FuZZy SetsandSystems8(1982)253-283 North-Holland PublishingCompany 253 COMP ARI S ONOFF UZZYREAS ONI NGMETHODS *Ma s a h a r u MI Z UMOT O InformationScienceCenter,OsakaElectro-CommunicationUniversity,Neyagawa,Osaka572, Japan Ha n s - J f i r g e n Z I MME R MANN LehrstuhlfiirUnternehmensforschung,RWTHAachen,IV.Germany ReceivedMarch1981 Revised June1981 L.A.Zadeh,E.H.Mamdani,andM.Mizumotoetal.haveproposedmethodsforfuzzy reasoning in whichtheantecedentinvolvesafuzzyconditional proposition ' If xis Athenyis B,' withAandBbeingfuzzyconcepts.Mizumotoetal.haveinvestigated theproperties of theirmethodsinthecaseof'generalized modusponens' . Thispaperdealswiththepropertiesoftheirmethodsinthecaseof'generalizedmodus tollens' , andinvestigatestheothernewfuzzyreasoningmethodsobtainedbyintroducing the implicationrulesofmanyvaluedlogics)~tems.Finally,thepropertiesofsyllogismand contrapositive areinvestigated undereachfuzzyreasoningmethod. Keywords:Fuzzyreasoning,Fuzzyconditionalinference,Generalizedmodusponens, Generalized modustollens, Syllogism,Contrapositive. 1.Introduction I no u r d a i l y l i f ewe o f t e n ma k e i n f e r e n c e s wh o s e a n t e c e d e n t s a n d c o n s e q u e n c e scont ai nf uz z yc o n c e p t s . Su c h ani n f e r e n c e c a nn o t b e ma d e a d e q u a t e l y b y t h eme t hods wh i c h a r e b a s e d e i t h e r o n cl as s i cal t wov a l u e d l ogi c o r o n ma n y v a l u e dlogic. I no r d e r t oma k e s uc hani n f e r e n c e , Z a d e h [ 1] s u g g e s t e d a ni n f e r e n c e r u l ecal l ed' c o mp o s i t i o n a l r u l e of i n f e r e n c e ' . Us i n g t hi s i n f e r e n c e r ul e , he , Ma md a n i[2]a ndMi z u mo t o e t al . [ 3 - 6 ] s u g g e s t e d s e v e r a l me t h o d s f or f uz z yr e a s o n i n g i n whi cht h e a n t e c e d e n t c o n t a i n s ac o n d i t i o n a l p r o p o s i t i o n wi t hf uz z yc o n c e p t s :An t l : I f x i s At h e n y i s BAn t 2: x i s A' (1) Cons' : yisB '* ThisworkwasattainedwiththeassistanceoftheAlexander yonHumboldtFoundation. 0 1 6 5 - 0 1 1 4 / 8 2 / 0 0 0 0 - 0 0 0 0 / $ 0 2 . 7 5 91982No r t h - Ho l l a n d254M. Mizumoto, H.