Similarity in fuzzy reasoning
Abstract
Fuzzy set theory is based on a `fuzzification' of the predicate in (element of), the concept of membership degrees is considered as fundamental. In this paper we elucidate the connection between indistinguishability modelled by fuzzy equivalence relations and fuzzy sets. We show that the indistinguishability inherent to fuzzy sets can be computed and that this indistinguishability cannot be overcome in approximate reasoning. For our investigations we generalize from the unit interval as the basis for fuzzy sets, to the framework of GL--monoids that can be understood as a generalization of MV--algebras. Residuation is a basic concept in GL--monoids and many proofs can be formulated in a simple and clear way instead of using special properties of the unit interval.
Full text
Math w are&SoftC om puting 2( 19 95)197-228 S im ilarity in Fuzz yReasoning F rankKl a w onn a andJ ua nLuisC astr o b a D ept.Com puterS ci ence .U ni v.ofBraunsc hw eig 38106Braun sc hw ei g,Ge rm an y b Dpto.Cienc iasCo m putac i oneI n t.Art. ETSII ng enie r aInform a ti ca.Univ .G ra nada, 18071 Granada,Spai n Abs tr act F uzzys ettheo ryisba sedo na`fuzzi cation'ofthepredicat e 2 (elemen tof), t heco nceptofm em bershipdegr eesisconsideredasfundamen ta l.Int hispaperw eeluci datet heco nnection bet w eenin distinguis ha bilit ymodelle db yf uzzyequiv alencerela tio nsa ndf uzzy sets.W esho wthattheindisti nguishabilit y inheren ttofuz zy setscanbecom putedandthatt hisindist inguishabilit yca nno tb eo ve r co meina ppro xima ter easoni ng. F oro urin vest iga tio nsw egeneralizef romt heunitin terv al ast heb asis forfuzzyset s,totheframeworko fGL{monoids thatca nbeund erst oodasag enera lizat iono fM V{algebras. Residuat io ni sa basicconceptinGL{monoid sa nd ma n yproof s canbefo rm ulatedinasim pleandclearw a yins t eado fusing sp eci a l prop e r ti es of th e un it i n terv al. 1In tro duction Fuzzy set theory is bas edontheideathatmany n on{math ematical properties cann ot b e described interms of crisp sets comprising those 197
198 F.Kla w onn&J .L .C astro el em en tsthatfull lagi ve np ro pe rt y .The reforethenot io no fm e m - bershipiscons idered asagradualprope rt yf or fuzzysets. Sim il ar it yo ri ndi sti ngui shabil ity is anot he rim portan tconcept for whic hacri spm odelisofte ni na de quate .Apossibl eform aliz at i onof sim il ari t yor indi st ingui shabil it yisthe notion off uz zyequi v al enceo r eq ua lity r elation. Inthispape rw eex am inet he relati onsbet w eenfuzzy eq uiv alenc e rel ationsandf uzzy se ts.Astheunde rlyi ngf ormalf ram ew ork,w eabstractf romtheuni ti n terv altoG L{ mo noids{astructurei nwhich resi duati onpla y safunda m en talrole .Exam pl esforG L{ m onoidsare com pl eteMV{al gebrasortheuni tin terv alendo w e dwi ththeusual orderinganda left con ti n uoust{norm . Sect ion2pro vi dest he form albasi sforthi spaper by ab ri efrevi ew ofthede n i tionofGL{m onoi dsi ncludingsom euse fulproperti esand ex am plesaswell asthedeni tionsoffuzz ye quiv ale ncerel ati onsand ex tensional it ywhich r equire sthatafuz zyse tbe ha v esw e llwithrespe ct toagi v enf uz zyeq uiv alence relati on. InSecti on3w etak eac lose rl ookatt heextensi onali t ypropert y andsho who wasui table fuzzyeq ui v al encere lation, whic hde sc ribes theim pli citindi sti ngui sh abil it yi nfuz zysets,canbed e riv edfroma givenc oll ecti onoffuz zysets. Ourm ainresultsarepre se n tedinSec tion4 ,whe rew epro v ethat thei ndi stinguishab i li t yinhe ren ttofuzzys etsc annotbeo v ercom eb y theu su alappro xim atereasonings che m essothatfuzzyeq uivalenc e rel ations pro vi deausefuldescri ption of thisindi sti ngui shabil it y. 2GL {Monoids ,F uz zy Se ts andSim ilarit y Relations In thisSec tion w ebri ey in tro du c ethe no ti o ns on whic h our cons i derations are based. The underl yi n g s truc ture for o ur in vestigations are GL {m ono i ds. F rom [ 8],w e rec al l the deniti on of a G L{ m o noid and so m epro p erti es w hi chw ewi ll use. Denition 2.1 ( L; ; 3 ) is a GL{monoid i 1. ( L; ) is a compl ete l attice,
