An Architecture for Making Judgments Using Computing with Words 1

Transcription

1 Mendel, J. M., "An Archtecture fr Makng Judgments Usng Cmputng Wth Wrds," Int. J. Appl. Math. Cmput. Sc., vl. 12, N. 3, pp , An Archtecture fr Makng Judgments Usng Cmputng wth Wrds 1 Jerry M. Mendel Integrated Meda Systems Center Unversty f Suthern Calfrna Ls Angeles, CA mendel@sp.usc.edu Telephne: (213) Fax: (213) Abstract Our thess s that cmputng wth wrds needs t accunt fr the uncertantes asscated wth the meanngs f wrds, and that these uncertantes requre usng type-2 fuzzy sets. Dng ths leads t a prpsed archtecture fr makng judgments by means f cmputng wth wrds a perceptual cmputer the Per-C. The Per-C ncludes an encder, a type-2 rule-based fuzzy lgc system, and a decder. It lets all human-cmputer nteractns be perfrmed usng wrds. In ths paper, a quanttatve language s establshed fr the Per-C, and many pen ssues abut the perceptual cmputer are descrbed. 1 Ths paper s an expanded versn f Mendel (2001b). 1

2 Mendel, J. M., "An Archtecture fr Makng Judgments Usng Cmputng Wth Wrds," Int. J. Appl. Math. Cmput. Sc., vl. 12, N. 3, pp , INTRODUCTION Zadeh (1996, 1999) cned the phrase cmputng wth wrds [see, als, Wang (2001)]; but ur thess s that wrds mean dfferent thngs t dfferent peple and s there s uncertanty asscated wth wrds, whch means that fuzzy lgc (FL) must smehw use ths uncertanty when t cmputes wth wrds [Mendel (1999, 2001a)]. Type-1 FL handles uncertantes abut the meanngs f wrds by usng precse membershp functns (MFs) that the user beleves captures the uncertanty f the wrds. Once the type-1 MFs have been chsen, all uncertanty abut the wrds dsappears, because a type-1 MF s ttally precse. Because f that, type-1 MFs cannt handle the uncertantes abut wrds. We mantan that cmputng wth wrds requres usng type-2 fuzzy sets 2. Tday, cmputng wth wrds must stll be dne usng numbers, and, therefre, numerc ntervals must be asscated wth wrds. An earler paper [Mendel (1999)] reprted n an emprcal study that was perfrmed t determne hw the scale 0 10 can be cvered wth wrds (r phrases). In typcal engneerng applcatns f FL, we dn t wrry abut ths, because we chse the number f fuzzy sets that wll cver an nterval arbtrarly, and then chse the names fr these sets just as arbtrarly (e.g., zer, small pstve, medum pstve, and large pstve). Ths wrks fne fr many engneerng applcatns, when rules are extracted frm data. Hwever, t s questnable practce when rules are extracted frm peple. Put anther way, machnes dn t care abut wrds, but peple d. In Mendel (1999), we frst establshed a vcabulary f 16 canddate wrds r phrases terms that we thught wuld let us cver ths nterval. Thse terms are: nne, very lttle, a small amunt, a lttle bt, a bt, sme, a mderate amunt, a far amunt, a gd amunt, a cnsderable amunt, a szeable amunt, a large amunt, a substantal amunt, a lt, an extreme amunt, and a maxmum amunt. We then surveyed students and asked them t prvde the end-pnts fr ntervals n the scale f 0 10 that they asscated wth each term. The 16 terms were randmzed n the survey and we cllected 70 useable surveys. We then cmputed mean and standard devatn values fr the tw end-pnts f each f the 16 term s nterval, and pltted the nterval fr each term. These results are depcted n Fgure 1. 2 A bref ntrductn t type-2 fuzzy sets s prvded n Appendx A. 2

3 Mendel, J. M., "An Archtecture fr Makng Judgments Usng Cmputng Wth Wrds," Int. J. Appl. Math. Cmput. Sc., vl. 12, N. 3, pp , nne a lttle bt a small amunt very lttle sme a bt a cnsderable amunt a far amunt a gd amunt a mderate amunt a substantal amunt a large amunt a szeable amunt a maxmum amunt an extreme amunt a lt Fgure 1: All 16 labels and ther ntervals and uncertanty bands. Sld lnes are drawn between the sample means fr the nterval end-pnts and dashed lnes are fr ±1 standard devatn abut each mean end-pnt. One f the mst strkng cnclusns drawn frm ths prcessed data s: lngustc uncertanty appears t be useful n that t lets us cver the 0 10 range wth a much smaller number f terms than wthut t. Fgure 2 depcts ths fr fve terms [Fgure 2-3 n Mendel (2001a) demnstrates cverage f the 0 10 range fr three terms]. Put n the cntext f frng rules, n a rule-based fuzzy lgc system 3 (FLS), uncertanty can fre rules. Ths cannt ccur n the framewrk f a type-1 FLS; but t can ccur n the framewrk f a type-2 FLS. Uncertanty can therefre be used t cntrl the rule explsn that s s cmmn n a FLS. If, fr example, we gnred uncertanty, and had rules wth three antecedents, each f whch s descrbed by sx terms, t culd take 216 rules t cmpletely 3

4 Mendel, J. M., "An Archtecture fr Makng Judgments Usng Cmputng Wth Wrds," Int. J. Appl. Math. Cmput. Sc., vl. 12, N. 3, pp , descrbe the fuzzy rule base. On the ther hand, usng three terms fr each antecedent requres a rule base wth nly 27 rules. Ths s an 87.5% reductn n the sze f the rule base. nne t very lttle sme a mderate amunt a maxmum amunt a large amunt Fgure 2: Althugh fve labels cver 0 10, there s nt much verlap between sme f them. It s when the standard devatn nfrmatn s used that suffcent verlap s acheved. Fnally, as cnjectured n Mendel (2001a) uncertanty s gd n that t lets peple make decsns (albet cnservatve nes) rapdly. Perhaps ths s why sme peple can make decsns very quckly and thers cannt. The latter may have parttned ther varables nt s many fne sets that they get hung up amng the resultng enrmus number f pssbltes. They are the eternal prcrastnatrs. Ths cnjecture s supprted by Klr and Werman (1998) wh state: Uncertanty has a pvtal rle n any effrts t maxmze the usefulness f systems mdels. Based n these prelmnary deas, n the rest f ths paper a specfc archtecture s prpsed fr makng judgments by cmputng wth wrds (by judgment we mean an assessment f the level f a varable f nterest). Such a cmputer wll be called a Perceptual Cmputer Per-C fr shrt. We beleve that a generc all-purpse methdlgy can be fund fr makng judgments by cmputng wth wrds, but that the specfc detals wll be cntext dependent. S, fr example, makng judgments by cmputng wth wrds fr dagnstc medcne wll have detals that are dfferent frm thse fr makng judgments by cmputng wth wrds fr accuntng. There are many nterestng and pen ssues asscated wth cmputng wth wrds, sme f whch are psed belw. 3 See Appendx B fr a hgh-level descrptn f a FLS. 4

