From Computing with Numbers to Computing with Words From Manipulation of Measurements to Manipulation of Perceptions

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1 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 45, NO. 1, JANUARY From Computng wth Numbers to Computng wth Words From Manpulaton of Measurements to Manpulaton of Perceptons Lotf A. Zadeh, Lfe Fellow, IEEE (Invted Paper) Dedcated to the memory of Sdney Darlngton Abstract Computng, n ts usual sense, s centered on manpulaton of numbers and symbols. In contrast, computng wth words, or CW for short, s a methodology n whch the objects of computaton are words and propostons drawn from a natural language, e.g., small, large, far, heavy, not very lkely, the prce of gas s low and declnng, Berkeley s near San Francsco, CA, t s very unlkely that there wll be a sgnfcant ncrease n the prce of ol n the near future, etc. Computng wth words (CW) s nspred by the remarkable human capablty to perform a wde varety of physcal and mental tasks wthout any measurements and any computatons. Famlar examples of such tasks are parkng a car, drvng n heavy traffc, playng golf, rdng a bcycle, understandng speech, and summarzng a story. Underlyng ths remarkable capablty s the bran s crucal ablty to manpulate perceptons perceptons of dstance, sze, weght, color, speed, tme, drecton, force, number, truth, lkelhood, and other characterstcs of physcal and mental objects. Manpulaton of perceptons plays a key role n human recognton, decson and executon processes. As a methodology, computng wth words provdes a foundaton for a computatonal theory of perceptons a theory whch may have an mportant bearng on how humans make and machnes mght make percepton-based ratonal decsons n an envronment of mprecson, uncertanty and partal truth. A basc dfference between perceptons and measurements s that, n general, measurements are crsp whereas perceptons are fuzzy. One of the fundamental ams of scence has been and contnues to be that of progressng from perceptons to measurements. Pursut of ths am has led to brllant successes. We have sent men to the moon; we can buld computers that are capable of performng bllons of computatons per second; we have constructed telescopes that can explore the far reaches of the unverse; and we can date the age of rocks that are mllons of years old. But alongsde the brllant successes stand conspcuous underachevements and outrght falures. We cannot buld robots whch can move wth the aglty of anmals or humans; we cannot Manuscrpt receved May 10, 1998; revsed August 17, Ths work was supported n part by NASA under Grant NAC2-1177, by the Offce of Naval Research under Grant N , by the Army Research Offce under Grant DAAH , and by the BISC Program of Unversty of Calforna at Berkeley. Ths paper s an expanded and updated verson of a paper enttled Fuzzy Logc = Computng wth Words, whch appeared n the IEEE TRANSACTIONS ON FUZZY SYSTEMS, vol. 4, , The author s wth the Graduate School, Berkeley Intatve n Soft Computng (BISC), and also wth the Computer Scence Dvson and the Electroncs Research Laboratory, Department of Electrcal Engneerng and Computer Scences, Unversty of Calforna at Berkeley, Berkeley, CA USA. Publsher Item Identfer S (99) automate drvng n heavy traffc; we cannot translate from one language to another at the level of a human nterpreter; we cannot create programs whch can summarze nontrval stores; our ablty to model the behavor of economc systems leaves much to be desred; and we cannot buld machnes that can compete wth chldren n the performance of a wde varety of physcal and cogntve tasks. It may be argued that underlyng the underachevements and falures s the unavalablty of a methodology for reasonng and computng wth perceptons rather than measurements. An outlne of such a methodology referred to as a computatonal theory of perceptons s presented n ths paper. The computatonal theory of perceptons, or CTP for short, s based on the methodology of CW. In CTP, words play the role of labels of perceptons and, more generally, perceptons are expressed as propostons n a natural language. CW-based technques are employed to translate propostons