Sim ilari t yinFuzz yR easo ni ng 199 2. ( L; 3 ) isac ommutativemonoid,i .e.theop er ation 3 : L 2 L 0 ! L isasso ciative an dc omm utativeandhasaunit 11 2 L , 3. ( L; 3 ) hasazero el ement 0 0 2 L fullling 3 0 0=00 , 4. 3 isisotonic,i .e. = ) 3 3 ; 5. ( L; ; 3 ) isinte gr al ,i .e. 1 1= W L isa lsotheuniver sal upperb ound of L , 6. ( L; ; 3 ) isth edualofadivisibil itymonoid,i.e. implie sthee xistenc eof 2 L su ch that = 3 ; 7. ( L; ; 3 ) is residuate d,me aningthatthere existsabina ryop er ation ! on L sa tis fying 3 ( ) ! ; (1) 8.t heinnitedistributivelawh olds,i.e. 3 _ i 2 I i = _ i 2 I ( 3 i ) : F ortherestoft hi sp ap erw eassum e( L; ; 3 )tobe aG L{ M onoid. L wi llbeconsi deredast he seto f\truth"{v al uesofam an y{ val ue d logic. The binary operation ! isuniq uelyd eterminedb ytheadj unc tion pro pert y(1): ! = _ f 2 L j 3 g : (2) ! can b e vie wed as the v al uati o n functi o n for the i mpli cation (a sso ciate d with th e conjuncti on 3 ). F ro m this i mpl icationw e can de riv ei n a canoni cal w aya v al ua ti o n fo r the nega ti on b y dening : = ! 00. Note tha t in a GL {m ono i d the ze ro ele men tof 3 isa lso the u ni versa l lo wer b ound, i .e .0 0= V L .W ede n e the bii mplication $ : L 2 L 0! L by $ =( ! ) ^ ( ! ) :
200 F.Kla w onn&J .L .C astro In teresti ngforappli cationsi st hecasewhen L i stheuniti nt erv al withtheusuall inearo rde ring. 3 canbean ycontin uo ust{norm(acom - m utativ e, associati v e ,non{dec reas ingb inaryoperationon[0 ; 1] ha ving 1as unit). 3 i sunderstood as an alte rnati v etothelatti ce op erati on ^ (i nthec aseof th eu ni tin te rv alsi m plym in) fort he v aluationfunction of ac onjunction. NotethataGL{m onoidi sacom ple teMV {al ge bra(see, e. g. ,[1,4] ) i fandonl yifth e` ne gati on' ! 0 0i sin v olutory ,i .e. :: =( ! 0 0) ! 0 0= .Ho w e v e r,forpracti calapplica ti onswiththeunitin tervala sthe underl yingl atti ce L ,c on side ringonlyMV{algebras is v eryrestri cti v e, sinc e 3 =m ax f + 0 1 ; 0 g istheonl yc hoi cefort heoperation 3 uptoisom orphis m. Thi sholdsforthefoll ow ingr e ason.B ecause : is ani n v ol utionandno n{incre asi ng,inac cordance to thei sotonici t yof 3 , : i salso con ti n uous.F orMV{algebras , 3 sati se salsotheequation (cf .Le m m a1.4 (6)of[ 8] ) 3 ^ i 2 I i = ^ i 2 I ( 3 i ) ; sothattogetherwi ththeinni tedistri buti vi t yof 3 o v erarbi traryjoi ns , w eobtai nthat 3 isal soc on tinuous.Thisim pl ie sthat ! isc on tin uous, sinc ew eh a v e(c f.Le m m a1. 4(3)of[ 8] ) ! = : ( 3: )=( 3 ( ! 0 0)) ! 00 : Bu ti ft heuni ti n ter v alistheunderl yi nglattic eoft heGL{monoid whoseop eration ! iscon ti n uous,then ([0 ; 1] ; ; 3 )isisom orphi cto theGL{m onoid (MV{algebra)([ 0 ; 1] ; ; 3 L )with 3 L =max f + 0 1 ; 0 g ( Luk asiewi czconjunctio n) (see ,e .g. , [19 ] ). Ex ample 2.2 . Le t L =[ 0 ; 1] b e t he unitin terv al with the usua l ordering. Th e n $ =m ax f ; g! min f ; g holds [ 22 ] . The following ta ble provides so meexa mples fo r the o p eration 3 and the in duced o p erations ! ; $ ,and : (cf. [12, 18]).
Sim ilari t yinFuzz yR easo ni ng 201 3 max f + 0 1 ; 0 g min f ; g 1 ! mi n f 1 0 + ; 1 g 1if otherw ise 1if otherw ise $ 1 0j 0 j 1if = mi n f ; g otherw ise 1 i f = m in f ; g max f ; g otherwise : 1 0 1if =0 0otherwise 1if =0 0otherwise Ifw ec hoose 3 = ^ ,then( L; ; 3 )i saH eyti ngalgebra,w he rew e obtainf or L =[0 ; 1]theG odelim pl ica ti on (seethethirdcolum ni n thet abl eabo v e)astheoperation ! . Le tusc ol lec tsom eusefulpropert iesofGL{m onoi dsw hic hw enee d int hisp aper. Le m m a2. 3. In anyGL {monoid ( L; ; 3 ) ,thefo ll owingpr op erties ar evalid: (a) ^ = 3 ( ! ) , (b) 3 ^ i 2 I i ^ i 2 I ( 3 i ) , (c ) ( ! ) 3 ( ! ) ! , (d) ( $ ) 3 ( $ ) $ , (e ) ) ! ! , (f ) ( 3 ) ! = ! ( ! ) , (g ) ) ! ! , (h) ( ! ) 3 ! ( 3 ) , (i) ! ( ^ )= ( ! ) ^ ( ! ) , (j) ( 1 $ 1 ) ^ ( 2 $ 2 ) ( 1 ^ 2 ) $ ( 1 ^ 2 ) . Pro of. (a), (f ), and ( i) are pro ved in the L em mas 1 . 1(2) an d1. 2(3) of [8]. (b) The mon otonicityof 3 implies 3 V i 2 I i 3 j fo r all j 2 I .