5 Mendel, J. M., "An Archtecture fr Makng Judgments Usng Cmputng Wth Wrds," Int. J. Appl. Math. Cmput. Sc., vl. 12, N. 3, pp , FRAMEWORK FOR A PERCEPTUAL COMPUTER The archtecture fr a perceptual cmputer s depcted n Fgure 3. Perceptns (.e., granulated terms, wrds) actvate the Per-C and are als utput by the Per-C; s, t s pssble fr a human t nteract wth the Per-C just usng a vcabulary wrds. The mappng f wrds nt wrds ccurs wthn the Per-C and s accmplshed usng the mathematcs f type-2 rule-based FLSs. The humans that nteract wth the Per-C d nt have t be cncerned wth the mathematcal detals, althugh the desgner f the Per-C must be. Perceptual Cmputer (Per-C) Perceptns (wrds) W Perceptns (wrds) W d Encder E D Decder Type-2 Fuzzy set W R Numbers Type-2 Fuzzy Lgc System O Fgure 3: Archtecture f a Perceptual Cmputer (Per-C). We let W dente the th wrd frm a vcabulary, V, f N V wrds (terms),.e., N V = { W } V =1 (1) Ths vcabulary s cntext-dependent, and may cntan sme terms that are cmmn acrss cntexts r that are used as adjectves (e.g. sme, a lt f, a maxmum amunt f). V s the vcabulary that a human uses t nterface wth the Per-C and whch the Per-C uses t cmmuncate ts fndngs back t a human (e.g., the 16 terms n Fgure 1). Each wrd n V must have a type-2 MF asscated wth t, whch suggests that nterval survey nfrmatn must be btaned fr t. New wrds can be added t V, just as a chld cntnues t add wrds t hs r her vcabulary, and nterval nfrmatn abut each wrd n V may als change ver tme. 5

6 Mendel, J. M., "An Archtecture fr Makng Judgments Usng Cmputng Wth Wrds," Int. J. Appl. Math. Cmput. Sc., vl. 12, N. 3, pp , The encder transfrms lngustc perceptns nt type-2 fuzzy sets that actvate a rule-based type-2 FLS, dented R. We dente the type-2 fuzzy set utput f the encder as W. Nte that the encder s the same as a fuzzfer; hwever, t always utputs a type-2 fuzzy set, whereas a general-purpse fuzzfer culd utput sngletn, type-1 r type-2 fuzzy sets [e.g., Mendel (2001a)]. Ths s why we dstngush between an encder and a fuzzfer. A type-2 FLS s rule-based, and ts rules are cntext dependent. The rules are IF-THEN rules (whch can nclude a rch varety f rules, e.g., ncmplete IF rules, mxed rules, fuzzy statement rules, cmparatve rules, unless rules, and quantfer rules [Wang (1997)]) whse antecedents and cnsequent are wrds that are frm antecedent and cnsequent vcabulares, V A and V C, where V A = { W } A =1 N A! V (2) and V C = { W C } = 1 N C! V (3) In ths paper, we assume that the wrds used n V A and V C are subsets f the wrds n V, and, that N A << N V and N C << N V (e.g., V mght be the 16 terms n Fgure 1, whereas V A and V C mght be the fve terms n Fgure 2). The latter assumptns mean that there s a carser granulatn asscated wth the antecedents and cnsequent than n the verall vcabulary. Ths s ne way t cntrl rule explsn. It s mprtant, hwever, t allw the human t nteract wth the Per-C usng the larger vcabulary s that ths nteractn s as natural as pssble. Because the wrds n V A and V C are subsets f the wrds n V, they wll each have a type-2 MF asscated wth them. We dente the ttalty f antecedent wrd type-2 MFs as V A, and the ttalty f cnsequent wrd type-2 MFs as V C. The utput f the type-2 FLS s a number, O, whch s a result f a sequence f nternal peratns nference, type-reductn, and defuzzfcatn whch we shall descrbe n Sectn 3. At a very hgh 6

7 Mendel, J. M., "An Archtecture fr Makng Judgments Usng Cmputng Wth Wrds," Int. J. Appl. Math. Cmput. Sc., vl. 12, N. 3, pp , level, we can descrbe ths utput as O = f [ W, V A, V C ], where the exact nature f the nn-lnear functn f [!] depends n many specfc chces that have t be made wthn the type-2 FLS (e.g., knd f nference, type-reductn, and defuzzfcatn). The decder, D, maps O = f [ W, V A, V C ] nt a wrd W d,.e., W d = D(O)! V (4) Hw t actually d ths s als brefly dscussed n Sectn SOME DETAILS OF THE PERCEPTUAL COMPUTER The man gal f ths sectn s t arrve at an nput-utput frmula fr the Fgure 3 Per-C. T d ths we must prvde sme detals fr the elements f the Per-C. A. Vcabulary Cntent experts need t be part f the prcess f establshng a meanngful vcabulary fr specfc judgments. Ths vcabulary shuld be as large as pssble n rder t prvde a human wth as much flexblty as pssble. After a behavr f nterest s dentfed fr whch judgments wll be made (e.g., flrtatn), the ndcatrs f that behavr must be establshed (e.g., eye cntact, tuchng, actng wtty, smlng, cmplementng, prmpng). A small subset f the ndcatrs needs t be establshed, because rules wll be establshed fr that small subset. Ths can be dne by means f an asscated survey n whch ndcatrs are rank-rdered. Scales then need t be establshed fr each ndcatr and the behavr f nterest 4. Names and nterval nfrmatn need t be establshed fr each f the ndcatr s fuzzy sets and behavr f nterest s fuzzy sets. Dng ths leads t the vcabulary V, and the antecedent and cnsequent vcabulares V A and V C. It als leads t the type-2 MFs that are used fr the wrds n all vcabulares. Althugh the nterval surveys descrbed n Sectn 1 cntan uncertanty nfrmatn abut each term, hw t use that nfrmatn t derve an 7

