)),whilea2(k) := kfork {0, 1}.3)hidingtwoparametersd N,s Zusedin[24],whichared = 2, s = 0inourcase7Thesequencea2was introduced in[3]asF: N N0,withF(k) = k 1fork {1, 2} and thesamerecursion law,whichyieldsF(k) = a2(k 1)fork N.Furthermore, usingF(1) =F(2) =1as initial conditions does not changeanythingelse,andthissequenceistheOEIS entryhttp://oeis.org/A046699.Itisshown(inournotation):Lemma16([3]).Fork Nandp := log2(k +1):a2(k) = 2p1+a2(k +1 2p).Lemma16yieldsafastcomputationofa2(k).[10,Corollary2.9,Equation(1)]determinestheplateausizes:Lemma17([10]).a2isa1-DeltasequencewithP(a2(k))kN= ru.We cannowshowa2=12S2, whichhas been conjecturedonthe OEIS(http://oeis.org/A007843,byMichelMarcus):Theorem18. k N0: S2(k) = 2 a2(k).Proof. ByLemma14andLemma17,togetherwithS2(0) = a2(0) = 0. 4 RecursionsforSmarandacheprimitivenumbersInSubsection4.1weintroducethesequenceS2viaarecursiveprocess, whichdirectlytiesintoourmainapplicationinTheorem38forconstructingunsatis-ablehittingclause-setswithmanyfull clauses. Thisrecursivedenitionusesanindex,whichisstudiedinSubsection4.2.Thecentralhelperfunctionistheslack, studied in Subsection 4.3. Wethen prove a meta-Fibonacci recursion inTheorem31,andobtainS2= S2inTheorem33.4.1 Asimplecourse-of-valuesrecursionDenition19. Fork N0let1. S2(0) := 0,S2(1) := 2;andfork 2:2. S2(k) := 2(ki+1) for the minimal i {1, . . . , k1} with ki+1 S2(i).Notethattherecursionstepiswell-dened(thei exists), sincefori =k 1holds ki+1 =2, andS2(k1) =2for k =2, while for k 3holdsS2(k 1)=2 ((k 1) i+ 1)2 ((k 1) ((k 1) 1) + 1)=4. Theconditionk i + 1S2(i)isequivalenttok + 1i + S2(i). SomesimplepropertiesarethatS2(k)isdivisibleby2, S2(k)2fork1, andS2(2)=4andS2(k) 4fork 2.84.2 AnalysingtheindexDenition20. Fork 0letiS(k) := k + 1 S2(k)2 N.Simpleproperties(forallk 0):1. S2(k) = 2 (k iS(k) + 1).2. iS(0) = iS(1) = iS(2) = 1.3. iS(k) = 0 S2(k) = 2andiS(k) = 1 S2(k) = 0.Analternativecharacterisation ofiS(k):Lemma21. Fork 0:iS(k)istheminimal i N0withi + S2(i) k + 1.Proof. The assertionfollows bywhat has alreadybeensaidabove, plus theconsiderationof thecornercases: 0 + S2(0)=0 0,thenslS(k + 1) = slS(k) 1.2. IfslS(k) = 0,thenslS(k + 1) {0, 2}.4.4 Ameta-FibonaccirecursionWearereadytoproveaninterestingnestedrecursionforS2.Firstacombina-torial lemma, just exploitingthefact that theshapeof theslackrepeats thefollowing pattern (Corollary 29): a plateau of zeros, followed by a jump to 2 anda stepwise decrement to 0 again (where right at k = 0 we start with slS(0) = 2):Lemma30. Fork 2holds