202 F.Kla w onn&J .L .C astro (c )I nacc ordancetotheadjunctionpropert y ,w eha v etop ro ve that 3 ( ! ) 3 ( ! ) : U si ng(a),w eobtai n: 3 ( ! ) 3 ( ! )=( ^ ) 3 ( ! ) 3 ( ! ) = ^ : (d)B yapply ing(b)and(c ),w eget: ( $ ) 3 ( $ )= (( ! ) ^ ( ! )) 3 (( ! ) ^ ( ! )) (( ! ) 3 ( ! )) ^ (( ! ) 3 ( ! )) ( ! ) ^ ( ! ) = $ : (e )and(g)aredi rectconseq uencesofthei sot onic it yof 3 andeq uati on(2). (h)I na cc ordancetotheadjuncti onp ropert y ,w eha v etopro v ethat 3 ( ! ) 3 3 : Dueto(a),thele fthands ideo ft hi sinequali t yissimply ( ^ ) 3 whic hisl ess than orequalto 3 be causeof th ei sotoni cit yof 3 . (j) Us ing(i )and( e),w eobtain: ( 1 ^ 2 ) $ ( 1 ^ 2 )= (( 1 ^ 2 ) ! ( 1 ^ 2 )) ^ (( 1 ^ 2 ) ! ( 1 ^ 2 )) =(( 1 ^ 2 ) ! 1 ) ^ (( 1 ^ 2 ) ! 2 ) ^ (( 1 ^ 2 ) ! 1 ) ^ (( 1 ^ 2 ) ! 2 ) ( 1 ! 1 ) ^ ( 1 ! 1 ) ^ ( 2 ! 2 ) ^ ( 2 ! 2 ) = ( 1 $ 1 ) ^ ( 2 $ 2 ) :
Sim ilari t yinFuzz yR easo ni ng 203 2 Asw ehav ea lreadym en tioned,w ei n terpr et L asthese toftruth v al ues.Th usage nerali za tion of deniti onsofcl assic al conce ptsi s nece ssary .H erewe restrictoursel v e stothenotions of subsetsand equi va lencer e lations. Denit ion2. 4. A n L {fuzzy( sub)set( orsimplyafuzzys et)o ftheset X isama pping : X 0 ! L . Th ev alue ( x ) 2 L i sunderstooda sthede gre eortruthv al ueof x be inganelem en to fthe(sub)se t . Denit ion2. 5. A n L {fuzzye quivalenc er elation (orsimp lyafuzzy eq uivalenc er elation)(withresp e cttoth eop er ation 3 )o nth es et X is am apping E : X 2 X 0 ! L satisfy ingtheaxi om s: (E1) E ( x ;x )=1,(ree xiv it y ) (E2) E ( x ;y )= E ( y;x ),(sym m etry) (E3) E ( x ;y ) 3 E ( y;z ) E ( x;z ).(transi tiv it y) Note thatfor L = f 0 ; 1 g , E isafuzzye qui v al encerelati oni fa nd onl yi fi ti sthec haracteri sti cf unctionofanordi naryequi va lencer elation.De pe ndi ngon thec hoiceoftheoperation 3 ,so meti m es E i s al so call edasim i larit yrela ti on [24] ,indi sti ngui sha bility operator[ 21], fuzzy equalit y(rel ati on)[ 9,16]o rp ro xi m i t yrel ati on[ 6] . Exam pl e 2.6 . (i) If is an eq uiv alenc ere lation on X ,the nits c haracteri sti c function E R ( x; y )= ( 1 if x y; 0otherwise, isaf uz zy equi val ence rel ati on with re spec ttoan yc hoic e of the op eration 3 .Es p ecially the cha racteristic function of the crisp equalityon X is a fuzzy equivalence relation.