8 Mendel, J. M., "An Archtecture fr Makng Judgments Usng Cmputng Wth Wrds," Int. J. Appl. Math. Cmput. Sc., vl. 12, N. 3, pp , asscated type-2 MF s an pen ssue. Mendel (2001a) advcates transfrmng the nterval uncertantes nt a ftprnt f uncertanty (FOU) fr each term, but ths requres makng an a prr chce fr the shape f the prmary MF (e.g., a Gaussan prmary MF whse mean and/r standard devatn are uncertan). Chsng a FOU t mdel lngustc uncertanty s analgus t chsng a prbablty densty functn t mdel randm uncertanty. There s n unque chce fr a FOU, just as there s n unque chce fr a type-1 MF. Hpefully a Per-C wll be rbust t the chce f MF shapes, just as, e.g., a type-1 fuzzy lgc cntrller s. Recent results (Mendel, 2001d) abut FOUs have demnstrated that a smlar lkng FOU can be btaned fr a trangular prmary MF (where there can be uncertantes abut all three vertces), a trapezdal prmary MF (where there can be uncertantes abut all fur vertces), and a Gaussan prmary MF (where bth the mean and standard devatn are uncertan). It appears, therefre, that granulatn f a type-1 fuzzy set t a type-2 fuzzy set reduces the prblem f determnng the type-1 MFs. Because the type-2 FLS perates n numbers, scales must be establshed fr each ndcatr and the behavr f nterest 5. Cmmnly used scales are 1 thrugh 5, 0 thrugh 5, 0 thrugh 10, etc. The survey descrbed n Sectn 1 fr the 16 terms was perfrmed n a cntext-free stuatn. An pen ssue s whether r nt such cntext-ndependent results can be appled when the terms are used wthn a specfc cntext, e.g. f sme, n a cntext-ndependent stuatn, s lcated (n a scale 0 10) n the nterval [0.5, 6], then are sme tuchng and sme prmpng lcated ver that same nterval? There may als be stuatns where a natural scale already exsts fr an ndcatr (e.g., pressure, temperature). Is the nterval asscated wth, e.g. lw pressure prprtnately the same as the nterval asscated wth lw temperature? If nt, then s there a way t acheve a scale-nvarance fr wrds s that cntext-ndependent-ntervals can be used n cntext-dependent stuatns? Humans seem t understand certan terms n a cntext-free stuatn, and are able t apply them n cntextdependent nes, and als understand terms n a cntext and are able t adapt them t ther cntexts; s, we cnjecture that t ught t be pssble fr cmputers t d lkewse. The FOUs fr the wrds n V are pre-cmputed usng the wrd surveys descrbed abve. Anther pen ssue s hw much uncertanty ne shuld asscate wth the nterval end-pnts. Let! L and 4 Fr sme judgments (e.g., wealth), the ndcatrs and judgments wll have natural scales, whereas fr thers (e.g., flrtatn) n natural scales exst. Instead f usng scales, t may be pssble t use a lne f arbtrary length and percentages that are asscated wth that length. 5 See ftnte 4 fr an alternatve apprach. 8

9 Mendel, J. M., "An Archtecture fr Makng Judgments Usng Cmputng Wth Wrds," Int. J. Appl. Math. Cmput. Sc., vl. 12, N. 3, pp , 2002.! R dente the standard devatn fr the left and rght end-pnts f a wrd s nterval, respectvely, and! dente a fractn f uncertanty,.e. 0! "!1; then, when we use!" L r!" R fr nterval end-pnt uncertantes, what s!? If! s chsen t large then all MFs wll verlap t much. A thery s needed t gude us n hw much uncertanty shuld be used t characterze the uncertanty f wrds. B. Encder The encder transfrms a wrd nt a type-2 MF µ W. A type-2 MF s three-dmensnal. Each element f a FOU has a pssblty value a secndary MF assgned t t. The unn f all pssblty values defned ver the FOU cnsttutes the type-2 MF. The FOU fr each f the wrds n V s pre-cmputed, as descrbed n Sectn 3.A. The remanng ssue s what t chse fr the secndary MFs. Mendel (2001a, 2001b) advcates usng nterval sets because then all remanng peratns wthn the type-2 FLS are manageable. Addtnally, even fr a type-1 FLS there s n ne best chce fr the shape fr a MF, s why shuld we cmpund ths by tryng t chse secndary MFs as arbtrary shapes? Anther very mprtant reasn fr usng nterval type-2 fuzzy sets s that all set-theretc laws (e.g., DeMrgan s, dstrbutve, asscatve, etc.) are satsfed regardless f whch t-nrm r t-cnrm are used. Ths s nt the case when arbtrary type-2 fuzzy sets are used [see Appendx B n Mendel (2001a)], whch culd cause serus prblems n a rule-based system. C. Type-2 FLS As descrbed n Appendx B, a type-2 FLS cnssts f a fuzzfer and an nference engne fllwed by type-reductn and defuzzfcatn. The fact that all wrds are encded usng type-2 fuzzy sets means that the encder already accmplshes fuzzfcatn. Rules, whch establsh the detaled archtecture f the FLS, are the heart f any FLS. They reman the same regardless f whether we use type-2 r type-1 fuzzy sets n the nference engne. What changes s the way n whch we mdel the antecedent and cnsequent fuzzy sets. T begn, a specfc archtecture must be chsen fr the FLS. Ths wll depend n part n hw many ndcatrs f a judgment are cnsdered t be sgnfcant. Mre than tw ndcatrs cause a prblem, because 9

10 Mendel, J. M., "An Archtecture fr Makng Judgments Usng Cmputng Wth Wrds," Int. J. Appl. Math. Cmput. Sc., vl. 12, N. 3, pp , peple d nt lke t answer many questns and have great dffculty n crrelatng mre than tw thngs at a tme (e.g., a tw-antecedent rule can be nterpreted as prvdng the crrelatn between the tw antecedents). S, f mre than tw ndcatrs are requred, then sme new archtecture(s) wll be needed fr the type-2 FLS, e.g. a parallel ntercnnectn f sub-advsrs, r a herarchcal archtecture. T-date, n such archtectures have been publshed fr type-2 FLSs, s ths s anther pen ssue. Chces must be made t mplement the (cmpstnal) nference engne, namely the mplcatn (!) and t-nrm (!) t be used. Althugh Mamdan mplcatns and prduct r mnmum t-nrms are wdely used n many engneerng applcatns f a FLS, fr the Per-C the knd f mplcatn and t-nrm t use are pen ssues. In fact, any chces fr mplcatn r t-nrm dfferent frm thse just mentned wll requre new results be develped fr the extended sup-star cmpstn f a type-2 FLS, whch s yet anther pen ssue. Yager and Flev (1994), fr example, prpse that Mamdan mplcatn and lgc mplcatn ( ~ p! q ) are the lwer and upper bund fr the nference engne, and establsh a prcedure cmprmse fuzzy reasnng n whch a lnear cmbnatn (.e., a parametrc mdel) f these tw extremes s used. Further dscussns abut the many chces that have t be made fr the nference engne as cmpared t rule-related uncertantes are gven n Mendel and Wu (2002). Rules are type-2 cmpstns, and n Fgure 3 are dented as R,.e. R = R ( V A, V C ) = R ( V A, V C!) (5) Inferencng n a type-2 FLS s dne usng the extended sup-star cmpstn [Mendel (2001a)] whch transfrms W and R nt anther type-2 fuzzy set, I, and nvlves a t-nrm peratn,.e., I = I ( W, R ) = I ( W, V A, V C ) = I ( W, V A, V C!,!) (6) 10