expressed n a natural language nto what s called the Generalzed Constrant Language (GCL). In ths language, the meanng of a proposton s expressed as a generalzed constrant, s R, where s the constraned varable, R s the constranng relaton and sr s a varable copula n whch r s a varable whose value defnes the way n whch R constrans. Among the basc types of constrants are: possblstc, verstc, probablstc, random set, Pawlak set, fuzzy graph and usualty. The wde varety of constrants n GCL makes GCL a much more expressve language than the language of predcate logc. In CW, the ntal and termnal data sets, IDS and TDS, are assumed to consst of propostons expressed n a natural language. These propostons are translated, respectvely, nto antecedent and consequent constrants. Consequent constrants are derved from antecedent constrants through the use of rules of constrant propagaton. The prncpal constrant propagaton rule s the generalzed extenson prncple. The derved constrants are retranslated nto a natural language, yeldng the termnal data set (TDS). The rules of constrant propagaton n CW concde wth the rules of nference n fuzzy logc. A basc problem n CW s that of explctaton of, R, and r n a generalzed constrant, sr R, whch represents the meanng of a proposton, p, n a natural language. There are two major mperatves for computng wth words. Frst, computng wth words s a necessty when the avalable nformaton s too mprecse to justfy the use of numbers; and second, when there s a tolerance for mprecson whch can be exploted to acheve tractablty, robustness, low soluton cost and better rapport wth realty. Explotaton of the tolerance for mprecson s an ssue of central mportance n CW and CTP. At ths juncture, the computatonal theory of perceptons whch s based on CW s n ts ntal stages of development. In tme, /99$ IEEE

2 106 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 45, NO. 1, JANUARY 1999 t may come to play an mportant role n the concepton, desgn and utlzaton of nformaton/ntellgent systems. The role model for CW and CTP s the human mnd. I. INTRODUCTION IN THE Fftes, and especally late Fftes, crcut theory was at the heght of mportance and vsblty. It played a pvotal role n the concepton and desgn of electronc crcuts and was enrched by basc contrbutons of Darlngton, Bode, McMllan, Gullemn, Carln, Youla, Kuh, Desoer, Sandberg, and other poneers. However, what could be dscerned at that tme was that crcut theory was evolvng nto a more general theory system theory a theory n whch the physcal dentty of the elements of a system s subordnated to a mathematcal charactercaton of ther nput/output relatons. Ths evoluton was a step n the drecton of greater generalty and, lke most generalzatons, t was drven by a quest for models whch make t possble to reduce the dstance between an object that s modeled the modelzand and ts model n a specfed class of systems. In a paper publshed n 1961 enttled From Crcut Theory to System Theory, [33] I dscussed the evoluton of crcut theory nto system theory and observed that the hgh effectveness of system theory n dealng wth mechanstc systems stood n sharp contrast to ts low effectveness n the realm of humanstc systems systems exemplfed by economc systems, bologcal systems, socal systems, poltcal systems and, more generally, man-machne systems of varous types. In more specfc terms, I wrote: There s a farly wde gap between what mght be regarded as anmate system theorsts and nanmate system theorsts at the present tme, and t s not at all certan that ths gap wll be narrowed, much less closed, n the near future. There are some who feel that ths gap reflects the fundamental nadequacy of conventonal mathematcs the mathematcs of precselydefned ponts, functons, sets, probablty measures, etc. for copng wth the analyss of bologcal systems, and that to deal effectvely wth such systems, whch are generally orders of magntude more complex than man-made systems, we need a radcally dfferent knd of mathematcs, the mathematcs of fuzzy or cloudy quanttes whch are not descrbable n terms of probablty dstrbutons. Indeed, the need for such mathematcs s becomng ncreasngly apparent even n the realm of nanmate systems, for