204 F.Kla w onn&J .L .C astro (ii) Let L =[0 ; 1]andl et 3 be the Luk asie wi czconjunctio n. Then E isafuzz ye quivale ncerel ati onon X wi thres pe ct to 3 i fand onlyif1 0 E is ap se udo{m etr icon X .Thusp se ud o{m etri cs bou ndedb yoneandfuzz yequi valencerel ati onswi thres pe ct to the Luk asiewiczc onjunc tionaredualconcepts. Ifapseudo{ metric i snotbounde db yone,we cane nforcethi spropert y b yc onsideri ngthepse udo{ m etr ic =m i n f ( x ;y ) ; 1 g ,which co - i ncide swi th for \s m al l"di stance s.Th usan yp seudo{m etri c i nd uc es afuzz yeq uiv alenc ere lation on X wi threspecttothe Luk asie wi czc onjunctio nb y E ( x;y )= 1 0 ( x;y )= 1 0 mi n f ( x ;y ) ; 1 g : (i ii )We obtai nthesam edual it yasin( ii)whenw erepl acethenotion ofm etr icb yultra{m etri csandthe Luk asie wi czc onjuncti onb y them inim um . Whenw ei n terpretanordi naryequi v al encerel ati on ontheset X i nthesensetha tequi v al en tele m en tscannot bedistinguishedorm a y beiden tied,the nonlythoses ubsets M X satisfyi ng x 2 M and x y ) y 2 M (3) \be ha v ewell "w ithrespectt o .Inother words, asubset M full ls conditi on(3)ifando nl yifi tequalsauni onofequi va l enceclassesof . Thefoll o wi ngde ni tion generali zestheaxi om(3)for fuzz ye q uiv alenc e rel ationsandfuzz ysets. De ni tion2.7. Afuzzyset 2 L X isc alle dext ensio nalw.r.t.the fuzzy e quiv ale nc er el ation E on X i ( x ) 3 E ( x;y ) ( y ) holds for al l x; y 2 X . If a fuz zy set isnot extensi onal wi th resp e ct to the c o nside red fuzzyequivalencerelation E ,wemayconsider instead of the smallest extensiona l fuzzy set whichconta ins .
Sim ilari t yinFuzz yR easo ni ng 205 Denit ion2. 8. L et E b eafuzzy equi valenc er elationon X and le t 2 L X .Th efuzzyset ^ = ^ f j and isexte ns iona lw.r. t. E g isc alle dtheextens ionalhullof w.r. t. E . Propo siti on2.9. L et E b eafuzzye quiv alenc er elationon X a ndl et 2 L X .Th en (i) ^ ( x )= W f ( y ) 3 E ( x ;y ) j y 2 X g , (ii) ^ isextensionalwithr esp e ctto , (iii) ^ ^ =^ . Proof. (i)Le tusabbrevi atetherigh thandsideof(i )b y~ .~ i sex tensional w.r. t. E bec ause ~ ( x ) 3 E ( x ;y )= E ( x ;y ) 3 _ f ( z ) 3 E ( y;z ) j z 2 X g = _ f ( z ) 3 E ( x;y ) 3 E ( y;z ) j z 2 X g _ f ( z ) 3 E ( y;z ) j z 2 X g =~ ( y ) : W ealsoha ve that ~ ( x )= _ f ( y ) 3 E ( x; y ) j y 2 X g ( x ) 3 E ( x ;x )= ( x ) ; whic hi mpli es ~ ^ . In order to pro ve~ ^ , let be an e xtensiona l fuzz y set of X w.r. t. E suc hthat .Since ( x ) ( y ) 3 E ( x ;y ) ( y ) 3 E ( x;y ) ho l ds for a n y y 2 X ,w eha veal so ~ ^ . (ii) In the pro of o f (i) wehave already shown tha t ~ =^ is extensiona l w.r.t. E .
212 F.Kla w onn&J .L .C astro Th us w ecanre wri te 3 inthef orm ( ! ( ! 00 )) ! 0 0= 3 (( ! 00) ! 0 0) ! 0 0 ! 0 0= 3 ; sothat(c )isal soful l le d. (d)isjust(i i).Usi ng Equation,(8)w ecan rewri te ! by ! =( 3 ( ! 0 0)) ! 0 0 : Sinc ew eha v ealre ad ys ho wnthat A sati se s(c ),we ha v ealsopro v ed (e). 2 Theorem3. 4sho wsthat when theunderl yi ngGL{m onoidi sa com - ple te MV{algebra,thealgebraic characte rizati onofthe se to fe xtensional fu z zysetssi m pli estotheclosednesswithrespe cttoarbi trary unionsandtothegenerali zedcom pl em e n tati on ! . Inthi sSe ctionw eha v ediscussedconnec tionsbet w eenfuzzy sets andfuzzyequi v al encerelations.F or agiv enc ol lec tionoffuzzyse ts, thefuzzyequiv al encerel ati on(5)c harac teri zestheindi sti ngui shabil it y i nhe ren ttothe sefuzz ysets.I nthefol lo wingsecti onw ewi llsho wthat thisindi stingui shabil it ycannotb eo v e rcom ei nt ypic alappro xi mate reasoni ngsituationswi thfuz zysets. A tthee ndofSe ction2we ha v eex plainedho wcri sp po in tsand setsinducefuz zysets in th ep re se nceofafuzzye quivale ncerel ati on. Since iti ssuc ien tf orourpurposestoc harac teriz ethei nd istingui shabil it yi nhe rent to ag iv encoll ect ionoff uz zy se ts ,i nthispape rw edo notpursuethequestio nw hena giv encol le ctionoffuzzys etsc anbe i nterprete dasextensi on al hullso fc ri spel em en tsors etsw ithr espe ct to a su i tab l e fuzz y equiv alence rel ati on . This question is treated in [13, 18, 20 ], for a n o vervi ew se e[14] . 4 Sim ilarityRe la tions in F uz zy Reasonin g In [15, 18],itwas show n that fuzzy control can b e interpreted as interpolation in th e presence o f ind istinguisha bilitycha racterizedby fuzzy