2i=1 slS(k i) =

2i=1 i min(1, slS(k i)).Proof. Thereare0 p 2and1 q 3suchthattheleft-handsideisp + (p 1) + + 1 + 0 + + 0 + 2 + (2 1) + + q;forp=0theinitial partisempty,forq=3thenalpartisempty.Letr0bethenumber of zeros; sor =0ip=2(andthenalsoq =3). Wehavep + r + (2 q + 1) = 2,i.e.,p + r + 1 = q.Nowtheright-handsideis1 + 2 + + p + 0 + + 0 + q + (q + 1) + + 2,andweseethatbothsidesareequal. Theorem31. Fork 2holdsS2(k) =2

i=1S2(iS(k i))(notethatbyLemma24holdsiS(k i) < k).Proof. ByLemma26andLemma30holds2

i=1S2(iS(k i)) = (2

i=1slS(k i)) + S2(k) 122

i=1(S2(k) S2(k i)) =S2(k) + (2

i=1i min(1, slS(k i))) 122

i=1i1

j=0S2(k + i j),10wherenowbyCorollary28holds

2i=1

i1j=0 S2(k + i j) = (S2(k 1)) +(S2(k 2) + S2(k 1))=

2i=1 i S2(k 1)=2

2i=1 i min(1, slS(k)),whichcompletestheproof. NowweseethatS2isbasicallythesameasa2(recallSubsection3.3):Corollary32. k N0: S2(k) = 2 a2(k).Proof. Forthepurposeoftheproofleta2(k):=12S2(k)forkN0. Sowegeta2(k) = kfork {0, 1},whileiS(k) = k + 1 a2(k),andthusfork 2:a2(k) =12S2(k) =122