11 Mendel, J. M., "An Archtecture fr Makng Judgments Usng Cmputng Wth Wrds," Int. J. Appl. Math. Cmput. Sc., vl. 12, N. 3, pp , Inferencng s very straghtfrward fr an nterval type-2 FLS 6. Fllwng the nference engne s type-reductn (TR) 7. TR maps a type-2 fuzzy set nt a type-1 fuzzy set. Just as there are many defuzzfcatn methds fr a type-1 FLS, there are many TR methds fr a type-2 FLS. Whether r nt ne TR methd s best fr a Per-C s yet anther pen ssue. Whether fred utput sets shuld be cmbned prr t r as part f TR s yet anther pen ssue (as t s fr defuzzfcatn). Wthut gng nt the detals f hw t perfrm TR, we can smply vew t as a nn-lnear peratr n I,.e., fr an nterval type-2 FLS, TR = f TR [ I ( W, V, V A C!,!)] = [y l, y r ] (7) where [y l, y r ] s an nterval type-1 fuzzy set, and y l = y l [ I ( W, V, V A C!,!, f TR )] (8) and [ ] (9) y r = y r I ( W, V, V A C!,!, f TR ) The fnal peratn wthn the type-2 FLS s defuzzfcatn, whch s the mappng f the type-1 fuzzy set TR nt a number, O, a type-0 fuzzy set. The bvus chce fr defuzzfcatn s the center f gravty (COG) f type-1 set TR,.e. { } [ I ( W, V, V A C!,!, f TR )] y r [ I ( W, V, V A C!,!, f TR )] O = COG(TR) = COG f TR I [ ( W, V, V!,!) ] A C = 1 2 y l (10) 6 See Appendx D fr nferencng results fr sngletn nterval type-2 FLSs. Mre general results fr nn-sngletn nterval type-2 FLSs (n whch measurements that actvate the FLS are mdeled ether as type-1 r type-2 fuzzy sets) can be fund n Mendel (2001a, Chapters 11 and 12). 7 See Appendx C fr a bref dscussn abut type-reductn methds. 11

12 Mendel, J. M., "An Archtecture fr Makng Judgments Usng Cmputng Wth Wrds," Int. J. Appl. Math. Cmput. Sc., vl. 12, N. 3, pp , Equatn (10) s the nput-utput frmula fr the type-2 FLS wthn the Per-C. D. Decder The decder, D, perates n O t prvde a wrd W d as n (4),.e., W d = D(O) = D[ COG(TR) ] { [ ]} = D" COG f TR I ( W, # $ V, V!,!) A C % &' ( V (11) Observe that the utput f the decder s a wrd frm the vcabulary V. Hw t g frm MF numercal values fr a varable t a lngustc descrptn f that varable fr type-1 fuzzy sets s well knwn; hwever, hw t d ths fr type-2 fuzzy sets s nt s well knwn. Cnsder, fr example, the type-1 stuatn depcted n Fgure 4 at x = x!. Ths value f x nly generates a nn-zer membershp value n the fuzzy set F 4 = Medum Pstve; hence, x = x! can be descrbed lngustcally, wthut any ambguty, as Medum Pstve. The stuatn at x = x! dfferent, because ths value f x generates a nn-zer membershp value n tw fuzzy sets F 4 = Medum Pstve and F 5 = Very Pstve. It wuld be very awkward t speak f x! s as beng Medum Pstve t degree µ F4 ( x!!) and Very Pstve t degree µ F5 ( x!!). Peple just dn t cmmuncate ths way. Instead, we usually cmpare µ F4 ( x!!) and µ F5 ( x!!) t see whch s larger 8, and then assgn x! speak f x! as beng Medum Pstve. t the set asscated wth the larger value; hence, n ths example, we wuld We can frmally descrbe what we have just explaned, as fllws. Gven P fuzzy sets F wth MFs µ F (x), = 1,, P. When x = x!, evaluate all P MFs at ths pnt, and then cmpute [ ] " µ Fm (! max µ F1 ( x!), µ F2 ( x! ),, µ ( x! ) FP Then, L( x!) " F m,.e., x ). Let L( x!) dente the lngustc label asscated wth x!. 8 There s a lterature that deals wth ther ways fr dng ths; hwever, all ther ways are mre cmplcated than the present way, and usually rely n the avalablty f truth data. Such data s usually nt avalable when cmputng wth wrds. 12

13 Mendel, J. M., "An Archtecture fr Makng Judgments Usng Cmputng Wth Wrds," Int. J. Appl. Math. Cmput. Sc., vl. 12, N. 3, pp , [ ] (12) L( x!) = arg max "F µ F1 ( x!), µ F2 ( x! ),, µ ( x! ) FP Cnsder the type-2 stuatn depcted n Fgure 5 at x = x!. Ths value f x nly generates a nnzer membershp n the type-2 fuzzy set W 4 = Medum Pstve; hence, x = x! can be descrbed lngustcally, wthut any ambguty, as Medum Pstve. The stuatn at x = x! s qute dfferent, because ths value f x generates a range f nn-zer secndary MF values n the tw type- 2 fuzzy sets W 4 = Medum Pstve and W 5 = Very Pstve. It wuld be extrardnarly dffcult t cmmuncate ths lngustcally. An apprach (nt necessarly an ptmal ne, but ne that generalzes ts type-1 cunterpart t type-2 fuzzy sets), whch we descrbe next, s t frst cnvert the ntersectn f the vertcal lne at x = x! wth the FOUs nt a cllectn f numbers, after whch we can chse the lngustc label at x = x! usng the algrthm descrbed belw. µ F ( x) F 4 F 5 Medum Pstve Very Pstve x! x!! Fgure 4: Returnng t a lngustc label fr type-1 fuzzy sets. x The type-2 MFs µ W 4 (x,u) and µ W 5 (x,u) are characterzed by ther (shaded) FOUs, FOU( W 4 ) and FOU( W 5 ), respectvely. The upper and lwer MFs fr W 4 are µ W 4 (x) and µ W 4 (x), whereas the cmparable quanttes fr W 5 are µ W 5 (x) and µ W 5 (x). Cnsder, fr example, the vertcal lne at x = x!, and ts ntersectns wth the FOU fr W 4 (see Fgure 5). Asscated wth the nterval [ µ W 4 ( x!! ), µ W 4 ( x!!)] s the secndary MF f! [ ] x (u), u " µ W 4 ( x! ), µ W 4 ( x! ). Let the center f gravty f 13