Sim ilari t yinFuzz yR easo ni ng 213 equi v al encerelati onswhe nevert hefuzz ysetsuse dfort he fu zzypartiti onssati sf ys om er e asonabl erestric tions .T he correspondingfuzzy equi v al ence re lationsc anbecom putedo nt he basi sofform ula(5). Int hisSecti onw ewi llsho wt ha ti nt ypicalappli cationso ff uzzy reasoni ng{notonlyf or fuzz yc on trol{thesefuzzy equiv ale nc er elationsa re of im portanc esi nc ethe ycharac teri zeanindisti nguishabil it y thatcannotbeo v e rcom e. Inappro x im atere as oningonehasoft entode alwithi f{thenrul es oftheform If is A; then is B; (9) where and arev ari abl esw ithdom ains X and Y ,respectiv e ly . A and B arel i nguisti cte rm sli k e positiv ebig or appro x im atel yzero (see,e .g., [17]). Thesel inguisticte rm sareusua llym ode ll edb ysui tabl efuzzy sets,sa y A 2 L X and B 2 L Y . The rule softheform(9)re presen tge neralk nowl edgeaboutt he consi deredprobl em .Intheactualsituation,i naddi tiontothisgene ral kno wle dge,the inform ation i s A 0 (10) isgiv en,w here A 0 i sre presen te db ythefuzz yset A 0 2 L X (orsi mply b y 2 L X ) .O fcourse,a saspe cialcase A 0 canalsostand foracr isp va lue x 0 in whic hc ase A 0 ( x )w oul dyi eldthev alueonef or x = x 0 and zer oo therwi se .Th us,ani nf erence sc hem eisneededthatderiv es from thefuzzyse ts A ; B ,appearing in therule s,andthefuzzyset A 0 , represe n tingthe actual i nform ati on,afuzzys et conclu sion thatdesc ribes the restri cti onorpo ss ible valu e sf orthev ari abl e undert he giv enrul es andtheactualinputi nform ation(10). Le tusrstonl y consider a si ng l erul e of the fo rm (9) .Av ery con venie nt p ossibil ityto re presen t suc ha rul eisi n the f o rm o f a fuzzy rel ati on % 2 L X 2 Y . Usual ly is dened o n the b asis o f o ne of the o p erations ^ ; 3 ; and ! ,i.e. % ( x; y )= % ( x; y )= A ( x ) B ( y )(11) where 2 f^ ; 3 ; !g . For a given input info rmatio n in the form of the fuzzy s et A 0 2 L X , the \ output" fuzzy set conclusion is computed
214 F.Kla w onn&J .L .C astro as thec om posi tionofthefuzzyr elati on % an dt hefuzz yset A 0 . Thec om posi tiono fafuzzyrel ati onandafuzzys et is de ne dasa generalizationofthecom posit ionof an or di naryrel ati on R X 2 Y withanordinaryset M X . M R = f y 2 Y j ( 9 x 2 X )( x 2 M and( x;y ) 2 R ) g V aluat ingthee xiste n tialquan tie ri nthisform ula,asusuall y, by the suprem umandtheconjunc tionb ytheoperation u2f^ ; 3g ,w eobtain thefoll owingd e nitionofthecom positio no fafuz zyre lation % 2 L X 2 Y and afuzzyset 2 L X : ( u % )( y )= _ x 2 X f ( x ) u % ( x ;y ) g (12) foral l y 2 Y ,where u2f ^ ; 3 g (c f.,e .g., [5,7,17]). Ex ample 4.1. Le t L =[0 ; 1]withtheus ualorde ring.F or u = ^ = m i nand = ^ ,(12)bec om es ( % )( y )= _ x 2 X m in f ( x ) ; A ( x ) ; B ( y ) g ; (13) i .e. % ( x;y )=m i n f A ( x ) ; B ( y ) g : (14) Thisi stheusuali nferences chem eforasi ngl eruleinfuzzyc on trol, whereinm ostc as es i sthec haract eristi cf uncti onofase twitha singleel em e n trepres en ti ngthecri spi nputv alue x 0 2 X .Inthisc ase, (13 )sim plie sto ( x 0 % )( y )=m in f A ( x 0 ) ; B ( y ) g : Another i ntere sti ng e xam ple is u = ^ =m in, 3 = ^ ,and = ! . In thi s case is theG odel i mpl icati o n, so that weha ve % ( x; y )= ( 1 if A ( x ) B ( y ) ; B ( y )otherwise. (1 5) Bo th fuzz y relations (14) a nd (1 5) pla yan i mp o rtan t role for the so - lution of fuzzy relation equations of the form A % = B (1 6)