-J. Zimmennann whereA, A' , B, B' arefuzzy concept s. Anexampl eofthefuzzy reasoni ngisthe following. Ant 1:Ifat omat oisredt hent het omat oisripe Ant 2:Thi st omat oisveryred(2) Cons:Thi st omat oisveryripe In[4-6]wehavepoi nt edout t hat fort het ypeoffuzzy reasoni ngin(1)called ' general i zedmodusponens' , t heconsequencesi nferredbyZadeh' s andMam-dani ' smet hodsarenot alwaysreasonabl eandsuggest edseveralnewmet hodsR~, Rg,R~,Rsg,Rg,andRgg whichcoincidewi t hour i nt ui t i onwi t hrespecttoseveral criteria.Ascont i nuat i onofour studies, thispaper investigatesthepropert i esoftheir fuzzy reasoni ngmet hodsinthecaseof' general i zedmodustollens' . Mor eover , by i nt roduci ngt hei mpl i cat i onrulesofmanyval uedlogicsyst ems[7-9], wediscuss thenewlyobt ai nedfuzzyreasoni ngmet hodsint hecasesofgeneral i zedmodus ponensandgeneral i zedmodustollens.Fi nal l y, wediscussthepropert i esof syllogismandcont raposi t i veunder eachfuzzyreasoni ngmet hod.2.Fuzzy reasoningmethods Weshallfirstconsi dert hefollowingformofi nferenceinwhichafuzzy condi t i onal proposi t i oniscont ai ned.Ant 1:I f x i s At heny i s B Ant 2:x i s A' (3) Cons:yisB'wherexandyaret henamesofobj ect s, andA, A' , BandB' arefuzzy Concepts r epr esent edbyfuzzysetsinuniversesofdi scourseU,U,VandV,respectively.Thisformofi nferencemaybevi ewedasageneralizedmod, s ponenswhich reducestomodusponenswhenA' = AandB' = B.Mor eover , thefollowing formofi nferenceisalsopossiblewhichalsocont ai nsa fuzzycondi t i onal proposi t i on.Ant 1:I f x i s At heny i s B Ant 2:y i s B' (4) Cons: xisA'Thi si nferencecanbeconsi deredasageneralizedmodustollenswhichreducesto modustollenswhenB'=notBandA' =, ot A.TheAnt 1oft heform" I f xisAt henyisB" in(3)and(4)mayrepresent a certainrel at i onshi pbet weenAandB.Fr omthispoi nt ofview,severalmet hods wereproposedforthisformoffuzzycondi t i onal proposi t i on: ' IfxisAt hen) isB' .Comparison o[ [uzzy reasoning methods2 5 5Let AandBbefuzzy set sinUandV,respect i vel y, whi charer e pr e s e nt e das A = ~ t x a ( u ) / u , B = ~ t t u ( v ) / v (5) andletx , U, f-I,--1and9becart esi anpr oduc t , uni on, i nt er sect i on, c ompl e -ment andb o u n d e d - s u mf or fuzzyset s, respect i vel y. The nt hefol l owi ngfuzzy relationsinUxVcanbeder i vedf r omt hefuzzycondi t i onal pr opos i t i on" I f xis At henyisB " inAn t 1of (3)and(4).Th e fuzzyr el at i onsR,,,andR, wer e pr oposedbyZa d e h [1],RcbyMa md a n i [2],andR~,Rg,R,g,Ugg, Rg~andR~sare byMi zumot oet al.[3-6].R, , = ( A xB)U ( - h AxV) [(I~A(tt)AtXI3(V))V(I--I.tA(tt))/(tt, V).( 6 )" O x VR,~ = ( - hAxV ) ~ ( U x B )[1 ^ ( 1 - t x a ( u ) + t . t u ( v ) ) / ( u , v).(7) 9i ox VRr= A x B= ~u~, , , (u)^uB(v)l(u,v).(8) where R s = A x V = ) , U x BS =I u[m,(u)~u .(v)]/Cu. v ) .m, (u)V m,(v)= {lo~A( u) u . ( v ) .