i=1S2(iS(k i)) =2

i=1a2(iS(k i)) =2

i=1a2(k i + 1 a2(k i)),andsotheassertion followsbytheequationsofDenition15. Weobtainthemainresultofthissection:Theorem33. S2= S2(recall Denition9).Proof. ByCorollary32andTheorem18. 5 Onthenumberoffull clausesFirst wereviewfull subsumptionresolution, C {v}, C {v} C, anditsinversion, calledextensioninSection5.1, wheresomecareis needed, sincewe needcomplete control. Fromaclause-set F withmanyfull clauses wecanproducefurther clause-setswithmanyfull clausesbyfull subsumptionextensiondone inparallel, andthis process of full expansionis presentedinDenition36. Therecursivecomputationof S2viaDenition19capturesmaximisationforthisprocess,andsowecanshowinTheorem38,thatwecanconstruct examples of unsatisablehittingclause-sets Fkof deciencykandwithS2(k)manyfull clauses. ItfollowsthatS2yieldsalowerboundonFCH(Conjecture48saysthislowerboundisactuallyanequality).5.1 FullsubsumptionresolutionAsstudiedin[23,Section6]insomedetail:Denition34([23]). Afull subsumptionresolutionforFCLScanbeperformed,if there isa clause C/ Fwith C{v}, C{v} Fforsome variablev,andreplacesthetwoclausesC {v}, C {v}bythesingleclauseC.Forthestrictform, theremust exist athirdclauseDF\ {C {v}, C {v}}withvvar(D), whileforthenon-strictformtheremust NOTexist suchathirdclause.11If Fis obtained from Fby one full subsumption resolution, then c(F) = c(F)1;wehavethestrictformin(F)=n(F),or,equivalently, (F)=(F) 1,while we have the non-strict form i n(F) = n(F) 1, or, equivalently, (F) =(F). Averyoldtransformationof aCNF(DNF)intoanequivalentoneusestheinverseoffullsubsumptionresolution4):Denition35([23]). Afull subsumptionextensionfor FCLSandaclause CF canbeperformed, if thereis avariable v VA \ var(C) withC {v}, C {v}/ F, andreplacesthesingleclauseCbythetwoclausesC{v}, C {v}.Forthestrictformwehavev var(F),whileforthenon-strictformwehavev/ var(F).If we consider F MUand C F, then we can always perform a non-strict fullsubsumptionextension,whilewecanperform thestrictformiCisnotfull.Ifwe denote the result by F, then for F UHIT we have again F UHIT , butfor general F MUwe might have F/ MU;see [23,Lemma 6.5] for an exactcharacterisation.5.2 FullexpansionsWenowperformfullsubsumptionextensionsinparalleltomfullclausesofF,rstusinganon-strictextension, andthenreusingtheextensionvariableviastrictextensions:Denition36. For F CLS and m N, where fc(F) m, a full m-expansionofFissomeG CLSobtainedby1. choosingsomeF F A(var(F))withc(F) = m,2. choosingsomev VA\ var(F)(theextensionvariable),3. andreplacingtheclausesCFinFbytheirfull subsumptionextensionwithv(recall Denition35).Thechoiceof vinDenition36isirrelevant, whilethechoiceof FmighthaveaninuenceonfurtherpropertiesofG,butisirrelevantforouruses.Thefollowingbasicpropertiesallfollowdirectlyfromthedenition:Lemma37. ConsiderthesituationofDenition36.1. Thereisalwaysafull m-expansionG(uniqueforanyxedF,v).2. IfF UHIT ,thenG UHIT .3. n(G) = n(F) + 1,c(G) = c(F) + m.4. (G) = (F) + m1.5. fc(G) = 2 m.Weturntotheconstructionof unsatisablehittingclause-setswithmanyfullclauses(foragiven deciency):4)Boole introduced in [2], Chapter 5, Proposition II, the general expansion f(v, x) =(f(0, x)v)(f(1, x)v) for boolean functions f, where for our application f(v, x) C.Thiswastakenupby[26],andisoftenreferredtoasShannonexpansion.12Theorem38. Fork NwerecursivelyconstructFk UHIT=kasfollows:1. F1:= {{1}, {1}}.2. Fork 2letFkbeafull a2(k)-expansionofFiS(k).Thenwehavefc(Fk) = S2(k).Thus k N : S2(k) FCH(k).Proof. If theconstruction is well-dened,then we get fc(Fk) = 2 a2(k) = S2(k)and (Fk) = (FiS(k)) +a2(k) 1 = iS(k) +a2(k) 1 = k for k 2 by Lemma 37(usingTheorem 33freely),whilethesetwopropertiesholdtriviallyfork = 1.Itremainstoshowthat1 iS(k)k 1anda2(k)fc(FiS(k))fork2.TherststatementfollowsbyCorollary24,whilethesecondstatementfollowsbyLemma21. 