n most practcal cases the a pror data as well as the crtera by whch the performance of a man-made system are judged are far from beng precsely specfed or havng accuratelyknown probablty dstrbutons. It was ths observaton that motvated my development of the theory of fuzzy sets. startng wth the 1965 paper Fuzzy Sets [34], whch was publshed n Informaton and Control. Subsequently, n a paper publshed n 1973, Outlne of a New Approach to the Analyss of Complex Systems and Decson Processes, [37] I ntroduced the concept of a lngustc varable, that s, a varable whose values are words rather than numbers. The concept of a lngustc varable has played and s contnung to play a pvotal role n the development of fuzzy logc and ts applcatons. The ntal recepton of the concept of a lngustc varable was far from postve, largely because my advocacy of the use of words n systems and decson analyss clashed wth the deep-seated tradton of respect for numbers and dsrespect for words. The essence of ths tradton was succnctly stated n 1883 by Lord Kelvn: In physcal scence the frst essental step n the drecton of learnng any subject s to fnd prncples of numercal reckonng and practcable methods for measurng some qualty connected wth t. I often say that when you can measure what you are speakng about and express t n numbers, you know somethng about t; but when you cannot measure t, when you cannot express t n numbers, your knowledge s of a meagre and unsatsfactory knd: t may be the begnnng of knowledge but you have scarcely, n your thoughts, advanced to the state of scence, whatever the matter may be. The depth of scentfc tradton of respect for numbers and derson for words was reflected n the ntensty of hostle reacton to my deas by some of the promnent members of the scentfc elte. In commentng on my frst exposton of the concept of a lngustc varable n 1972, Rudolph Kalman had ths to say: I would lke to comment brefly on Professor Zadeh s presentaton. Hs proposals could be severely, ferocously, even brutally crtczed from a techncal pont of vew. Ths would be out of place here. But a blunt queston remans: Is Professor Zadeh presentng mportant deas or s he ndulgng n wshful thnkng? No doubt Professor Zadeh s enthusasm for fuzzness has been renforced by the prevalng clmate n the U.S. one of unprecedented permssveness. Fuzzfcaton s a knd of scentfc permssveness; t tends to result n socally appealng slogans unaccompaned by the dscplne of hard scentfc work and patent observaton. In a smlar ven, my esteemed colleague Professor Wllam Kahan a man wth a brllant mnd offered ths assessment n 1975: Fuzzy theory s wrong, wrong, and perncous. says Wllam Kahan, a professor of computer scences and mathematcs at Cal whose Evans Hall offce s a few doors from Zadeh s. I cannot thnk of any problem that could not be solved better by ordnary logc. What Zadeh s sayng s the same sort of thngs Technology got us nto ths mess and now t can t get us out. Well, technology dd not get us nto ths mess. Greed and weakness and ambvalence got us nto ths mess. What we need s more logcal thnkng, not less. The danger of fuzzy theory s that t wll encourage the sort of mprecse thnkng that has brought us so much trouble. What Lord Kelvn, Rudolph Kalman, Wllam Kahan, and many other brllant mnds dd not apprecate s the fundamental mportance of the remarkable human capablty to perform a wde varety of physcal and mental tasks wthout

3 ZADEH: FROM COMPUTING WITH NUMBERS TO COMPUTING WITH WORDS 107 Fg. 1. Informal and formal defntons of a granule. (a) Fg. 2. Examples of crsp and fuzzy granulaton. any measurements and any computatons. Famlar examples of such tasks are parkng a car; drvng n heavy traffc; playng golf; understandng speech, and summarzng a story. Underlyng ths remarkable ablty s the bran s crucal ablty to manpulate perceptons perceptons of sze, dstance, weght, speed, tme, drecton, smell, color, shape, force, lkelhood, truth and ntent, among others. A fundamental dfference between measurements and perceptons s that, n general, measurements are crsp numbers whereas perceptons are fuzzy numbers or, more generally, fuzzy granules, that s, clumps of objects n whch the