(9) where R g = A x V O U g = Iu[ ~ ( . ) ~u . ( v l ] l ( u , v), 1IxA ( u ) ~< gu ( v ) ,~( u)7u. ( v) =~.(v)U~(U)>U.(V). R~g = ( A xVOU x B ) N ( T A xV~Ux--1B) sg = I o [ ~ A ( u ) 9u B ( v ) ] ^ [ 1 - ~ A ( u ) ~ 1 - u B ( v ) ] / ( . , v). (10) (11) Rgg=( AV~U x B ) N ( " n A xV~U x ~ B )gg =I u [~A (u) - * Uu ( v ) ] ^ [ 1 - UA(u) g--> 1 - gu (v)]/(u, v). (12) 256M. Mizumoto,H.-I.Zi mmennann R~,= ( A xV~U x B ) N ( - 1 A xVOU x ~ B )gs =L [~A(u) g-~ p.D(V)]A[1--/ZA(U)--~,1--I.tB(V)]I(u,V).(13) R~= ( A xV~U x B ) f q ( ' - n A xV~U x ~ B )S$ =Iu[/.&~Cu)--)/..tBCv)]A[I--/.tA(U)T) 1-/ . t B(v)]/ (u, v).(14) s Not et hat t heimplicationsa- ~banda--~ baret hei mpl i cat i onrul esin' St andard g sequence' Ssand' GSdel i ansequence' G~,respect i vel y[7].R a isbasedonthe i mpl i cat i onrul einLukasi ewi cz' slogicL~. Inaddi t i ont ot heabovefuzzy rel at i ons(6)-(14), it isalsopossiblet odefi nenew fuzzyrel at i onsfort heproposi t i on" I f xisAt henyisB" byi nt r oduci ngthe i mpl i cat i onrul esofmanyval uedlogic systems[7-9]. Thesei mpl i cat i onrul esand t hei mpl i cat i onrul esusedin(6),(7),(9)and(10)arediscussedindet ai l in[7-9]. Int hefollowingweshalldiscusss omenewfuzzyrelations.R b = ( ~ A x V ) U ( U x B )=[(1--/XA(U))V~t~(V)/(U,V).(15] a ux vR a = A 2 1 5I u [p.A(u) a-~ ~n(v)]/ (u, v), (16 wher e 1~A(U) , B ( o ) .wher e RA =AxV~UxB A =J~x,,,[~A(u) .---, ~B(v)]/Cu, v), (17 /"tA (I'l) ~~B(I)) = [/"J"A(") ~/IB ('0)] A [1 - - l-tB(I)) "~' 1 -P,A(u)] I"m, ( 0l - , ~, ( u)1 A ~ A p. A( u) >O, 1 - - / . t B( U)>O,=Vi A( u ) 1- / z B( v),1IZA(U)----O or 1 - - m] ( v ) =O.R , = A x V ~ , U x B=L ~ x [~A(u)~~ ( O ] / ( u , v ) ,(1~ where where where Comparisonof fl~zzyreasoning methods 257 /XA(U) .---> /~B (V) =1 -- ~A (U) +/~A (U)~ZB(V). R, z=AXV~UxB -u[[ w A ( u ) y . ~ .( v ) ] / ( u ,v), (19) tLa (U) - ~tLB (V) =(/XA(U) ^/XB (V)) V (1 -- tLA (U) A 1 -- tLB(V)) V (tLB(V) ^1 --/xa (u)) =( 1 -t x a ( u ) vl ab ( v ) ) ^(txa ( u ) v1- - / x a ( u ) )^(gB (v) v1 -- ~B (v)). Rr n=Ax V~UxB [] = s o [ [ t ~ , ( u ) - j t ~ ( v ) ] / ( u , v ) ,(20) {~~ A( u ) < l or/ ZB(V)=I,OA(U)-~~B(V)=/~A(U)=i , OB( V) - -2 270 M. Mizumoto,H.-J.Zimmennann 11 9lab= ' 7I . \ 0101 F i g . 3 . T h e w a y o f o b t a i n i n g B[ , a n d A[ , : ( a ) ~s~ a t / x A . = I x 2 ; ( b ) P-Af, a t gs' =1 -Ix 2 .Ther ef or e, weobt ai n {3 - ~ 3 - x / 52' g " ' =3 -',/5 /-tsP.s ~ - ~ ,whichleadst o 3-4"5 /aBe'=2.vp.s. Thesamemet hodisapplicabletoA"= A, moreorlessA, andnot A.