14 Mendel, J. M., "An Archtecture fr Makng Judgments Usng Cmputng Wth Wrds," Int. J. Appl. Math. Cmput. Sc., vl. 12, N. 3, pp , x (u),!u " [ µ W ( x!! ), µ ( x!!) W ] be dented f cg 4 4 x! ( W 4 ). In a smlar manner, we can cmpute f!! f cg x! ( W 5 ). We can then cmpare f cg x! ( W 4 ) and f cg x! ( W 5 ). If f cg x! ( W 4 ) > f cg x!! ( W 5 ), then we wuld speak f x! as beng Medum Pstve ; therwse we wuld speak f x! as beng Very Pstve. We can frmally descrbe what we have just explaned as fllws. Gven N V type-2 fuzzy sets W wth MFs µ W (x,u), = 1,, N V. These MFs are characterzed by ther FOUs, FOU( W ), whse upper and lwer MFs are µ W (x) and µ W (x) ( = 1,, N V ), respectvely. Cnsder an arbtrary [ ]! f " value f x, say x = x!, and cmpute max f cg x! ( W ), f cg ( W ),, f cg ( W 1 x! 2 x! N v ) [ ] the center f gravty f the secndary MF f x! (u),"u # µ W ( x! ), µ W ( x!) lngustc label asscated wth x!. Then, L( x!) " W m,.e., cg x ( W m ) where f cg x! ( W ) s. Let L( x!) dente the L( x!) = arg max " [ cg x ( W ), f cg ( W ),, f cg ( W 1 x! 2 x! N v )] (13) W f! Fr nterval secndary MFs, t s easy t cmpute f cg x! ( W ), as cg f x! ( W [ ] (13) ) = 1 2 µ W ( x! ) + µ ( x!) W u W 4 Medum Pstve W 5 Very Pstve µ W 4 ( x!!) µ W 5 ( x!!) µ W ( x!!) 4 x! x!! µ W ( x!!) 5 Fgure 5: Returnng t a lngustc label fr type-2 fuzzy sets. The shaded regns are the FOUs fr the tw type-2 fuzzy sets. x 14

15 Mendel, J. M., "An Archtecture fr Makng Judgments Usng Cmputng Wth Wrds," Int. J. Appl. Math. Cmput. Sc., vl. 12, N. 3, pp , CONCLUSIONS We have presented a specfc archtecture fr makng judgments by cmputng wth wrds the perceptual cmputer (Per-C) and have argued that type-2 fuzzy sets must be used fr cmputng wth wrds. The fllwng ssues (ncludng thers) must be reslved befre the Per-C can be mplemented: 1. Hw shuld survey nterval nfrmatn abut wrds be used t derve an asscated type-2 MF? 2. Can cntext-ndependent results be appled when terms are used wthn a specfc cntext? Is there a way t acheve scale-nvarance fr wrds s cntext-ndependent ntervals can be used n cntext-dependent stuatns? Or, s there a way t map frm cntext-ndependent ntervals t cntext-dependent ntervals? 3. Hw much uncertanty shuld be asscated wth the nterval end-pnts. A methdlgy fr chsng r desgnng the uncertanty factr needs t be develped. 4. What are new archtectures fr type-2 FLSs that can be used n a Per-C? Can a specfc archtecture be valdated? 5. Whch mplcatns and t-nrms are mst apprprate fr a Per-C? What are the asscated extended sup-star cmpstn results fr them? It seems that, at the very least, we must accunt fr the uncertantes present n all rule-wrds, ncludng cnnectr wrds. Ths can be accmplshed by usng type-2 fuzzy sets fr antecedent and cnsequent wrds and parametrc peratrs fr cnnectr wrds. It may then be necessary t accunt fr the uncertantes asscated wth mplcatn and cmbnng f rules. Whether r nt t s necessary t parameterze the enrmus numbers f chces that are avalable fr the peratr mdels f mplcatn, unn, ntersectn, cmplement, t-nrm and t-cnrm, as n Yager and Flev (1994) remans t be explred. See Wu and Mendel (2001) fr mre dscussns abut ths. 6. Hw are fred rule utputs cmbned by peple? The engneerng lterature n FLSs has n adequate answer t ths questn. Yager and Flev (1994) ntrduce the ntn f sft rule aggregatn n whch rules are cmbned usng ether the SOWA-OR r the SOWA-AND. Ths apprach remans t be examned n the cntext f type-2 fuzzy sets. 7. Whch type-reductn methd s best fr a Per-C? 15

16 Mendel, J. M., "An Archtecture fr Makng Judgments Usng Cmputng Wth Wrds," Int. J. Appl. Math. Cmput. Sc., vl. 12, N. 3, pp , Is there a best way t return t a lngustc label fr a type-2 fuzzy set,.e., an ptmal decder? In the future, a cmputer-archtecture may cme nt beng that wll let us actually cmpute wth wrds. In the meantme, f we are t cmpute wth wrds, t must be dne wthn the framewrk f exstng cmputer archtectures, all f whch cmpute wth numbers. The Per-C lets us make judgments 9 by cmputng wth wrds usng exstng cmputer archtectures. APPENDIX A: Backgrund Knwledge abut Type-2 Fuzzy Sets In ths appendx we cllect sme mprtant defntns abut type-2 fuzzy sets. Fr mre detals abut such fuzzy sets, as well as many examples that llustrate the defntns, see Mendel (2001a). Defntn A-1: A type-2 fuzzy set, dented A, s characterzed by a (three-dmensnal) type-2 membershp functn µ A (x,u),.e., A = " (x,u) / (x,u) J x # [0,1] (A-1) µ x!x " u!j A x where!! dentes unn ver all admssble x and u, and 0! µ A (x,u)! 1. At each fxed value f x!x, J x s the prmary membershp f x and x s called the prmary varable. Defntn A-2: At each value f x, say x = x!, the 2D plane whse axes are u and µ (! A x,u) s called a vertcal slce f µ (x,u). A secndary membershp functn s a vertcal slce f µ A A (x,u). It s µ (x =! A x,u) fr x! "X and!u "J x # $[0,1],.e., µ (x = x!,u) " µ (! A A x ) = $ f (u) u x! J x! % [0,1] (A-2) u #J x! 16