6 ApplicationsWestartbysharpeningtheupperboundfromLemma12:Theorem39. Fork NholdsS2(k) nM(k) k + 1 +log2(k).Proof. ByTheorem38andTheorem3. We can also provide an independent proof of thelower bound of Lemma 12:Lemma40. Fork NholdsS2(k) k + 1.Proof. Weprovetheassertionbyinduction. Fork=1wehaveS2(1)=2, soconsider k 2. We use Corollary 22, and so we need i N with k+1 2(ki+1),i.e.,i k+12.Sowechoosei := k+12 N.Wehavei < k,andsowecanapplythe induction hypothesis to i: i +S2(i) = k+12 +S2(k+12) k+12 +k+12 +1 = 2k+12 + 1 > 2(k+121) + 1 = k,andthusi + S2(i) k + 1. Whenupperandlowerboundcoincide,thenweknowall fourfundamentalquantities;rstwenamethesetsofdeciencies(recallTheorems3,4):Denition41. SNM:={kN: S2(k) =nM(k)}, SNM1:={kN:S2(k) = nM1(k)}.ByS2 VDM nM1 nMwegetSNM SNM1and:Theorem42. Fork SNM1holdsS2(k) = FCH(k) = FCM(k) = VDH(k) =VDM(k) = nM1(k).Weprovenowthatthespecial deciencies2n n, 2n n 1(n1; note(An) = 2nn)considered in[23,Lemmas12.10,12.11], wherewehaveshownthat for them the four fundamental quantities coincide, are indeed in SNM, andthatfurthermorethespecialdeciencies2n n + 1(n 3),wherenM1diersfromnM,areinSNM1:Lemma43. Considern N.131. S2(2nn) = 2n,andfork N0holdsS2(k) = 2n 2nn k 2n1.2. 2nn SNM,while2nn + 1, . . . , 2n1/ SNM.3. Assumen 2now.Then2nn 1 SNMwithS2(2nn 1) = 2n2.4. Forn 3holds2nn + 1 SNM1.Proof. By [23, Corollary 7.24] we have nM(2nn) = 2n, while nM(2nn1) =2n 2(rememberthatthejumpsfornMhappensatthedeciencies2n n).Thus S2(2n n) 2nandS2(2n n 1) 2n 2. Sincefor thevalue2nthesequenceS2has aplateauof lengthn(Lemma10), whilenMis strictlyincreasing,forParts1,2,3itremainstoshowS2(2n n)2n. Weshowthisbyinduction: Forn=1wehaveS2(1)=2=21,whileforn2byinductionhypothesis we have (2nn) (2n1(n1)) +1 = 2n1 S2(2n1(n1)),thusbyCorollary22S2(2n n)2 2n1=2n. Finally, forPart4wenoteS2(2nn+1) = S2(n) = 2nby Part 1, while nM1(k) diers from nM(k) exactlyat thepositions k = 2nn+1 for n 3, where then nM1(k) = nM(k) 1 = 2n([23,Theorem14.7]). SothelowerboundofLemma40issharpforinnitelymanydeciencies:Corollary44. WehaveS2(k) = k + 1forall k = 2n1,n N.7 Initial valuesofthefourfundamental quantitiesThe task of this penultimate section is to prove the values in Table 1 (in Theorem47;ofcourse,onlythefourfundamentalquantitiesareopen).k 1 2 3 4 5 6 7 8 9 10 11 12 13nM(k) 2 4 5 6 8 9 10 11 12 13 14 16 17nM1(k) 2 4 5 6 8 8 10 11 12 13 14 16 16VDM(k) 2 4 5 6 8 8 10 11 12 13 14 16 16VDH(k) 2 4 5 6 8 8 10 11 12 13 14 16 16FCM(k) 2 4 4 6 8 8 9 10 12 12 14 16 16FCH(k) 2 4 4 6 8 8 8 10 12 12 14 16 16S2(k) 2 4 4 6 8 8 8 10 12 12 14 16 16Table1.Valuesforthefundamental quantitiesfor1k13; inboldthecolumnsnotinSNM1,whiletheverticalbarsareleftofthespecialdeciencies2nn,n 2.Strengthening[23,Corollary 12.13],rstweshowstrongpropertiesformin-imallyunsatisableclause-setsF suchthat thenumber of full clauses equalsthemin-var-degree, i.e.,thereisavariable whichoccurs onlyin thefullclauses.We use varvd(F) :={v var(F) : vdF(v) =vd(F)}for F CLSwithn(F) > 0 (the set of variables with minimal degree). Furthermore we use DPv(F)14forFCLSandvvar(F)fortheresultofreplacingtheclausescontainingvariablevbytheirresolventsonv;indeedthespecialuseinLemma45willbejusttheinverseoftheexpansion processfromDenition36.Lemma45. ConsiderF MUwithfc(F) = vd(F)(andthusn(F) > 0).1. varvd(F)isthesetofall v var(F)whichoccuronlyinfull clausesofF.2. fc(F)iseven.3. Forv varvd(F)andF:= DPv(F)wehaveF MU=(F)fc(F)2+1.4. fc(F) 2 FCM((F) fc(F)2+ 1).Proof. Considervvar(F)withvdF(v)=vd(F). Theoccurrencesof varenow exactly in the full clauses of F(Part 1). Every full clauses must be resolvableon v, and thus the full clauses of Fcan be partitioned into pairs {v} C, {v} Cforfc(F)2many clauses C. This shows Part 2. Parts 3, 4 now follow by consideringF:= DPv(F):Fis obtained by replacing thefull clauses of Fby the clauses C(i.e., performs a full subsumption resolution, which are all strict except of the lastone,whichisnon-strict).ThenewclausesCarefullinF(thoughtheremightbeotherfullclausesinF).ObviouslyF MUand(F) = (F) fc(F)2+ 1.Fordeciencyk = 7wehavetherstcaseofFCH(k) < FCM(k):Lemma46. FCM(7) = 9 = nM(7) 1,whileFCH(7) = 8 = S2(7).Proof. ByS2(7) =8 we have FCH(7) 8. ByLemma 45, Part 4 andbyFCM(3) =4theassumptionof FCM(7) =10=nM(7) yields thecontradic-tion10 2 FCM(7 5 +1) = 2 4 = 8,andthusFCM(7) 9.ByLemma1weobtainFCH(7) = 8.Aclause-set F MU=7withfc(F) = 9(andn(F) = 4)isgivenbythefollowingvariable-clause-matrix: + + + +0+ + + + 0+ + + 0+ + + + + + + + +0 Letthevariablesbe1, . . . , 4, asindicesof therows. Nowsettingvariable4tofalseyields A3, where one non-strict subsumptionresolutionhas beenper-formed,whilesettingvariable4totruefollowedbyunit-clausepropagationof{3} yields A2. So both instantiations yield minimally unsatisable clause-sets,whenceby[23,Lemma3.15,Part2]F MU.5)Wearereadytoprovethenalmainresultofthispaper:Theorem47. Table1iscorrect.5)[23,Lemma3.15] containsatechnicalcorrectionover[21,Lemma1].15Proof. Thevaluesfor1k6havebeendeterminedin[23, Section14]. Weobservethat1, 2, 4, 5, 6, 9, 11, 12, 13 SNM1,andthusbyTheorem42nothingistobedoneforthesevalues,andonlythedeciencies7, 8, 10remain.By Lemma 45, Part 2, we get that FCH(8) = FCM(8) = 10 (since nM(8) = 11is odd), and also FCH(10) = FCM(10) = 12. By Lemma 46 it remains to provideunsatisablehittingclause-sets witnessingVDH(7) =10, VDH(8) =11andVDH(10) = 13.Fordeciency7considerF7:=0+ + + + +0 + + + + + + + + + + +0 + + +0+ + +.F7 has 4 variables and 11 clauses, thus (F7) = 114 = 7; the hitting property ischecked by visual inspection, and F7 is unsatisable due to 824+223+22=12+14+14=1, whilenallyeveryrowcontainsexactlyone0, andthusF7isvariable-regular ofdegree10 = nM(7).FinallyconsiderA4with(A4)=16 4=12andvd(A4)=16: performfour strict full subsumption resolutions on variables 1, 2, 3, 4, and obtain elementsofUHIT ofdeciency11, 10, 9, 8 withmin-var-degree 14, 13, 12, 11. 8 ConclusionandOutlookIn this paper we have improved the understanding of the four fundamental quan-tities, by supplying the lower bound S2 FCH. The recursion dening S2shedsalsolight onS2=S2, andwe gainedadeeper understandingof S2=2a2.Moreover webelieveConjecture48. k N : S2(k) = FCH(k).Thiswouldindeedgivean unexpectedpreciseconnectionofcombinatorial SATtheory and elementary number theory. On the upper bound side, by Conjectures12.1, 12.6 in [23](see Figure1 therefor asummary of therelations between thefourfundamentalquantities)weget:Conjecture49. k N : nM(k) 1 FCM(k) VDM(k) = VDH(k).RecallthatVDM(k) nM(k);sowebelievethatthreeofthefourfundamentalquantitiesareveryclosetonM(k). ThisisincontrasttonM(k) S2(k)beingunbounded, andindeedS2(k) =k+1for innitelymanyk (Corollary44),whilebyLemma43wealsoknowS2(k) =nM(k) for innitelymanyk, andthus S2oscillates between the linear function k +1 and the quasi-linear functionnM(k). Toeventuallydeterminethefourfundamentalquantities(which,ifourconjecturesaretrue, boil downtoVDMandFCM, whileVDH=VDMandFCH = S2),detailed investigations likethosein Section7 