Relations between Protofuzzy Concepts Crisply Generated Fuzzy Concepts and Interval Pattern - PDF document

RelationsbetweenProto-fuzzyConcepts,CrisplyGeneratedFuzzyConcepts,andIntervalPatternStructuresVeraV.PankratievaandSergeiO.KuznetsovStateUniversityHigherSchoolofEconomics,Pokrovskiybd.,11,Moscow101000,Russia,vera.pankratieva@gmail.com,skuznetsov@hse.ruAbstract.Relationshipsbetweenproto-fuzzyconcepts,crisplygener-atedfuzzyconcepts,andpatternstructuresareconsidered.Itisshownthatproto-fuzzyconceptsarecloselyrelatedtocrisplygeneratedfuzzyconceptsinthesensethatthemappingsinvolvedinthede¯nitionsco-incideforcrispsubsetsofattributes.Moreover,aproto-fuzzyconceptdeterminesacrispsubsetofattributes,whichgeneratesa(crisplygen-erated)fuzzyconcept.However,thereverseistrueonlyinpart:givenacrispsubsetofattributes,onecan¯ndaproto-fuzzyconceptwhoseintentincludes(butnotnecessarilycoincideswith)thegivensubsetofattributes.Intervalpatternconceptsareshowntoberelatedtocrisplygeneratedformalconcepts.Inparticular,everycrisplyclosedsubsetofobjectsisanextentofanintervalpatternconcept.1IntroductionVariousapproachestoextendingthebasicde¯nitionsoftheFormalConceptAnalysistonumericalandmorecomplexdatarepresentationswereproposedinthelastthreedecades.Animportantdirectionofthisresearchisrelatedtofuzzyconceptanalysis,see[2]foranoverviewofvariousapproachestofuzzyconceptlatticesandsomerelationshipsbetweenfuzzyconceptlatticesandcon-ceptlatticesofl-cutsofformalfuzzycontexts.ArelationshipbetweenordinaryGaloisconnectionsandfuzzyGaloisconnectionswasgivenin[3].Inthispa-perwetrytoestablishrelationshipsbetweenmostimportantnotionsofseveralapproachestofuzzyconceptanalysis.Weshowhowcrisplygeneratedfuzzycon-cepts(R.Belohlavek,V.Sklenar,andJ.Zacpal[4]),proto-fuzzyconcepts(in-troducedbyO.KridloandS.Krajciin[5]),andpatternstructures(B.GanterandS.O.Kuznetsov[6]),inparticularintervalpatternstructures,arerelated.Itisshownthatproto-fuzzyconceptsarecloselyrelatedtocrisplygeneratedfuzzyconceptsinthesensethatthemappingsinvolvedinthede¯nitionscoincideforcrispsubsetsofattributes.Also,eachproto-fuzzyconceptdeterminesacrispsubsetofattributes,whichgeneratesa(crisplygenerated)fuzzyconcept.How-ever,thereverseistrueonlyinpart:givenacrispsubsetofattributes,onecan¯ndaproto-fuzzyconceptwhoseintentincludes(butnotnecessarilycoincideswith)thegivensubsetofattributes. RelationsbetweenConcepts51Intervalpatternconcepts,whichareparticularcaseofpatternstructures,areshowntoberelatedtocrisplygeneratedformalconcepts.Inparticular,everycrisplyclosedsubsetofobjectsisanextentofanintervalpatternconcept.Thepaperisorganizedasfollows.Thesecondsectioncontainsmainde¯ni-tionsofFCAandtheirextensionstofuzzydata.Thethirdsectiondescribesre-lationsbetweencrisply-generatedfuzzyconceptsandproto-fuzzyconcepts.Therelationshipbetweenfuzzyconceptsandintervalpatternconceptsisdescribedinthefourthsection.2FuzzyContextsandFormalConceptsWestartwithasetXofobjects,asetYofattributes,andafuzzyrelationIbetweenXandY.ThismeansthatatruthdegreeI(x;y)L,whereListhesetofvaluesofsomecompleteresiduatedlatticeL,see,e.g.,[1],isassignedtoeachpair(x;y),xX,yY.TheelementI(x;y)isinterpretedasthedegreetowhichattributeyappliestoobjectx.ThetriplehX;Y;Iiiscalledaformalfuzzycontext.Inpaper[3]thefollowingmappingsareintroduced.FuzzysetsALandBLYaremappedintofuzzysetsA"LYandB#LXaccordingtotheformulasA"(y)=V2X(A(x)I(x;y));B#(x)=Vy2Y(B(y)I(x;y))foryYandxX.De¯nition1([4]).AformalfuzzyconcepthA;BiconsistsofafuzzysetAofobjects(theextentoftheconcept)andafuzzysetBofattributes(theconceptintent)suchthatA"BandB#A.ThesetofallformalfuzzyconceptsisdenotedbyB(X;Y;I).De¯nition2([4]).AformalfuzzyconcepthA;Bi2B(X;Y;I)issaidtobecrisplygeneratedifthereexistsacrispsubsetBcYsuchthatAB#c(thus,BB#"c).Strictlyspeaking,suchcrisplygeneratedconceptsshouldbecalledcrisplyattribute-generatedfuzzyconceptsincontrasttocrisplyobject-generatedfuzzyconceptswhichcanbede¯nedinasimilarway.Thereisanotherapproachtogeneralizationofformalconceptanalysistothecaseoffuzzycontexts,whichwasdevelopedinpaper[5].GivenafuzzycontexthX;Y;Ii,de¯neitsl-cutsIl,lL,asIlf(x;y)XY:I(x;y)lgandintroducethemappings"l:2X2Yand#l:2Y2X:"l(A)=fyY:(xA)I(x;y)lg;AX;#l(B)=fxX:(yB)I(x;y)lg;BY: 52VeraV.Pankratieva,SergeiO.KuznetsovDe¯nition3([5]).ApairhA;Bi22X2Yiscalledanl-conceptifandonlyif"l(A)=Band#l(B)=A,whichmeansthatthepairhA;Biisaformalconceptinthel-cuthX;Y;Ili,whichisabinarycontext.Thesetofallconceptsinthel-cutisdenotedbyKl.De¯nition4([5]).TripleshA;B;li22X2YLsuchthathA;Bi2Sk2LKkandl=supfkL:hA;Bi2Kkgarecalledproto-fuzzyconcepts.Thesetofallproto-fuzzyconceptsisdenotedbyKP.De¯nition5([5]).LetBYbeanarbitrarysetofattributes.Wede¯nethecontractionofthesetofproto-fuzzyconceptssubsistenttothesetBasKPBfhA;li22XL:(B0B)hA;B0;li2KPg:Thus,theelementsofthesetKPBareproto-fuzzyconcepts(inthefuzzycontexthX;Y;Ii)\contractedtothesubsetB."De¯nition6([5]).De¯nethemappings*:LX2Y;+:2YLXasfollows:foranysubsetAofobjectsandanysubsetBofattributeslet*(~A)==fBY:(xX)(9hA;li2KPBxAl~A(x)g;+(B)(x)=supflL:(9hA;li2KPB);xAg:Themapping*takesafuzzysubsetofobjects~Atoa(crisp)setofat-tributesBY.Themapping+takesacrispsubsetofattributesBYtoafuzzysubsetofobjects~A.3Relationshipbetweencrisplygeneratedfuzzyconceptsandproto-fuzzyconceptsConsiderafuzzycontexthX;Y;Iiwithkobjectsandnattributes(i.e.,jXjk,jYjn)b1b2¢¢¢bn a1 d11 d12 ::: d1n a2 d21 d22 ::: d2n ... ... ak dk1 dk2 ::: dkn RelationsbetweenConcepts53andtakesome(crisp)intentBYtowhichsomeproto-fuzzyconceptsmaybecontracted.Tocompute+(B),wewritethesetBintheformofann-tuple(b1;b2;:::;bn),wherebi=1ifandonlyiftheintentBcontainstheithattribute.Then+(B)(x)=supflL:(9hA;li2KPB)xAg:Evaluating+(B)attheelementx1gives+(B)(x1)=supflL:(9hA;li2KPB)x1Ag:Thismeansthatitisnecessaryto¯ndacutthatcorrespondstothemaximumvalueoflandcontainsaproto-fuzzyconcept(includingtheelementsofthetoprow)whichmaybecontractedtoB.Inorderthatallentriesd1jthatcorrespondtounitjthattributesbecon-tainedinsomel-cut,itisnecessarythattheconditiond1jlbesatis¯edforallsuchindicesj.Ifwetakethein¯mumdVbj=1d1joveralljsuchthatbj=1,thenalld1jarenotlessthand.Therefore,alld1jarecontainedinthed-cutand,hence,theybelongtosomeproto-fuzzyconceptinthiscut.Suchproto-fuzzyconceptmayconsistofthetoprowcontractedtoB,unlessitmaybeextendedtootherrowstoformarectangular(i.e.,ifthecontractionofanyotherrowtoBcontainselementswhicharelessthand).Itmayalsohappenthatthed-cutcontainsarectangularthatconsistsofatleasttworowsandcanbecontractedtoB.Suchrectangularpresentsaproto-fuzzyconcept,sincethisisaformalconceptwhichdoesnotoccurhigherthanatthed-cut.Thus,a1+(B)(x1)=Vbj=1d1j.Thesameforallotherobjectsxi.Asaresult,wearriveatafuzzyset+(B)=(a1;a2;:::;ak)=0@bj=1d1j;bj=1d2j;¢¢¢bj=1dkj1A:Nowletuscomputetheresultofthemapping#appliedtothecrispsetofattributesB=(b1;b2;:::;bn):B#(x)=y2Y(B(y)I(x;y)):Hereandinwhatfollows,computationsareperformedintheframeworkoftlatticewiththeÃLukasiewiczlogicalconnectives(however,mostofthestatementsseemtoholdalsoforthegeneralcase),soab=minf1ab;1g,andab=minfa;bg.LetuscomputeB#(x1)|theresultoftheevaluationofthemapping#attheelementx1:B#(x1)=y2Y(B(y)I(x1;y))=(b1d11;b2d12;:::;bnd1n):Thevaluesbjthatareequaltozeroareinessential,sincethecorrespondingimplicationevaluatesto1:minf10+d1j;1g=1.Forthevaluesbjwhichare 54VeraV.Pankratieva,SergeiO.Kuznetsovequalto1,wehavebjd1j=11+d1jd1j.Thus,todeterminetheresultofthemapping,itisnecessaryto¯ndthein¯mumoverallelementsofthetoprow.Therefore,B#(x1)=Vbj=1d1j.Thesameforallotherobjects.Asaresult,weobtainB#=(a1;a2;:::;ak)=0@bj=1d1j;bj=1d2j;:::;bj=1dkj1A:Thus,wearriveatTheorem1.Themapping+coincideswiththemapping#restrictedtocrispsubsetsofattributes.1Foranarbitraryk-tupleA=(a1;:::;ak)bylA=(la1;:::;lak)wedenotethel-cutofA,i.e.,thecrispk-tuplewhosecomponentsarede¯nedaslaj(1;ajl;0;ajl:Theorem2.LetA=(a1;:::;ak)=B#c,BB#"c.DenotelWB#cWaiandconsiderthesetZ=(z1;:::;zk)suchthatzi=0ifailandzi=1ifail.Then(z1;z2;:::;zk)1Bcorrespondstosomeproto-fuzzyconceptinthel-cutthatmaybecontractedto1B.Proof.Considertherowsthatcorrespondtoail.Theelementsthatstayattheintersectionoftheserowswith1Bbelongtothel-cut.Thisrectangularcannotbeextendedtootherrows,sinceanyotherrowcontractedto1Bcontainselementswhicharelessthanl.However,itmayhappenthatitmaybeextendedtocolumnswhichcorrespondtozerocomponentsofBc.Inthecaseofsuchextensionweobtaininthel-cutaproto-fuzzyconceptwhichmaybecontractedtoBc.utRemark1.Thestatementthatthisproceduregivesaproto-fuzzyconcept(ratherthanacontractionofone)is,ingeneral,notvalid.Example.Considerafuzzycontextandits0.5-cutgivenbythefollowingtables:10.710.70.5-cut 0.2 0.2 0.2 0.5 0.7 X X 0.3 0.3 0 0.7 0.8 X X 0.5 0.5 0.8 0.8 0.2 X X X 0.5 0.5 0.7 0.7 0.4 X X X 1Whenthistextwasalreadypreparedforpublication,theauthorsdiscoveredthework[7],whichcontainsrelatedresults. RelationsbetweenConcepts55TakeBc=(1;0;1;0).ThenB#cA=(0:2;0:3;0:5;0:5);A"B=(1;0:7;1;0:7);B#=(0:2;0:3;0:5;0:5);Wai=0:5;Z=(0;0;1;1).Asaresult,ZBc=(0;0;1;1)(1;0;1;0)correspondstothecontext 0 0 0 0 0 0 0 0 1 0 1 0 1 0 1 0 