(ii)Thecaseof RbatB ' =notveryB TheconsequenceA~,whichisobt ai nedbyt aki ng t hecomposi t i onofRbandB'asin(26)-(29), isgivenby A' b=Rbo B'=[ ( TAxV ) U( U B' .Themember shi pfunct i onofA~,atB' = not veryB(35)is /x&, =V{( 1 -P-A ) V P-B] A (1 -- p2)}(46) t~ byomi t t i ng' (u)' and' (v)' . Theexpressi onin(46) [(1 - / ~A)vtaB]A (1 --/X 2)(471 at/xa =0 . 2 ( ~ ( x / 5 - - 1 ) / 2 = 0 . 6 1 8 0 . . . ) isshownbyt heline''inFig.3(b whichcomesfromt herightfigureofFig.l(x). Themaxi mumval ueofthislin~ becomes0.8( =1-0. 2). Wh e n / xa=0.7( ~ ( x/ 5- 1)/2),(47)isi ndi cat edbyt helin~ ' - . - . ' anditsmaxi mumvalueis( x/ 5- 1) / 2. ThuswehaveingeneralI , / g - 11 - / xa/xA 4 . 2 - - ~.LAI~ T Namely, ~ " ~ b ' = Comparison o[ [uzzyreasoning methods x/ - 5- 1v(1 _~A). 2 271 Thesamewayisappl i cabl et oB' = not B, not mor eorlessB, andB. Example. Usi ngTabl es1and2andFig.2,weshallpr es ent asi mpl eexampl eoffuzzyr easoni nginFig.4.Fig.4(a)showsf uzzyset sAandB, andFig.4(b) includesfuzzy sets' not A ' , ' not ver yA ' , . . . . ' u n k n o w n ' inor der t ocompar ewi t h thei nf er enceresul t sof Fig.4(c)-(j ).Int hef or msof f uzzycondi t i onal i nf er ences(3)and(4),itseemsaccor di ngt o ouri nt ui t i onst hat t herel at i onsbet weenA' inAnt 2andB' inConsoft he generalizedmodus ponens (3)ought t obesatisfiedasshowninTabl e3.Similarly,therel at i onsbet weenB' inAnt 2andA' inConsof t hegener al i zedmodustollens(4)ought t obesatisfiedasinTabl e4. Rel at i onIinTabl e3cor r espondst ot hemodus ponens. Rel at i onI1-2hasa consequence di f f er entf r omt hat of Rel at i onI I - 1, but if t her eisnot ast r ongcausal relationbet ween" x isA" and" y isB" int hepr oposi t i on" I f xisAt henyis B", t hesat i sfact i onof Rel at i onI I - 2willbeper mi t t ed. Rel at i onIV-1assertst hatwhenxisnot A , a nyi nf or mat i onabout yisnot conveyedf r omAnt 1.Th esatisfactionof Rel at i onI V- 2isde ma nde dwhent hef uzzy pr oposi t i on" I f xisA thenyisB" means t aci t l yt hepr oposi t i on" I f xisAt henyisBelseyisnot B " .Althoughthisr el at i onmaynot beaccept edinor di nar ylogic,inour dai l ylifewe oftenencount er t hesi t uat i oninwhi chthisr el at i oncanhol d. Rel at i onVcor r e-spondst omodus tollens. Rel at i onVI I I isdi scussedasint hecaseof Rel at i onIV.InTabl e5,t hesat i sfact i on(0)or fai l ure(x)of eachcri t eri oninTabl es3and4 undereachf uzzyr easoni ngme t hodisi ndi cat edbyusi ngt hecons equenceresul t s of Tabl es1and2. Under t hesecri t eri aitisf oundt hat RmandRaar enei t her ver y sui t abl ef or t he fuzzycondi t i onal i nf er enceint hecaseof gener al i zedmodust ol l ensnor int he caseofgener al i zedmodus ponens. Rcisnot bad. R~,Rg,R~g. . . . . R~ar e satisfactory.Rb . . . . . Rrnar enot ver ygood.PA 12 1 jPB o 34 Fig. 4(a).Fuzzysets Aand B. 272 . 