transton from membershp to nonmembershp s gradual rather than abrupt. The fuzzness of perceptons reflects fnte ablty of sensory organs and the bran to resolve detal and store nformaton. A concomtant of fuzzness of perceptons s the preponderant partalty of human concepts n the sense that the valdty of most human concepts s a matter of degree. For example, we have partal knowledge, partal understandng, partal certanty, partal belef and accept partal solutons, partal truth and partal causalty. Furthermore, most human concepts have a granular structure and are context-dependent. In essence, a granule s a clump of physcal or mental objects (ponts) drawn together by ndstngushablty, smlarty, proxmty or functonalty (Fg. 1). A granule may be crsp or fuzzy, dependng on whether ts boundares are or are not sharply defned. For example, age may be granulated crsply nto years and granulated fuzzly nto fuzzy ntervals labeled very young, young, mddle-aged, old, and very old (Fg. 2). A partal taxonomy of granulaton s shown n Fg. 3(a) and (b). In a very broad sense, granulaton nvolves a parttonng of whole nto parts. Modes of nformaton granulaton (IG) n whch granules are crsp play mportant roles n a wde varety of methods, approaches and technques. Among them are: nterval analyss, quantzaton, chunkng, rough set theory, dakoptcs, dvde and conquer, Dempster Shafer theory, Fg. 3. (b) (a) Partal taxonomy of granulaton. (b) Prncpal types of granules. machne learnng from examples, qualtatve process theory, decson trees, semantc networks, analog-to-dgtal converson, constrant programmng, mage segmentaton, cluster analyss and many others. Important though t s, crsp IG has a major blnd spot. More specfcally, t fals to reflect the fact that most human perceptons are fuzzy rather than crsp. For example, when we mentally granulate the human body nto fuzzy granules labeled head, neck, chest, arms, legs, etc., the length of neck s a fuzzy attrbute whose value s a fuzzy number. Fuzzness of granules, ther attrbutes and ther values s characterstc of ways n whch human concepts are formed, organzed and manpulated. In effect, fuzzy nformaton granulaton (fuzzy IG) may be vewed as a human way of employng data compresson for reasonng and, more partcularly, makng ratonal decsons n an envronment of mprecson, uncertanty and partal truth. The tradton of pursut of crspness and precson n scentfc theores can be credted wth brllant successes. We have sent men to the moon; we can buld computers that are capable of performng bllons of computatons per second; we have constructed telescopes that can explore the far reaches of the unverse; and we can date the age of rocks that are mllons of years old. But alongsde the brllant successes stand conspcuous underachevements and outrght falures. We cannot buld robots whch can move wth the aglty of anmals or humans; we cannot automate drvng n heavy traffc; we cannot translate from one language to another at the level of a human nterpreter; we cannot create programs whch can summarze nontrval stores; our ablty to model the be-

4 108 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 45, NO. 1, JANUARY 1999 (a) Fg. 4. Conceptual structure of computatonal theory of perceptons. havor of economc systems leaves much to be desred; and we cannot buld machnes that can compete wth chldren n the performance of a wde varety of physcal and cogntve tasks. What s the explanaton for the dsparty between the successes and falures? What can be done to advance the fronters of scence and technology beyond where they are today, especally n the realms of machne ntellgence and automaton of decson processes? In my vew, the falures are conspcuous n those areas n whch the objects of manpulaton are, n the man, perceptons rather than measurements. Thus, what we need are ways of dealng wth perceptons, n addton to the many tools whch we have for dealng wth measurements. In essence, t s ths need that motvated the development of the methodology of computng wth words (CW) a methodology n whch words play the role of labels of perceptons. CW provdes a methodology for what may be called a computatonal theory of perceptons (CTP) (Fg. 4). However, the potental mpact of the methodology of computng wth words s much broader. Bascally, there are four prncpal ratonales for the use of CW. 1) The don t know ratonale: In ths case, the values of varables and/or parameters are not known wth suffcent precson to justfy the use of conventonal methods of numercal computng. An example s decson-makng wth poorly defned probabltes and utltes. 