Theorem3.LetA=(a1;:::;ak)=B#candBB#"c.Foranyi,1ik,specifyak-tupleZibytheformulazji(1;ajai;0;ajai:ThenthecontextZiBccorrespondstoaproto-fuzzyconceptintheai-cutoftheoriginalcontext,whichmaybecontractedtoBc.TheproofofthistheoremissimilartothatofTheorem2.Theorem4.Supposethatthel-cutcontainsaproto-fuzzyformalconcept.Thenthereexistsann-tupleBcsatisfyingthefollowingconditions:1.B#c=(a1;:::;ak),whereailifthisproto-fuzzyconceptinvolvestheithobjectandail,otherwise;2.then-tupleBcisclosedwithrespecttocrispcomponents(orjustcrisplyclosed);thatis,1(B#"c)=Bc.Proof.De¯nethen-tupleBc=(b1;:::;bn)asfollows:bi=1ifandonlyifthegivenproto-fuzzyconceptinvolvestheithattribute.Then,inaccordancewiththede¯nitionofaproto-fuzzyconcept,wehaveB#cA=(a1;:::;ak),whereailifthisproto-fuzzyconceptinvolvestheitheobjectandail,otherwise.Letusshowthatthen-tupleBcobtainedinthiswaysatis¯esProperty2.Suppose,bycontradiction,thatthereisanattributejsuchthatBc(j)=bj=0andB#"c(j)=1:Thenthereexistsanobjectithatbelongstothegivenformalconceptandsatis¯estheconditiondijl(orelsetheattributejbelongstothegivenformalconceptandBc(j)=bj=1).Now,takingintoaccounttheinequalityB#c(i)l,weobtainB#"c(j)1B#c(i)+dij1ldij1ll=1:Thisestimatemeansthatthen-tuple1(B#"c)hasnononzerocomponentsotherthanthoseofBc.ut 56VeraV.Pankratieva,SergeiO.KuznetsovByTheorem3thecontextlABcmaybeconsideredacontractionofsomeproto-fuzzyconcepttoBc.Inorderto¯ndoutwhetherthecontextlABcdeterminesaproto-fuzzyconceptoracontractionofsomeproto-fuzzyconcept,weconsidertheclosureBA"cofthecrispn-tuplelAAc.FollowingthelinesoftheproofofTheo-rem3onecanshowthatthecontextAclBisacontractionofsomeproto-fuzzyconcepttoAc.Ontheotherhand,then-tupleAclAwasde¯nedasthel-cutofthentupleA,whichistheclosureofthecrispn-tupleBcofattributes.Forthisreason,then-tupleAccannotbemajorizedbyagreater(crisp)n-tuple\possessing"theattributesBcand,therefore,thecontextAclBcannotbeconsideredanontrivialcontractionofsomeproto-fuzzyconcepttoAc.4RelationshipbetweenfuzzyformalconceptsandpatternstructuresDe¯nition7.LetGbeasetofanarbitrarynature,(D;)beameet-semilattice,andlettherebeamapping:GD:Thetriple(G;D ;±);whereD =(D;),iscalledapatternstructure,providedthattheset(G):=f()jGggeneratesacompletesubsemilattice(D;)of(D;).Eachcompletesemilatticeofthiskindhaslowerandupperbounds,whichwedenoteby0and1,respectively.Forapatternstructure(G;D ;±)wede¯nethederivationoperatorsactingonsubsetsAGasA:=lg2A()andonthesemilatticeelementsdDasd:=fGjd()g:Example.Considerthefollowingfuzzycontext: object1 0.2 0.2 0.5 0.7 object2 0.3 0 0.7 0.8 object3 0.5 0.8 0.8 0.2 object4 0.5 0.7 0.7 0.4 RelationsbetweenConcepts57ThesetGisthesetofallobjects.Theclosureoperatorappliedtoasub-setAofobjectsgivesann-tuple(thelengthofwhichisequaltothenumberofattributes|inthiscase,n=4)ofshortestintervalswhichcoverthevaluesofthecorrespondingattributesforallobjectsofthesubsetA.Forinstance,forthesubsetAf1;2gofobjectswehavef1;2gd12f[0:2;0:3];[0;0:2];[0:5;0:7];[0:7;0:8]gandd12f1;2g.Asinthebinarycase,apair(A;d),AG,dD,satisfyingtheconditionsAd,dA,issaidtocompriseapatternconcept.Thus,thepair(f1;2g,f[0:2;0:3];[0;0:2];[0:5;0:7];[0:7;0:8]g)isapatternconcept.Nowletuscomputef1;3g.Wehavef1;3gd13f[0:2;0:5];[0:2;0:8];[0:5;0:8];[0:2;0:7]gandd13f1;3;4g.Subsetsofobjectsmaybeconsideredascrispn-tuples(ofobjects),wherenjGj,extentsarethenclosedcrispn-tuples.Objectimplicationisde¯nedasusual:forsetsA;CGwewriteACi®AC,whereisanaturalsubsumptionrelationassociatedto:XYi®XYX.Inparticular,thereisanobjectimplicationaiaj(ai;ajG)iftherowofthecontextcorrespondingtoai(i.e.,ai)iscomponent-wisesmallerthantherowofthecontextcorrespondingtoaj(i.e.,aj).Theorem5.Ifthesetofallextentsofintervalpatternconceptscoincideswiththesetofallcrisplyclosedn-tuples,thenthecontextcontainsnoimplications.Proof.Assumethatall(crisp)n-tuplesthatcorrespondtoclosedsubsetsofobjects(i.e.,suchthatAA)arecrisplyclosedandtherearenoothercrisplyclosedn-tuples.Thensuppose,bycontradiction,thattheformalcontextcontainsanimplicationaiaj.Sincethecontextcontainsnoidenticalrows,allobjectsareclosed.Inparticular,aiai.Considerthecrispn-tupleAithatcorrespondstotheobjectai(0;:::;0| {z }i1;1;0;:::;0| {z }ni)anditsimageA"iunderthemapping".ItiseasilyseenthatA"iisafuzzyntuplethatcoincideswiththeithrowofthecontext.Sinceeverycomponentofthefuzzyn-tupleA"iisnotgreaterthanthecorrespondingcomponentofthejthrow,theclosureA"#icontains1atthejthcomponent.Thus,then-tupleAiisnotcrisplyclosed.Thecontradictionobtainedprovesthetheorem.utTheorem6.Foranycrisplyclosedn-tuplethecorrespondingsubsetofobjectsisclosed. 58VeraV.Pankratieva,SergeiO.KuznetsovProof.Consideranarbitrarycrisplyclosedn-tupleA.ThetupleA"consistsoftherowminimaforthesupportofthecrispn-tupleA.ThefactthatAiscrisplyclosedmeansthatA"#A,thatis,noneoftherowsdominateA".Hence,thecorrespondingextentisclosed,sincenootherrowfallswithintheintervalbetweentheminimumandthemaximumvalues.utThus,thesetofcrisplyclosedn-tuplesiscontainedinthesetofextentsofintervalformalconcepts.Corollary1.Inorderthatthesetofcrisplyclosedn-tuplescoincidewiththesetofextentsofintervalpatternconceptsitisnecessarythateachpatternconceptsatisfythefollowingcondition:theminimaoftheintervalscompriseann-tuplewhichisnotlessthananyobjectintentofanobjectnotcontainedintheextentoftheintervalpatternconcept.Proof.FollowsdirectlyfromTheorems5and6.ut5ConclusionsInthiswork,westudiedrelationsbetweenvariousgeneralizationsofformalcon-ceptanalysistothecaseofnumericalvalues.Theknownapproachesincludeproto-fuzzyconcepts,crisplygeneratedfuzzyconcepts,andpatternstructures.Itwasshownthatproto-fuzzyconceptsandfuzzyconceptshavemuchincommon:themappingsinvolvedintheirde¯nitionscoincideforcrispsubsetsofattributes.Also,eachproto-fuzzyconceptdeterminesacrispsubsetofattributes,whichgeneratesa(crisplygenerated)fuzzyconcept.However,thereverseistrueonlyinpart:anycrispsubsetofattributesisacontractionoftheintentofsomeproto-fuzzyconcept,butnotnecessarilytheintentitself.Intervalpatternconcepts,whichareparticularcaseofpatternstructures,wereshowntoberelatedtocrisplygeneratedformalconcepts.Inparticular,everycrisplyclosedsubsetofobjectsisanextentofanintervalpatternconcept.Su±cientconditionsarederivedunderwhichnootherextentsofintervalpatternconceptsexist.Furtherresearchinthisareawouldgointhefollowingdirections:{introduce\two-sided"(double)patternstructuresandstudytheirrelation-shipwithfuzzyformalconcepts;{¯ndaminimal(canonical)fuzzycontextthatinducesalatticeoffuzzycon-ceptsisomorphictoagivenone;{elaboratenewmethodsforfuzzydataanalysisonthebasisoftheresultsobtained.References1.H¶ajek,P.,MetamathematicsofFuzzyLogic,Kluwer,Dordrecht,1998. 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