5 1 . 5 3~ .5 M. Mizumoto,H.-J.Zimmennann unknown not very A - - ~ ~ A F//~-~ess A/ //not more or I12 U unknown 34 Fi g. 4(b). not A, notvery A, notmore or lessA, not B,very B,moreor lessB,a ndunknown. ~ B '/ / ~ B S . B w. / ~ B ~ g , B ~ s , B ~ s34 iL~aB, 0v 34 Fi g. 4(c). I nf e r e nc e r e s ul t s at A' =A.laB , ~ ' ~ ' B ~ B~'~2,~gs 34 . 5 34 Fi g. 4(d). I nf e r e nc e r e s ul t s at A' =very A.qaB , / / \.\ B~g,B~s,Bw / / o- \ \o 34 lab, . . . . . . . . .34 Fi g. 4(e). I nf e r e nc e r es ul t s at A' =more or lessA.PB' i .5 i PA' .5 0 .5 . 5 Compar i s onof[ u z z y reasoni ngmet hods 273 Bfi, Bfi,Bfi,B~,Bb,BA,B~ ,B~, ',~," ~Bwt z \ B 2 s /' , .34 ~ A '1 .5 0 Aft~ A ~ , A ~ A DFAA TA" rA~ ~ A . ~ A f i / *--_ A ~"~/~ iJ , ~ s , A w12 Fig. 4(0.Inference results at A'=notA ; (g) at B'=B. A ,A wA~,A~.A~g, A~s,AD,A~_ ~ 12 ~ A '1 .5 O Aft" ~ A / ~ , A ~. S / / / s 9r! 12 Fig. 4(h).Inference results atB'=not B.~ A ' ~ A 'I A~,A~g,A~s ~ . ~ - - AV~ - l" ' / ~ " ~ .4.fi,Aii 9 5s J/ A/ 9i A f g , A f s / ",,Aftst 9~.~I i21 Fig. 4(i).Inference results atB' = not veryB.o 9 ~----- AA U ,}~A' ~A~,A~g,A~s.5 3 - EA~ , AwN%A~..U0 12 VA' s9 /~-~ A As# 9 ~ JI2 Fig. 4(j).Inference results atB'=not mor eor less B.2 7 4I~LMi z umot o, H. - J . Zi nnne r mann T a b l e 3 . R e l a t i o n s b e t we e n An t 2a n d C o n s u n d e r An t 1f o r t h eg e n e r a l i z e d mo d u s p o n e n s i n( 3)xi sA' ( An t 2) yi sB ' ( Co n s )R e l a t i o n Ixi sAyi sB ( mo d u s p o n e n s )R e l a t i o n I I - 1 xi sv e r y Ayi sver yB R e l a t i o n 1I - 2xi sv e r y Ayi sB R e l a t i o n I I I - 1 xi smor e orl essAyi smor e orl essB R e l a t i o n 1 I I - 2 xi smor e orl essAyi sB R e l a t i o n I V- I xi snot Ayi sunk nown R e l a t i o n I V- 2 xi snot Ayi snot B T a b l e 4 . R e l a t i o n s b e t we e n An t 2a n d C o n s u n d e r An t 1f o r t h eg e n e r a l i z e d mo d u s t o l l e n s i n( 4)yi sB ' ( An t 2) xi sA' ( Co n s )y i s n o t B xi snot AR e l a t i o n V ( mo d u s t o l l e n s )R e l a t i o n VIR e l a t i o n VI IR e l a t i o n VI I I - 1R e l a t i o n VI I I - 2yi snot v e r y Bxi snot v e r y A yi snot mor e orl essBxi snot mor e orl essA yi sBxi sunk nown yi sBxi sA T a b l e 5. S a t i s f a c t i o n o f e a c h R e l a t i o n i nT a b l e s 3a n d 4u n d e r e a c h me t h o dAn t 2Cons RmRol ~R, R, R, , R**R, , R. . RbRz~RAR . R.