2) The don t need ratonale: In ths case, there s a tolerance for mprecson whch can be exploted to acheve tractablty, robustness, low soluton cost, and better rapport wth realty. An example s the problem of parkng a car. 3) The can t solve ratonale: In ths case, the problem cannot be solved through the use of numercal computng. An example s the problem of automaton of drvng n cty traffc. 4) The can t defne ratonale: In ths case, a concept that we wsh to defne s too complex to admt of defnton n terms of a set of numercal crtera. A case n pont s the concept of causalty. Causalty s an nstance of what may be called an amorphc concept. The basc dea underlyng the relatonshp between CW and CTP s conceptually smple. More specfcally, n CTP per- (b) Fg. 5. (a) Examples of reasonng wth perceptons. (b) Examples of ncorrect reasonng. ceptons and queres are expressed as propostons n a natural language. Then, the propostons and queres are processed by CW-based methods to yeld answers to queres. Smple examples of lngustc characterzaton of perceptons drawn from everyday experences are: Robert s hghly ntellgent Carol s very attractve Hans loves wne Overeatng causes obesty Most Swedes are tall Berkeley s more lvely than Palo Alto It s lkely to ran tomorrow It s very unlkely that there wll be a sgnfcant ncrease n the prce of ol n the near future Examples of correct conclusons drawn from perceptons through the use of CW-based methods are shown n Fg. 5(a). Examples of ncorrect conclusons are shown n Fg. 5(b). Perceptons have long been an object of study n psychology. However, the dea of lnkng perceptons to computng wth words s n a dfferent sprt. An nterestng systemtheoretc approach to perceptons s descrbed n a recent work of R. Vallee [31]. A logc of perceptons has been descrbed by H. Rasowa [26]. These approaches are not related to the approach descrbed n our paper. An mportant pont that should be noted s that classcal logcal systems such as propostonal logc, predcal logc and modal logc, as well as AI-based technques for natural language processng and knowledge representaton, are concerned n a fundamental way wth propostons expressed n a natural language. The man dfference between such approaches and CW s that the methodology of CW whch s based on fuzzy logc provdes a much more expressve language for

5 ZADEH: FROM COMPUTING WITH NUMBERS TO COMPUTING WITH WORDS 109 knowledge representaton and much more versatle machnery for reasonng and computaton. In the fnal analyss, the role model for computng wth words s the human mnd and ts remarkable ablty to manpulate both measurements and perceptons. What should be stressed, however, s that although words are less precse than numbers, the methodology of computng wth words rests on a mathematcal foundaton. An exposton of the basc concepts and technques of computng wth words s presented n the followng sectons. The lnkage of CW and CTP s dscussed very brefly because the computatonal theory of perceptons s stll n ts early stages of development. II. WHAT IS CW? In ts tradtonal sense, computng nvolves for the most part manpulaton of numbers and symbols. By contrast, humans employ mostly words n computng and reasonng, arrvng at conclusons expressed as words from premses expressed n a natural language or havng the form of mental perceptons. As used by humans, words have fuzzy denotatons. The same apples to the role played by words n CW. The concept of CW s rooted n several papers startng wth my 1973 paper Outlne of a New Approach to the Analyss of Complex Systems and Decson Processes, [37] n whch the concepts of a lngustc varable and granulaton were ntroduced. The concepts of a fuzzy constrant and fuzzy constrant propagaton were ntroduced n Calculus of Fuzzy Restrctons, [39], and developed more fully