Sim ilari t yinFuzz yR easo ni ng 215 with gi v e nfuzzysets A an d B andu nk no wnfuzz yrelati on % .For theusualsup{m in co mp ositi on,theoperation is dene donthebasi s of u = ^ .I ft he fuzz yrel ati onEquati on (1 6) hasatl eastonesol ution % ,t hen(14)and(15)areal so soluti ons(see ,e.g.,[ 7]).T he great est sol ution(15) is also v eryusefulforsy stem so ff uzzyrel ati one quati ons , since, by takingthem ini m umo vert he so luti onsofthesi ngl ee quat ions, oneobtai nsasol uti on of thesy ste mofequationswhe nt hereex istsa sol utionofthe systematall . Le tusre m ark thati tisnotreasonabletoconsiderallpossible com binati onsofthec hoice sfor u and in(12)wi th % = % inEquation (11),asthef ol lo wingl em m asho ws. Le m m a4. 2. L et A 2 L X b eano rmalfuzzyset,i .e .t her eexists x 2 X suchthat A ( x )=1 1 hold s,andlet B 2 L Y .Dene % asin Equation (11) .T he nthee quation ( A u % )( y )= _ x 2 X f ( x ) u % ( x;y ) g = B ( y )(17) iss ati se dforall y 2 Y forthe c om binations = ! and u = 3 , = 3 and u = 3 , = 3 and u = ^ , = ^ and u = 3 , = ^ and u = ^ . Proof. (i) = ! and u = 3 . T ak ingthenorm alit yof A andLe m m a2. 3(a )i ntoaccoun t,w e obt ai n: ( A u % )( y ) = _ x 2 X f A ( x ) 3 ( A ( x ) ! B ( y )) g = _ x 2 X f A ( x ) ^ B ( y ) g = B ( y ) : (i i) = 3 and u = 3 . The no rm al it yof A immedi ate ly yi el ds ( A u % )( y )= _ x 2 X f A ( x ) 3 ( A ( x ) 3 B ( y )) g = B ( y ) :
216 F.Kla w onn&J .L .C astro (i ii ) = 3 and u = ^ . Thenorm al it yo f A giv es ( A u % )( y )= _ x 2 X f A ( x ) ^ ( A ( x ) 3 B ( y )) g = B ( y ) : (iv ) = ^ and u = 3 . A ga in usingthenorm ali t yof A w eo btain: ( A u % )( y )= _ x 2 X f A ( x ) 3 ( A ( x ) ^ B ( y )) g = B ( y ) : (v ) = ^ and u = ^ . Fi nal ly ,thenorm al it yo f A al soensures ( A u % )( y )= _ x 2 X f A ( x ) ^ ( A ( x ) ^ B ( y )) g = B ( y ) : 2 Ex am ple4.3. F orthem issingcom bi nati on = ! and u = ^ in Lem m a4. 2e quati on(17)doe si ngeneralnothold.Let L =[0 ; 1] beendo w e dwi th theusualorderi ngandthe Luk asie wiczconjunction 3 .Consi der X = Y = f a;b g , A ( a )=1, A ( b )=0 : 9, B ( a )=1, B ( b )=0 : 8.The nw eobtain: ( A u % )( b )=m ax f A ( a ) ^ ( A ( a ) ! B ( b )) ; A ( b ) ^ ( A ( b ) ! B ( b )) g =m ax f 0 : 8 ; 0 : 9 g =0 : 9 > B ( b ) : W eareno w prepared to f orm ulatet wointere sting a nd im p ortant theorem sabou t apply ing i f{ the n rule sandt he i nd i stinguisha bi li ty inhere nt to the corresp o ndi ng fuzzy se ts . Theore m4.4. Let ; A 2 L X , B 2 L Y . Furthermo re, le t E be a fuzzy eq uival encere lation on X such that A is e xte nsional w .r.t. E .Let % be den ed as in Equat ion (11 ) . Then for the combin ations
Sim ilari t yinFuzz yR easo ni ng 217 = ! and u = 3 , = 3 and u = 3 , = ^ and u = 3 ,thee quation (cf .Equati on (12 ) ) ( u % )=( ^ u % ) is va lid. Proof. ^ ,togethe rwi ththei sotonici t yof u ,im pl ie sthat ( u % ) (^ u % ) foran yc hoic eof and u .Thereforew eonl yha v etopro v etheo the r ine qual it y . (i) = ! and u = 3 . (^ u % )( y )= _ x 2 X f ^ ( x ) 3 ( A ( x ) ! B ( y )) g = _ x; x 0 2 X n ( ( x 0 ) 3 E ( x;x 0 )) 3 ( A ( x ) ! B ( y )) o _ x; x 0 2 X n ( x 0 ) 3 E ( x;x 0 ) 3 ( E ( x;x 0 ) 3 A ( x 0 )) ! B ( y ) o = _ x; x 0 2 X n ( x 0 ) 3 E ( x;x 0 ) 3 E ( x;x 0 ) ! ( A ( x 0 ) ! B ( y )) o = _ x; x 0 2 X n ( x 0 ) 3 E ( x;x 0 ) ^ ( A ( x 0 ) ! B ( y )) o _ x 0 2 X n ( x 0 ) 3 ( A ( x 0 ) ! B ( y )) g = ( u % )( y ) : In the s e cond l ine we used the i nnite di stributiv elaw, in the third th e ex tensional ityof A to gether with Lem ma 2. 3(e), i n the fourth L emma 2.3(f ), in the fth L emma2.3(a ), and in the sixth the isoton icityof 3 .