needtobecontinued.As FCH(k) and S2(k) are closely related via (boolean) hitting clause-sets, viageneralised(non-boolean)(hitting-)clause-sets(see[19,20]forthebasictheory)we can establish a close connection to the Sp(k) for all prime numbers p in forth-comingwork(Sp(k)isthesmallestnN0suchthatpkdividesn!,introducedin[28,UnsolvedProblem 49]).16References1. Ron Aharoni and Nathan Linial.Minimal non-two-colorable hypergraphs and min-imal unsatisable formulas.Journalof CombinatorialTheory, Series A, 43(2):196204,November1986. doi:10.1016/0097-3165(86)90060-9.2. GeorgeBoole. AnInvestigationof TheLaws of Thought, onwhicharefoundedTheMathematicalTheorieofLogicandProbabilities. Dover Publication, Inc., rstpublishedin1958. ISBN0-486-60028-9; printingoftheworkoriginallypublishedbyMacmillanin1854, withall correctionsmadewithinthetext. Availablefrom:http://www.gutenberg.org/ebooks/15114.3. B.W. Conolly. Meta-Fibonacci sequences (aletter fromB.W. Conolly). InFi-bonacci andLucasNumbers andtheGoldenSection, chapter12, pages127138.DoverPublications, Inc., 2008. Unabridgedrepublicationof theworkoriginallypublishedin1989byEllisHorwoodLimited,Chichester,England.4. YvesCramaandPeterL. Hammer. BooleanFunctions: Theory, Algorithms, andApplications, volume 142of Encyclopedia of Mathematics andIts Applications.CambridgeUniversityPress,2011. ISBN978-0-521-84751-3.5. Gennady Davydov, Inna Davydova, and Hans Kleine B uning. An e-cient algorithm for the minimal unsatisability problem for a subclass ofCNF. Annals of Mathematics andArticial Intelligence, 23(3-4):229245, 1998.doi:10.1023/A:1018924526592.6. Herbert Fleischner, Oliver Kullmann, and Stefan Szeider. Polynomialtimerecognition of minimal unsatisable formulas with xed clausevariable dif-ference. Theoretical Computer Science, 289(1):503516, November 2002.doi:10.1016/S0304-3975(01)00337-1.7. W.M. Gorman. More scope for qualitative economics. The Re-view of Economics Studies, 31(1):6568, January 1964. Available from:http://www.jstor.org/pss/2295936.8. Douglas R. Hofstadter. Godel, Escher, Bach: An eternal goldenbraid. Basic Books, 1979. Pdf version with 801 pages,md5sum=0cb32e8ea5dd2485f63842f5acb3f0 GEBen.pdf. Available from:http://www.physixfan.com/wp-content/files/GEBen.pdf.9. Henry Ibstedt. Computer Analysis of Number Se-quences. American Research Press, 1998. Available from:http://www.gallup.unm.edu/~smarandache/Ibstedt-computer.pdf.10. Brad Jackson and Frank Ruskey. Meta-Fibonacci sequences, binary treesand extremal compact codes. The Electronic Journal of Combinatorics,13(1), March 2006. R26; Corrigendum November 2007. Available from:http://www.emis.de/journals/EJC/Volume_13/Abstracts/v13i1r26.html.11. VictorKlee. Recursivestructureof S-matricesandanO(m2)algorithmforrec-ognizingstrongsignsolvability. LinearAlgebraanditsApplications, 96:233247,November1987. doi:10.1016/0024-3795(87)90347-8.12. Victor Klee andRichardLadner. Qualitative matrices: Strong sign-solvabilityandweaksatisability. InHarveyJ. Greenberg andJohnS. Maybee, editors,Computer-AssistedAnalysisandModel Simplication,pages293320, 1981. Pro-ceedings of the First Symposiumon Computer-Assisted Analysis and ModelSimplication, University of Colorado, Boulder, Colorado, March 28, 1980.doi:10.1016/B978-0-12-299680-1.50022-7.13. Victor Klee, Richard Ladner, and Rachel Manber. Signsolvability re-visited. Linear Algebra and its Applications, 59:131157, June 1984.doi:10.1016/0024-3795(84)90164-2.1714. Hans Kleine B uning.On subclasses of minimal unsatisable formulas.Discrete Ap-pliedMathematics,107(1-3):8398, 2000. doi:10.1016/S0166-218X(00)00245-6.15. Hans