n A Theory of Approxmate Reasonng, [45] and Outlne of a Computatonal Approach to Meanng and Knowledge Representaton Based on a Concept of a Generalzed Assgnment Statement, [49]. Applcaton of fuzzy logc to meanng representaton and ts role n test-score semantcs are dscussed n PRUF A Meanng Representaton Language for Natural Languages, [43], and Test-Score Semantcs for Natural Languages and Meanng-Representaton va PRUF, [46]. The close relatonshp between CW and fuzzy nformaton granulaton s dscussed n Toward a Theory of Fuzzy Informaton Granulaton and ts Centralty n Human Reasonng and Fuzzy Logc [53]. Although the foundatons of computng wth words were lad some tme ago, ts evoluton nto a dstnct methodology n ts own rght reflects many advances n our understandng of fuzzy logc and soft computng advances whch took place wthn the past few years. (See references and related papers.) A key aspect of CW s that t nvolves a fuson of natural languages and computaton wth fuzzy varables. It s ths fuson that s lkely to result n an evoluton of CW nto a basc methodology n ts own rght, wth wde-rangng ramfcatons and applcatons. We begn our exposton of CW wth a few defntons. It should be understood that the defntons are dspostonal, that s, admt of exceptons. As was stated earler, a concept whch plays a pvotal role n CW s that of a granule. Typcally, a granule s a fuzzy set of ponts drawn together by smlarty (Fg. 1). A word may be atomc, as n young, or composte, as n not very young Fg. 6. Words as labels of fuzzy sets. (Fg. 6). Unless stated to the contrary, a word wll be assumed to be composte. The denotaton of a word may be a hgher order predcate, as n Montague grammar [12], [23]. In CW, a granule, g; whch s the denotaton of a word, w; s vewed as a fuzzy constrant on a varable. A pvotal role n CW s played by fuzzy constrant propagaton from premses to conclusons. It should be noted that, as a basc technque, constrant propagaton plays mportant roles n many methodologes, especally n mathematcal programmng, constrant programmng and logc programmng. (See references and related papers.) As a smple llustraton, consder the proposton Mary s young, whch may be a lngustc characterzaton of a percepton. In ths case, young s the label of a granule young. (Note that for smplcty the same symbol s used both for a word and ts denotaton.) The fuzzy set young plays the role of a fuzzy constrant on the age of Mary (Fg. 6). As a further example consder the propostons and p 1 = Carol lves near Mary p 2 = Mary lves near Pat: In ths case, the words lves near n p 1 and p 2 play the role of fuzzy constrants on the dstances between the resdences of Carol and Mary, and Mary and Pat, respectvely. If the query s: How far s Carol from Pat?, an answer yelded by fuzzy constrant propagaton mght be expressed as p 3 ; where p 3 = Carol lves not far from Pat: More about fuzzy constrant propagaton wll be sad at a later pont. A basc assumpton n CW s that nformaton s conveyed by constranng the values of varables. Furthermore, nformaton s assumed to consst of a collecton of propostons expressed n natural or synthetc language. Typcally, such propostons play the role of lngustc characterzaton of perceptons. A basc generc problem n CW s the followng. We are gven a collecton of propostons expressed n a natural language whch consttute the ntal data set, or IDS for short. From the ntal data set we wsh to nfer an answer to a query expressed n a natural language. The answer, also expressed n a natural language, s referred to as the termnal

6 110 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 45, NO. 1, JANUARY 1999 Fg. 7. Computng wth words as a transformaton of an ntal data set (IDS) nto a termnal data set (TDS). Fg. 10. Fuzzy graph of a functon defned by a fuzzy rule-set. Fg. 8. Fg. 9. Fuzzy graph of a functon. A fuzzy graph of a functon represented by a rule-set. data set, or TDS for short. The problem s to derve TDS from IDS (Fg. 7). A few problems wll serve to llustrate these concepts. At ths juncture, the problems wll be formulated by not solved. 