218 F.Kla w onn&J .L .C astro (ii) = 3 and u = 3 . I nac cordancet ot he innitedistributi v ela wa nd theexte nsi onali t yof A ,w eha ve that (^ u % )( y )= _ x;x 0 2 X n ( x 0 ) 3 E ( x;x 0 ) 3 A ( x ) 3 B ( y ) o _ x 0 2 X n ( x 0 ) 3 A ( x 0 ) 3 B ( y ) o =( u % )( y ) : (i ii ) = ^ and u = 3 . (^ u % )( y )= _ x; x 0 2 X n ( x 0 ) 3 E ( x;x 0 ) 3 ( A ( x ) ^ B ( y )) o _ x ;x 0 2 X n ( x 0 ) 3 ( E ( x ;x 0 ) 3 A ( x )) ^ ( E ( x;x 0 ) 3 B ( y )) o _ x 0 2 X n ( x 0 ) 3 `( A ( x 0 ) ^ B ( y )) o =( u % )( y ) : I nthesec ondli newe ap pl iedLem m a2.3(b). 2 Whenw ei n terpre tT he or e m4.4inthesensethatthef uzz ysets A and B represen tthel inguisti ct erms A and B ofanif{thenrul e oftheform (9), the nitst atesthatf orth em en ti onedcom bi nati onsof ope rati onsfora gi ve ni nput theoutputfuz zy set u % infe rred byt he rul edoes notc hang e when w erepl ac e byi ts ex tensiona l h ull. This m eansthatt he i ndistinguisha bi li tyi nh eren t to the fuzz yset A canno t b e a voi ded, e ven if the input fuzz yset standsforac risp value. Note that if is a cri sp se t, the c hoice o f the op e ra ti on u ha s no i nuence i nE quation (1 2). From Theorem 4 . 4w ede riv e also a n a nsw er t otheq uestion of fuzz yi nputs . A fuzz ied input do es not c ha nge the outcom eof a rul e as long a s the fuzzy s et obta ined by the fuzzication iscontained in the extensio nal of the o rigina l crisp input value. From Theorem 4 .4we
Sim ilari t yinFuzz yR easo ni ng 219 al so learnthati tdoesnotm ak es ensetom e asurem oree xactlyt han theindi sti ngu is habil it yadm i ts. Thecase s, that areco v eredb yThe orem 4.4, incl udethem ostcom - m on andusefulform aliz at io ns of if{thenrul es(see[ 2, 3]), nam el ythe sc hem e = ^ and u = 3 with 3 = ^ asw el lasthesc he me = ! and u = 3 with 3 = ^ . Thefoll o wi ngexam ple showsthatThe ore m4.4doe sno th ol dfor theothe rc om binations. Exam pl e4.5. F orallc oun terexa mp l esw eassum e L =[ 0 ; 1]e ndow ed withtheusualorderingan dthe Luk asiewi czconjuncti on. (i) = ! and u = ^ . Let X = f x 1 ;x 2 g , Y = f y 1 ;y 2 g , A ( x 1 )=1 , A ( x 2 )=0 : 5, B ( y 1 )=0, B ( y 2 )=1, E ( x 1 ;x 2 )=0 : 5, ( x 1 )=1, ( x 2 )=0. Thenw eha v e(^ u % )( y 1 )=0 : 5 > 0=( u % )( y 1 ). (i i) = 3 and u = ^ . Le t X = f 0 ; 0 : 5 ; 1 g , Y = f y 1 ;y 2 g , A ( x )= x , B ( y 1 )=1, B ( y 2 )=0, E ( x;x 0 )=1 0 m in fj x 0 x 0 j ; 1 g , (0)=1, (0 : 5)= (1)=0.Thenw eha v e(^ u % )( y 1 )=0 : 5 > 0=( u % )( y 1 ). (ii i) = ^ and u = ^ . Le t X = f x 1 ;x 2 ;x 3 g , Y = f y 1 ;y 2 g , A ( x 1 )=1, A ( x 2 )=0 : 5, A ( x 3 )=0, B ( y 1 )=0, B ( y 2 )=1, E ( x 1 ;x )= A ( x ), x 2 X . E ( x 2 ;x 3 )=0 : 5, ( x 3 )=1 , ( x 1 )= ( x 2 )=0 .Thenw eha v e (^ u % )( y 1 )=0 : 5 > 0=( u % )( y 1 ). The or em 4.4s ho wsthattheindistin guishabil it yinduce dby the fuzzy setre presen ti ngthel inguisticinthepremiseoftherul ec annot be o verc om e. T he sam ehol ds for the o utput o f the rule ,i. e. the output can nev er b e m o re prec ise tha n the indisti n guishabili tyi nduc ed b ythe fuzzyse tmodel li n g the li ng ui sti c term i n the c onc lusion of the rule . Theore m4. 6. Let ; A 2 L X , B 2 L Y . Fur t herm or e, let F be a fuzzy e q uival encer elat io non Y such that B is e xte nsional w .r.t. F .L et % bed ene d as in Equation (11 ) . Then for the c omb inations = ! an d u = 3 , = 3 and u = 3 , = ^ an d u = 3 , the fuzzy set ( u % ) (c f. Equat ion (1 2) ) is ext ension al w .r.t. F .