~5m ABxx0000000xxxxxx Rel at i onI ( modus ponens )Rel at i onI t - 1 Rel at i onI1-2 Rel at i onIII-1 Rel at i onI I I - 2 Rel at i onI V- 1 Rel at i onI V- 2 ~' e r yAr e r y / 3 xxx0x0xx0xxxxxx v e r y ABxx0x0x00xxxxxxx moreormoreorxxx000000xxxxx lessAlessB nl or e or Bxx0xxxxxxxxxxxx lessA notAunknown0000xx00000 notAnotBxxxxx0000xxxxxx Rel at i onVnotBnotA000 ( modus t ol l ens) Rel at i onVInotveryBnotveryAxxx0x0xx0xxxxx Rel at i onVl l not nmrenotmorex0x00xx orlessBorl e s s A Rel at i onVI I I - I Bunknownx00000000 Rel at i onVI I I - 2BAxx0xxxx00xxxxxx Comparison of fuzzy reasoning methods 4.Syllogismandcontrapositiveundereachfuzzyreasoningmethod 275 Inthissectionweshallinvestigatetwointerestingconceptsof'syllogism'and ' contrapositive' undereachfuzzyreasoningme t hodobtainedinSection2. LetP~,P2andP3befuzzyconditionalpropositionssuchas P~:IfxisAthenyisB, Pa:IfyisBthenzisC, P3:IfxisAthenzisC, whereA, BandCarefuzzy setsinU,VandW,respectively.Iftheproposition P3isdeducedfromthepropositionsP~andP2,i.e.thefollowingholds: PI:I f x i s Atheny i s B P2:IfyisBthenzisC(48) P3:I f x i s AthenzisC thenitissaidthatthesyllogismholds. LetR( A, B),R(B, C)andR( A, C)befuzzyrelationsinUx V,Vx Wand U xIV,respectively,whichareobt ai nedfromthepropositionsP~,P2andP3, respectively.Ifthefollowingequalityholds,thesyllogismholds: R( A, B) o R(B, C) = R( A, C).(49) Thatistosay, PI:IfxisAthenyisB~R( A, B)P2:IfyisBthenzisC--~R(B, C)(50) P3:I f x isAthenzisC~--R( A, B) oR( B, C)where' o' isthemax-mi ncompositionofR( A, B)andR(B, C),andthemember -shipfunctionofR( A, B) o R(B, C)isgivenby P-a(a.m~W)=V[P-R(A.m(U, V)^ ttn~B.C~(V, W)].(51) I) Now weshallobtainR( A, B)oR(B, C)undereachfuzzy reasoningmet hodand thenshow whet her thesyllogismholdsornot. Weshallbeginwiththemet hodR~. Thefuzzy relationsRa(A, B)andR, ( B, C) areobtainedfrompropositionsP~andP2byusing(7): R,,(A,B) = ("hAxV ) ~ ( UxB),R. ( 8 , C) =( ~ BW) @( VC).Thus,thecompositionofR,~(A, B)andR,~(B, C)willbe R,~(A,B) o Ra(B,C)= [ ( ~ AV ) ~ ( UB)] o [(-riBW) ~ ( V C)] 2 7 6 M. Mizumoto, t i . - 1 . Zimmerlnann anditsmember shi pfunct i onbecomesasfollows. tzRo(A,~)o~n.C)('I,W) =V{[1 ^(1 -Ix^ (u) +~D (v))] A [ 1 ^(1 -- f t ,(V) +~c(W)]} D = V { ( i ) ^ ( i i ) } . ( 5 2 )I) Thefunct i on(i),i.e.1 ^ ( 1 - txA(u)+ ~ ( v ) ) , canbedepi ct edbyusingt hepar ame-t ert-L^(U)asinFig.5(a)andt hefunct i on(ii),1 ^( 1 - ~t , ( v) +~ c ( W) ) , isshownby usingt hepar amet er t . t c ( w) asinFig.5(b).ThesefiguresbaseonFig.l(ii).From thesefigures,t hefunct i on(i )^(i i )in(52)wi t hbot hpar amet er sp. A(u)=aand t x c ( w) =cwillbeshownbyt hebr okenline' . . . . 