17 Mendel, J. M., "An Archtecture fr Makng Judgments Usng Cmputng Wth Wrds," Int. J. Appl. Math. Cmput. Sc., vl. 12, N. 3, pp , n whch 0! f x " (u)!1. Because! x " # X, we drp the prme ntatn n µ ( x!), and refer t µ (x) A A as a secndary membershp functn; t s a type-1 fuzzy set, whch we als refer t as a secndary set. Based n the cncept f secndary sets, we can renterpret a type-2 fuzzy set as the unn f all secndary sets,.e., usng (A-2), we can re-express A n a vertcal-slce manner, as: A = {(x,µ (x))!x "X} (A-3) A r as A = " µ (x) x = x!x A [ f x (u) / u x!x u!j x " " ] x J x # [0,1] (A-4) Defntn A-3: The dman f a secndary membershp functn s called the prmary membershp f x. In (A-4), J x s the prmary membershp f x, where J x! [0,1] fr!x "X. Defntn A-4: The ampltude f a secndary membershp functn s called a secndary grade. In (A-2) and (A-4), f x (u) s a secndary grade. Defntn A-5: An nterval type-2 fuzzy set s a type-2 fuzzy set all f whse secndary membershp functns are type-1 nterval sets,.e., f x (u) = 1,!u "J x # [0,1],!x "X. Interval secndary membershp functns reflect a unfrm uncertanty at the prmary membershps f x, and are the nes mst cmmnly used n type-2 FLSs. Nte that an nterval set can be represented just by ts dman nterval, whch can be expressed n terms f ts left and rght endpnts as [l, r], r by ts center and spread as [c - s, c +s], where c = (l + r) / 2 and s = (r! l) / 2. 9 We clam nthng else fr the Per-C, althugh t s archtecture may als be useful fr ther knds f cmputng wth wrds. 17

18 Mendel, J. M., "An Archtecture fr Makng Judgments Usng Cmputng Wth Wrds," Int. J. Appl. Math. Cmput. Sc., vl. 12, N. 3, pp , Defntn A-6: Assume that each f the secndary membershp functns f a type-2 fuzzy set has nly ne secndary grade that equals 1. A prncpal membershp functn s the unn f all such pnts at whch ths ccurs,.e., µ prncpal (x) = " u x where f x (u) = 1 (A-5) x!x and s asscated wth a type-1 fuzzy set. Fr nterval secndary membershp functns, we defne the prncpal membershp functn as ccurrng at the unn f all prmary membershp mdpnts. Nte that when all membershp functn uncertantes dsappear, a type-2 membershp functn reduces t ts prncpal membershp functn. Defntn A-7: Uncertanty n the prmary membershps f a type-2 fuzzy set, A, cnssts f a bunded regn that we call the ftprnt f uncertanty (FOU). It s the unn f all prmary membershps,.e., FOU( A ) =! (A-6) J x!x x The term FOU s very useful, because t nt nly fcuses ur attentn n the uncertantes nherent n a specfc type-2 membershp functn, whse shape s a drect cnsequence f the nature f these uncertantes, but t als prvdes a very cnvenent verbal descrptn f the entre dman f supprt fr all the secndary grades f a type-2 membershp functn. Defntn A-8: Cnsder a famly f type-1 membershp functns µ A (x p 1, p 2,..., p v ) where p 1, p 2,..., p v are parameters, sme r all f whch vary ver sme range f values,.e., p!p ( = 1,, v). A prmary membershp functn (MF) s any ne f these type-1 membershp functns, e.g., µ A (x p 1 = p 1!, p 2 = p 2!,..., p v = p v! ). Fr shrt, we use µ A (x) t dente a prmary membershp functn. It wll be subject t sme restrctns n ts parameters. The famly f all prmary membershp functns creates a FOU. 18

19 Mendel, J. M., "An Archtecture fr Makng Judgments Usng Cmputng Wth Wrds," Int. J. Appl. Math. Cmput. Sc., vl. 12, N. 3, pp , Tw examples f very useful prmary MFs are: Gaussan MF wth uncertan mean and certan standard devatn, and Gaussan MF wth certan mean and uncertan standard devatn. Defntn A-9: An upper membershp functn and a lwer membershp functn are tw type-1 membershp functns that are bunds fr the FOU f a type-2 fuzzy set A. The upper membershp functn s asscated wth the upper bund f FOU( A ), and s dented µ (x),!x "X. The lwer membershp functn s asscated wth the lwer bund f FOU( A ), and s dented µ A (x),!x "X,.e., A µ A (x)! FOU( A ) "x #X (A-7) and µ A (x)! FOU( A ) "x #X (A-8) Because the dman f a secndary membershp functn has been cnstraned n Defntn A-1 t be cntaned n [0, 1], lwer and upper membershp functns always exst. APPENDIX B. Rule-Based Fuzzy Lgc Systems A rule-based FLS cntans fur cmpnents rules, fuzzfer, nference engne, and utput prcessr that are nter-cnnected, as shwn n Fgure B-1. Once the rules have been establshed, a FLS can be vewed as a mappng frm nputs t utputs (the sld path n Fgure B-1, frm Crsp nputs t Crsp utputs ), and ths mappng can be expressed quanttatvely as y = f (x). Ths knd f FLS s very wdely used n many engneerng applcatns f FL, such as n FL cntrllers and sgnal prcessrs, and s als knwn as a fuzzy cntrller, fuzzy system, fuzzy expert system, r fuzzy mdel. 19