220 F.Kla w onn&J .L .C astro Proo f. (i ) = ! and u = 3 . T akingLemm a2 .3(g),(h)andtheexte nsio nali ty of B i n toaccoun t, w eobtain: ( u % ) ( y ) 3 F ( y;y 0 )= _ x 2 X n ( x ) 3 ( A ( x ) ! B ( y )) 3 F ( y;y 0 ) o _ x 2 X n ( x ) 3 A ( x ) ! ( B ( y ) 3 F ( y;y 0 )) o _ x 2 X n ( x ) 3 ( A ( x ) ! B ( y 0 )) o =( u % )( y 0 ) : (ii ) = 3 and u = 3 . ( u % ) ( y ) 3 F ( y;y 0 )= _ x 2 X n ( x ) 3 A ( x ) 3 B ( y ) 3 F ( y;y 0 ) o _ x 2 X n ( x ) 3 A ( x ) 3 B ( y 0 ) o =( u % )( y 0 ) : (i ii ) = ^ and u = 3 . U si ngLem m a2 .3(b),w eget ( u % ) ( y ) 3 F ( y;y 0 )= _ x 2 X n ( x ) 3 (( A ( x ) ^ B ( y )) 3 F ( y;y 0 ) o _ x 2 X n ( x ) 3 ( A ( x ) 3 F ( y; y 0 )) ^ ( B ( y ) 3 F ( y; y 0 )) o _ x 2 X n ( x ) 3 A ( x ) 3 B ( y 0 ) o =( u % )( y 0 ) : 2
Sim ilari t yinFuzz yR easo ni ng 221 In Theorem s4.4and4. 6w eha ve only consi deredasinglepa ir of fuzzysets A and B thatm ode lthel ingu is ti cte rmi ntheprem ise, respe ctiv elyt heconclusionofasingle if{t he nrul eoftheform(9) . How e v e r,ourre sul tscane asil yb ee xtendedtothem orer eali st iccase ofase tofif{thenr ule s. Consider thes etofr ul es If i s A i ; the n i s B ; ( i 2 I ) ; wheretheli nguisti cterm s A i and B i arem odel le db ythefuzz ys et A i 2 L X and B i 2 L Y . Gi v e nan \inputfuz zyset" 2 L X ,theoutputo fthissetofrul es iscom putedin thef oll o wingw a y:asarststep,a com binationofthe operations and u is chos en.Le tusassum ethatone of thet hree casesinThe orem s4. 4and4.6 is considere d.The nforeac hs ingl er ul e, thec or re spondi ngfuz zyre lation % i ( x;y )= A i B i anda lsoont hi s basi s,thecorrespondingoutputfuzz yset u % i i sc om pute d. Finall y , theseoutputsar eaggregated usual lyeithe rb ytak ingthei nm umor supre m um ,i. e.e ither ^ i 2 I ( u % i )(18) or _ i 2 I ( u % i ) : (19) Ge neralizing Theorem4. 4toasetofrules,t heresultrem ainsthe sam esincei nac cord ancetoTheorem4.4,re placin g b yitsex tensional h ull, doesnotc hangethefuzzy se t u % i sot ha tn eithe r(18)nor(19) isa ec ted. The result o f Theorem 4.6 is also v al id for a set of rul es, i.e .th e o utput fuzz ysetis ex tensiona l ,sinc e, due to Th e ore m3.2, i nm aand su pre mam ai ntain exte ns i on al it y. T he only i mp ortan tth ing i s that one ha s to consider fuzzy e quiv ale ncerel ations E and F on X ,respec ti vely Y ,suc h tha t all fuzz ys ets A i (resp ecti vely B i )ar eex tensiona l with re sp ec tto E (re spec tiv ely F ). Of course, the m ost i ntere sti ng fuz zy e quiv alenc ere lations are the coa rsest ones since they cha racterize the indisting uishabilityinherent to the g iven fuzzy sets and yield the greatest extensional hulls. The
228 F.Kla w onn&J .L .C astro [2 2] E.T ril las,L.V alv erde, On Im pli cati onandIndisti ng uishabil it yin theSettingofF uzzy Lo gic ,in:\Managem en tD ecis ionSupport Systems UsingF uzzySetsandP ossibil it yT heory ."J.Kac przyk, R.Y ager,e ds. ,V erlagT U VR heinl and,K ol n(1985), 19 8{ 212. [ 23 ]L .V al v e rde,OntheStructureo f F {I ndi stingui shabi lity Op erator s.F uzzySetsandSystems 17 (1985),313{328. [ 24 ]L . A.Zade h,Si m i larit yRela ti onsandF uzzyOrderings. In fo rm ationSc ie nc es3(1971), 17 7{ 20 0.