'inFig.5(c)anditsmaxi mum value(byvi rt ueof(52))is0.5 +(1 -a+c ) / 2 . Ont heot her hand, if t hepar amet ertxA(u)ist akent obea' asinFig.5(c),t hemaxi mumval ueofitsline' - . - . 'becomes1.Ther ef or e, ingeneral , forany par amet er saandc,t hemaxi mumvalue of(i)A(ii)isshownt obe1 ^ ( 0 . 5 + ( 1 - a + c ) 1 2 ) . Ther ef or e, t hemembershi p funct i on/~R,(A,B~.Ro(~.c(U, W)of(52)becomes:/xR,~A.B)~W) =1 ^(0.5 +(1 --/XA (U)+tZc (W))12). ~6(~)= .i .2 .3 .4 .5 .6 .7 , 8.9 a 1 bVc (w)= o1 liA(u)=a' ~ A ( u ) = aC 1~c(W)=C ,] ~ B ( v )1 Fi g. 5. T h e wa y o f o b t a i n i n g ( 5 2 ) ; ( a ) 1 , ' , ( 1 - p . A ( l l ) + p s ( v ) ) ; ( b) l ^ ( 1 - p n ( v ) + / X c ( W ) ) ; ( c) l ~(1 - / xA ( u )+~ s ( v ) )^(1 -~B ( v )+~ c ( W) ) .Comparisono[ fuzzy reasoningmethods277 Fromthisresult,wecanhave R,~(A, B) o R, ( B, C) [1A(0.5 +(1--#A(U)+I.tC(W))/2)/(U , W) JU'W 7~ Ra(A, C ) ( = Iu1 A (1 --1 "tA (It)q -I'g c(W ))/(ll'iV )).(53) Hence,wecanconcludethatthefuzzy reasoningmet hodR. doe s notsatisfythe syllogism. Similarly,wecanobtainR ( A , B ) o R ( B , C ) underotherfuzzyreasoning methodsandweshalllisttheminthefollowing. R,,,(A, B) o R. , ( B, C) [0-5 V (/-tA(/~) A/-tC(W)) V (1 -wA(u))/(u,w) JtJ xW 7s R m ( A ,C ) ( = IU x w( ~ A ( l i ) A I - t c ( W ) ) V ( 1 - - ~ A ( l t ) ) / ( I I ,W )) . (54) Re(A,B) o Rc(B,C) = ~uxw/~A (U) ^ttc(W)/(U,W) =R~(A,C).(55) R~(A, B) o R~(B,C) =[p,A(U)~t-tc(W)/(U, W) al l x~,V =R,( A, C) . ( 56)Rg(A,B) oRg(B,C)=IultA(u) ~tZc(W)/(u' w) = Rg(A, C).(57) R, g(A,B) o R,~(B,C) Ju[g,,,,(u) - ~~c(W)]^ [ 1 - ~A(u) ~1 - p,c(W)]/(u, W)= R, g(A, C).(58) Rgg(A, B) o Rgg(B, C) =[[tzA(u) g-->tXc(W)]^[1-t-t~,(u) g-~ 1 - t-tcCw)]/(u, w) Ju =Rgg(A,C).(59) Rg, ( A, B)o Rg~(B,C) =[[t xA(u)g-->~ c ( W) ] ^ [ 1 - t x A ( u ) ~ 1 - tlc(W)]/(u,w) Ju =Rg, ( A, C). ( 60)278 wh e r ewh e r eh i. l~lizumoto,H .-J.Zimmennann R~(A,B) oR~(B,C) I~[~A(u) ~gc(W)]^[1 -~A(u) ~1 -gc(w)]/(u,w) =n~,( A, C) .Rb( A, B) o Rb(B, C) f0. 5 v( 1 - ~ . ( u ) ) v t ~c( w) / ( u, w)at /x ~,v Rb(A,C ) ( = I U x W ( 1 - - b C A ( l l ) ) V i . t c ( W ) / ( I I ,W ) ) .RA(A, B) o R,,(B,C) = [[ ~ ( " ) 7 ~,~(w)]/(., w) at /~,v , Ra , A. C ) ( =Iu[~A(U'a--'~"c(W)]/(U'W))' 1 ~( . )t~c(W), I1m~ (u)