20 Mendel, J. M., "An Archtecture fr Makng Judgments Usng Cmputng Wth Wrds," Int. J. Appl. Math. Cmput. Sc., vl. 12, N. 3, pp , FLS Rules Crsp nputs x Fuzzfer Output Prcessr Crsp utputs y Fuzzy nput sets Inference Fuzzy utput sets y = f( x) Fgure B-1: Fuzzy lgc system. Rules are the heart f a FLS, and may be prvded by experts r can be extracted frm numercal data. In ether case, the rules that we are nterested n can be expressed as a cllectn f IF THEN statements. Fuzzy sets are asscated wth terms that appear n the antecedents r cnsequents f rules, and wth the nputs t and utput f the FLS. Membershp functns are used t descrbe these fuzzy sets. Tw knds f fuzzy sets can be used n a FLS, type-1 and type-2. Type-1 fuzzy sets are descrbed by membershp functns that are ttally certan, whereas type-2 fuzzy sets are descrbed by membershp functns that are themselves fuzzy. The latter let us quantfy dfferent knds f uncertantes that can ccur n a FLS. A FLS that s descrbed cmpletely n terms f type-1 fuzzy sets s called a type-1 FLS, whereas a FLS that s descrbed usng at least ne type-2 fuzzy set s called a type-2 FLS. A type-2 FLS whse MFs are nterval type-2 fuzzy sets s called an nterval type-2 FLS. In ths paper we assume the use f nterval type-2 FLSs. Returnng t the Fgure B-1 FLS, the fuzzfer maps crsp numbers nt fuzzy sets. It s needed t actvate rules that are n terms f lngustc varables, whch have fuzzy sets asscated wth them. The nputs t the FLS prr t fuzzfcatn may be certan (e.g., perfect measurements) r uncertan 20

21 Mendel, J. M., "An Archtecture fr Makng Judgments Usng Cmputng Wth Wrds," Int. J. Appl. Math. Cmput. Sc., vl. 12, N. 3, pp , (e.g., nsy measurements). Type-1 r type-2 fuzzy sets can be used t mdel the latter measurements. The nference engne f the Fgure B-1 FLS maps fuzzy sets nt fuzzy sets. It handles the way n whch rules are actvated and cmbned. Just as we humans use many dfferent types f nferental prcedures t help us understand thngs r t make decsns, there are many dfferent FL nferental prcedures. In many applcatns f a FLS, crsp numbers must be btaned at ts utput. Ths s accmplshed by the utput prcessr, and s knwn as defuzzfcatn. The utput prcessr fr a type-1 FLS cnssts nly f a defuzzfer; hwever, the utput prcessr f a type-2 FLS cntans tw cmpnents: the frst maps a type-2 fuzzy set nt a type-1 fuzzy set and s called type-reductn, and the secnd perfrms defuzzfcatn n the latter set. Type-reductn s vervewed n Appendx C. APPENDIX C: Type-Reductn The type-reduced set prvdes an nterval f uncertanty fr the utput f a type-2 FLS, n much the same way that a cnfdence nterval prvdes an nterval f uncertanty fr a prbablstc system. The mre uncertantes that ccur n a type-2 FLS, whch translate nt mre uncertantes abut ts MFs, the larger wll be the type-reduced set, and vce-versa. Fve dfferent type-reductn methds are descrbed n Mendel (2001a). Each s nspred by what we d n a type-1 FLS [when we defuzzfy the (cmbned) utput f the nference engne usng a varety f defuzzfcatn methds that all d sme srt f centrd calculatn] and are based n cmputng the centrd f a type-2 fuzzy set. Usng the Extensn Prncple, Karnk and Mendel (2001) defned the centrd f a type-2 fuzzy set; t s a type-1 fuzzy set. Cmputng the centrd f a general type-2 fuzzy set can be very ntensve; hwever, fr an nterval type-2 fuzzy set, an exact teratve methd fr cmputng ts centrd has been develped by Karnk and Mendel (2001). Ths was pssble because the centrd f an nterval type-2 fuzzy set s an nterval type-1 fuzzy set, and such sets are 21

22 Mendel, J. M., "An Archtecture fr Makng Judgments Usng Cmputng Wth Wrds," Int. J. Appl. Math. Cmput. Sc., vl. 12, N. 3, pp , cmpletely characterzed by ther left- and rght-end pnts; hence, cmputng the centrd f an nterval type-2 fuzzy set nly requres cmputng thse tw end-pnts. Center-f-sets, centrd, center-f-sums, and heght type-reductn can all be expressed as Y TR (x) = y l, y r f y = 1 # M (C-1) [ ] =!! 1 y # f # f M "[ f M, f ] 1 "[ f 1, f ] M " y M M y # 1 " y 1 1 [ [ 1 M l, y r ] l, y r ] M!! f =1 Fr the dfferent type-reductn methds y l, y r, f, f and M have dfferent meanngs, as summarzed n Table C-1. The Karnk-Mendel teratve prcedure fr cmputng y r s: 1. Wthut lss f generalty, assume that the pre-cmputed y r 1 2.e., y r! y r!!! y rm. are arranged n ascendng rder; M M 2. Cmpute y r as y r =! f r y r! f r by ntally settng f r =1 =1 = ( f + f ) 2 fr = 1,, M, where f and f have been prevusly cmputed usng the equatns gven n Appendx D, respectvely, and let y r! " y r. 3. Fnd R (1! R! M "1) such that y r R R! y r "! y +1 r. M M 4. Cmpute y r =! f r y r! f r wth f r y r! " y r. =1 =1 = f fr! R and f r = f fr > R, and let 5. If y r! " y! r, then g t Step 6. If y r! = y! r, then stp and set y r! " y r. 6. Set y r! equal t y r!, and return t Step 3. The prcedure fr cmputng y l s very smlar t the ne just gven fr y r. Just replace y r by y l, and, n step 3 fnd L (1! L! M "1) such that y L L +1 l! y l "! y l. Addtnally, n step 2 we nw 22

23 Mendel, J. M., "An Archtecture fr Makng Judgments Usng Cmputng Wth Wrds," Int. J. Appl. Math. Cmput. Sc., vl. 12, N. 3, pp , M cmpute y l as y l = f y M! l! f l l by ntally settng f r =1 = 1 = ( f + f ) 2 fr = 1,, M, and, n M step 4 we cmpute y l as y l = f y M! l! f l l wth f l =1 = 1 = f fr! L and f l = f fr > L. These tw fur-step teratve-prcedures (Steps 1 and 2 are ntalzatn-steps) have been prven by Karnk and Mendel (2001) t cnverge t the exact slutns n n mre than M teratns. Table C-1: Meanngs f y l, y r, f, f and M n (C-1) fr dfferent type-reductn methds a. Type-reductn methd center-f sets centrd b center-f-sums c heght y l and y r defned f and f defned d M defned left and rght end pnts f the lwer and upper frng degrees number f rules centrd f the cnsequent f f the th rule the th rule y l = y r = y, the th pnt n the lwer and upper membershp number f sampled pnts sampled unverse f dscurse f grades f the th sampled the FLS s utput utput f the FLS s utput y l = y r = y, the th pnt n the sums f lwer and upper number f sampled pnts sampled unverse f dscurse f membershp grades fr the th the FLS s utput sampled pnt f all rule utputs y l = y r = y, a sngle pnt n the lwer and upper frng degrees number f rules cnsequent dman f the th f the th rule rule, usually chsen t be the pnt havng the hghest prmary membershp n the prncpal MF f the utput set a. Cmparable results fr mdfed heght type-reductn can be fund n Mendel (2001a), Sectn b. Prr t calculatng the centrd type-reduced set, the fred type-2 fuzzy sets are unned. c. Prr t calculatng the center-f-sums type-reduced set, the membershp functns f the fred type-2 fuzzy sets are added (r a lnear cmbnatn f them s frmed). d. See Appendx D fr frmulas fr f and f, as well as fr rule utputs and unned rule utputs. 23