Kleine B uning and Oliver Kullmann. Minimal unsatisability and au-tarkies. InArminBiere, MarijnJ.H. Heule, HansvanMaaren, andTobyWalsh,editors, Handbook of Satisability, volume 185 of Frontiers inArticial Intelli-gence and Applications, chapter 11, pages 339401. IOSPress, February2009.doi:10.3233/978-1-58603-929-5-339.16. Oliver Kullmann. Anapplicationof matroidtheoryto the SATproblem. InProceedings of the15thAnnual IEEEConferenceonComputational Complexity,pages116124, July2000. doi:10.1109/CCC.2000.856741.17. OliverKullmann. Investigationsonautarkassignments. DiscreteAppliedMathe-matics,107:99137, 2000. doi:10.1016/S0166-218X(00)00262-6.18. Oliver Kullmann. Lean clause-sets: Generalizations of minimally un-satisable clause-sets. Discrete Applied Mathematics, 130:209249, 2003.doi:10.1016/S0166-218X(02)00406-7.19. Oliver Kullmann. Constraint satisfaction problems in clausal form I:Autarkies and deciency. Fundamenta Informaticae, 109(1):2781, 2011.doi:10.3233/FI-2011-428.20. Oliver Kullmann. Constraint satisfactionproblems inclausal formII: Minimalunsatisabilityandconictstructure. FundamentaInformaticae, 109(1):83119,2011. doi:10.3233/FI-2011-429.21. Oliver Kullmann and Xishun Zhao. On variables with few occurrences inconjunctive normal forms. In Laurent Simon and Karem Sakallah, edi-tors, Theory and Applications of Satisability Testing - SAT 2011, volume6695 of Lecture Notes in Computer Science, pages 3346. Springer, 2011.doi:10.1007/978-3-642-21581-0_5.22. OliverKullmannandXishunZhao. OnDavis-Putnamreductionsforminimallyunsatisable clause-sets. Theoretical Computer Science, 492:7087, June 2013.doi:10.1016/j.tcs.2013.04.020.23. OliverKullmannandXishunZhao. Boundsforvariableswithfewoccurrencesinconjunctive normalforms. TechnicalReport arXiv:1408.0629v3[math.CO],arXiv,November2014. Availablefrom:http://arxiv.org/abs/1408.0629.24. Frank Ruskey and Chris Deugau. The combinatorics of cer-tain k-ary meta-Fibonacci sequences. Journal of Integer Se-quences, 12, May 2009. Article 09.4.3. Available from:http://cs.uwaterloo.ca/journals/JIS/VOL12/Ruskey/ruskey6.html.25. Winfrid G. Schneeweiss. Boolean Functions with Engineering Applicationsand Computer Programs. Springer-Verlag, 1989. ISBN 3-540-18892-4.doi:10.1007/978-3-642-45638-1.26. Claude ElwoodShannon.A symbolic analysis of relay and switching circuits.Mas-ters thesis, Massachusetts Institute of Technology. Dept. of Electrical Engineering,1940. Availablefrom: http://dspace.mit.edu/handle/1721.1/11173.27. Neil J.A. Sloane. TheOn-LineEncyclopediaof IntegerSequences(OEIS), 2008.Availablefrom: http://oeis.org/.28. Florentin Smarandache. Only problems, not solutions! Xi-quan Publishing House, fourth edition, 1993. Available from:http://www.gallup.unm.edu/~smarandache/OPNS.pdf.29. Zhang Wenpeng, editor. Research on Smarandache Problems in NumberTheory (Collected papers). Hexis, Phoenix, USA, 2004. Available from:http://fs.gallup.unm.edu/wenpeng-book.pdf.1830. ZhangWenpengandLiuDuansen. OntheprimitivenumbersofpowerPanditsasymptotic property.In Jack Allen, Feng Liu, and Dragos Constantinescu, editors,SmarandacheNotions, pages173175. AmericanResearchPress, Rehoboth, NM,USA,2002. Availablefrom: http://dl.acm.org/citation.cfm?id=773309.31. Doron Zeilberger. How Berger, Felzenbaum and Fraenkel revolu-tionized covering systems the same way that George Boole revolu-tionized logic. The Electronic Journal of Combinatorics, 8(2):110,2001. The Fraenkel Festschrift volume; Article #A1. Available from:http://www.combinatorics.org/ojs/index.php/eljc/article/view/v8i2a1.