24 Mendel, J. M., "An Archtecture fr Makng Judgments Usng Cmputng Wth Wrds," Int. J. Appl. Math. Cmput. Sc., vl. 12, N. 3, pp , APPENDIX D: Fuzzy Inference Engne Results fr Interval Type-2 Fuzzy Sets Cnsder a type-2 FLS havng p nputs x 1! X 1,..., x p!x p and ne utput y!y. We assume there are M rules where the lth rule has the frm R l : IF x 1 s F l 1 and! and x p s F l p, THEN y s G l l = 1,...,M (D-1) Ths rule represents a type-2 relatn between the nput space X 1!!! X p, and the utput space, Y, f the type-2 FLS. Asscated wth the p antecedent type-2 fuzzy sets, F l, are the type-2 MFs µ F l (x ) ( = 1,, p), and asscated wth the cnsequent type-2 fuzzy set G l µ G l (y). s ts type-2 MF The majr result fr an nterval sngletn type-2 FLS s summarzed n the fllwng: Therem D-1: [Lang and Mendel (2000), Mendel (2001a)] In an nterval sngletn type-2 FLS usng prduct r mnmum t-nrm, fr nput x = x! : (a) The result f the nput and antecedent peratns, s an nterval type-1 set, called the frng set,.e., F l ( x!) = [ f l ( x!), f l ( x!)] " [ f l, f l ] (D-2) where f l ( x!) = µ F ( x! 1 l 1 )!!!µ F ( x! p l p ) (D-3) and f l ( x!) = µ ( x! F 1 1)!!!µ ( x! l F p l p ); (D-4) (b) The rule R l fred utput cnsequent set, µ B l (y), s the type-1 nterval fuzzy set 24

25 Mendel, J. M., "An Archtecture fr Makng Judgments Usng Cmputng Wth Wrds," Int. J. Appl. Math. Cmput. Sc., vl. 12, N. 3, pp , µ B l (y) = " 1 b l, y!y (D-5) [ ] b l! f l! µ G l ( y), f l! µ G l ( y) where µ G (y) and µ (y) are the lwer and upper membershp grades f µ l G l G (y). l (c) Suppse that N f the M rules n the FLS fre, where N! M, and the cmbned utput type-1 fuzzy set, µ B (y), s btaned by cmbnng the fred utput cnsequent sets by takng the unn f the rule R l fred utput cnsequent sets 10 ; then, µ B (y) = # 1 b, y!y (D-6) b! [[ f 1! µ G 1 ( y ) ] "! " [ f N! µ G N ( y ) ] [, f 1! µ G ] [ 1 ( y ) "! " f N! µ G ] ] N ( y ) A cmplete prf f ths therem can be fund n Lang and Mendel (2000) and n Mendel (2001a). Generalzatns f ths therem t the very mprtant case when the nput t the type-2 FLS s a type-2 fuzzy set whch wuld be the case when the wrds that actvate the Per-C are mdeled as type-2 fuzzy sets are als gven n thse references. 10 We d nt necessarly advcate takng the unn f these sets. Part c merely llustrates the calculatns f ne chses t d ths. 25

26 Mendel, J. M., "An Archtecture fr Makng Judgments Usng Cmputng Wth Wrds," Int. J. Appl. Math. Cmput. Sc., vl. 12, N. 3, pp , REFERENCES Karnk N. N. and Mendel J. M. (2001): Centrd f a type-2 fuzzy set. Infrmatn Scences, Vl. 132, pp Klr G. J. and Werman M. J. (1998): Uncertanty-Based Infrmatn. Hedelberg, Germany: Physca-Verlag. Lang Q. and Mendel J. M. (2000): Interval type-2 fuzzy lgc systems: thery and desgn. IEEE Trans. n Fuzzy Systems, Vl. 8, pp Mendel J. M. (1999): Cmputng wth wrds, when wrds can mean dfferent thngs t dfferent peple. Prc. f Thrd Internatnal ICSC Sympsum n Fuzzy Lgc and Applcatns, Rchester Unv., Rchester, NY, June, pp Mendel J. M. (2001a): Uncertan Rule-Based Fuzzy Lgc Systems: Intrductn and New Drectns. Upper Saddle Rver, NJ: Prentce-Hall. Mendel J. M. (2001b): On the mprtance f nterval sets n type-2 fuzzy lgc systems. Prc. f IFSA/NAFIPS Cnference, Vancuver, Canada, July, pp Mendel J. M. (2001c): The Perceptual Cmputer: an archtecture fr cmputng wth wrds. Prc. IEEE Cnference n Fuzzy Systems, Melburne, Australa, December, pp Mendel J. M. (2002): Uncertanty, type-2 fuzzy sets, and ftprnts f uncertanty. The 9 th Int l. Cnference n Infrmatn Prcessng and Management f Uncertanty n Knwledge-Based Systems, Annecy, France, July, pp Mendel J. M. and Wu H. (2002): Uncertanty versus chce n rule-based fuzzy lgc systems. IEEE WCCI, FUZZ Cnference, Hnlulu, HI, May, pp Wang L.-X. (1997): A Curse n Fuzzy Systems and Cntrl. Upper Saddle Rver, NJ: Prentce- Hall. Wang P. P. (Ed.) (2001): Cmputng Wth Wrds. New Yrk: Jhn Wley & Sns, Inc. Yager R. R. and Flev D. P. (1994): Essentals f Fuzzy Mdelng and Cntrl. New Jersey: Jhn Wley & Sns, Inc. Zadeh L. A. (1996): Fuzzy lgc = cmputng wth wrds. IEEE Trans. n Fuzzy Systems, Vl. 4, pp Zadeh L. A. (1999): Frm cmputng wth numbers t cmputng wth wrds frm manpulatn f 26

27 Mendel, J. M., "An Archtecture fr Makng Judgments Usng Cmputng Wth Wrds," Int. J. Appl. Math. Cmput. Sc., vl. 12, N. 3, pp , measurements t manpulatn f perceptns. IEEE Trans. n Crcuts and Systems I: Fundamental Thery and Applcatns, Vl. 4, pp