1) Assume that a functon, f;f: U! V; 2 U; Y 2 V; s descrbed n words by the fuzzy f then rules f: f s small then Y s small f s medum then Y s large f s large then Y s small. What ths mples s that f s approxmated to by a fuzzy graph f 3 (Fg. 8), where f 3 = small 2 small + medum 2 large + large 2 small: In f 3 ; + and 2 denote, respectvely, the dsjuncton and Cartesan product. An expresson of the form A2 B; where A and B are words, wll be referred to as a Cartesan granule. In ths sense, a fuzzy graph may be vewed as a dsjuncton of cartesan granules. In essence, a fuzzy graph serves as an approxmaton to a functon or a relaton [38], [51]. Equvalently, t may be vewed as a lngustc characterzaton of a percepton of f (Fg. 9). In the example under consderaton, the IDS conssts of the fuzzy rule-set f: The query s: What s the maxmum value of f (Fg. 10)? More broadly, the problem s: How can one compute an attrbute of a functon, f, e.g., ts maxmum value or ts area or ts roots f f s descrbed n words as a collecton of fuzzy f-then rules? Determnaton of the maxmum value wll be dscussed n greater detal at a later pont. 2) A box contans ten balls of varous szes of whch several are large and a few are small. What s the probablty that a ball drawn at random s nether large nor small? In ths case, the IDS s a verbal descrpton of the contents of the box; the TDS s the desred probablty. 3) A less smple example of computng wth words s the followng. Let and Y be ndependent random varables takng values n a fnte set V = fv 1 ; 111;vng wth probabltes p 1 ; 111;pn and q 1 ; 111;qn; respectvely. For smplcty of notaton, the same symbols wll be used to denote and Y and ther generc values, wth p and q denotng the probabltes of and Y; respectvely. Assume that the probablty dstrbutons of and Y are descrbed n words through the fuzzy f-then rules (Fg. 11): and P : f s small then p s small f s medum then p s large f s large then p s small. Q: f Y s small then q s large f Y s medum then q s small f Y s large then q s large. where granules small, medum and large are values of lngustc varables and Y n ther respectve unverses of dscourse. In the example under consderaton, these rule-sets consttute the IDS. Note that small n P need not have the same meanng as small n Q; and lkewse for medum and large. The query s: How can we descrbe n words the jont probablty dstrbuton of and Y? Ths probablty dstrbuton s the TDS.

7 ZADEH: FROM COMPUTING WITH NUMBERS TO COMPUTING WITH WORDS 111 Fg. 12. Canoncal form of a proposton. A fuzzy graph representaton of a granulated probablty dstrbu- Fg. 11. ton. For convenence, the probablty dstrbutons of and Y may be represented as fuzzy graphs: P : small 2 small + medum 2 large + large 2 small Q: small 2 large + medum 2 small + large 2 large wth the understandng that the underlyng numercal probabltes must add up to unty. Snce and Y are ndependent random varables, ther jont probablty dstrbuton (P;Q) s the product of P and Q: In words, the product may be expressed as [51] (P; Q): small 2 small 2 (small 3 large) + small 2 medum 2 (small 3 small) + small 2 large 2 (small 3 large) large 2 large 2 (small 3 large) where 3 s the arthmetc product n fuzzy arthmetc [14]. In ths example, what we have done, n effect, amounts to a dervaton of a lngustc characterzaton of the jont probablty dstrbuton of and Y startng wth lngustc characterzatons of the probablty dstrbuton of and the probablty dstrbuton of Y: A few comments are n order. In lngustc characterzatons of varables and ther dependences, words serve as values of varables and play the role of fuzzy constrants. In ths perspectve, the use of words may be vewed as a form of granulaton, whch n turn may be regarded as a form of fuzzy quantzaton. Granulaton plays a key role n human cognton. For humans, t serves as a way of achevng data compresson. Ths s one of the pvotal advantages accrung through the use of words n human, machne and man machne communcaton. The pont of departure n CW s the premse that the meanng of a proposton, p, n a natural language may be represented as an mplct constrant on an mplct varable. Such a representaton s referred to as a canoncal form of p; denoted as CF(p) (Fg. 12). Thus, a canoncal form serves to make explct the mplct constrant whch resdes n p: The concept of a canoncal form s descrbed n greater detal n the followng secton. Fg. 13. Conceptual structure of computng wth words. As a frst step n the dervaton of TDS from IDS, propostons n IDS are translated nto ther canoncal forms, whch collectvely represent antecedent constrants. Through the use of rules for constrant propagaton, antecedent constrants are transformed nto consequent constrants. Fnally, consequent constrants are translated nto a natural language through the use of lngustc approxmaton [10], [18], yeldng the termnal data set TDS. Ths process s schematzed n Fg. 13. In essence, the ratonale for computng wth words rests on two major mperatves: 1) computng wth words s a necessty when the avalable nformaton s too mprecse to justfy the use of numbers and 2) when there s a tolerance for mprecson whch can be exploted to acheve tractablty, robustness, low soluton cost and better rapport wth realty. In computng wth words, there are two core ssues that arse. Frst s the ssue of representaton of fuzzy constrants. More specfcally, the queston s: How can the fuzzy constrants whch are mplct n propostons expressed n a natural language be made explct. And second s the ssue of fuzzy constrant propagaton, that s, the queston of how can fuzzy constrants n premses,.e., antecedent constrants, be propagated to conclusons,.e., consequent constrants. These are the ssues whch are addressed n the followng. III. REPRESENTATION OF FUZZY CONSTRAINTS, CANONICAL FORMS, AND GENERALIZED CONSTRAINTS Our approach to the representaton of fuzzy constrants s based on test-score semantcs [46], [47]. In outlne, n ths semantcs, a proposton, p; n a natural language s vewed as a network of fuzzy (elastc) constrants. Upon aggregaton, the constrants whch are emboded n p result n an overall fuzzy constrant whch can be represented as an expresson of the form s R where R s a constranng fuzzy relaton and s the constraned varable. The expresson n queston s the canoncal form of p: Bascally, the functon of a canoncal form s to place n evdence the fuzzy constrant whch s mplct n p: Ths s represented schematcally as P! s R

8 112 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 45, NO. 1, JANUARY 1999 n whch the arrow! denotes explctaton. The varable may be vector-valued and/or condtoned. In ths perspectve, the meanng of p s defned by two procedures. The frst procedure acts on a so-called explanatory database, ED, and returns the constraned varable, : The second procedure acts on ED and returns the constranng relaton, R: An explanatory database s a collecton of relatons n terms of whch the meanng of p s defned. The relatons are empty, that s, they consst of relaton names, relatons attrbutes and attrbute domans, wth no entres n the relatons. When there are entres n ED, ED s sad to be nstantated and s denoted EDI. EDI may be vewed as a descrpton of a possble world n possble world semantcs [6], whle ED defnes a collecton of possble worlds, wth each possble world n the collecton correspondng to a partcular nstantaton of ED [47]. As a smple llustraton, consder the proposton p = M ary s not young: Assume that the explanatory database s chosen to be ED = POPULATION[Name; Age] + YOUNG[Age;] n whch POPULATION s a relaton wth arguments Name and Age; YOUNG s a relaton wth arguments Age and ; and + s the dsjuncton. In ths case, the constraned varable s the age of Mary, whch n terms of ED may be expressed as = Age(Mary) = Age POPULATION[Name=Mary]: Ths expresson specfes the procedure whch acts on ED and returns : More specfcally, n ths procedure, Name s nstantated to Mary and the resultng relaton s projected on Age, yeldng the age of Mary. The constranng relaton, R; s gven by R =( 2 YOUNG) 0 whch mples that the ntensfer very s nterpreted as a squarng operaton, and the negaton not as the operaton of complementaton [36]. Equvalently, R may be expressed as R = YOUNG[Age; ]: As a further example, consder the proposton p = Carol lves n a small cty near San F rancsco and assume that the explanatory database s: and In ths case, ED = POPULATION[Name; Resdence] + SMALL[Cty; ] + NEAR[Cty1; Cty2; ] = Resdence(Carol) = Resdence POPULATION[Name = Carol] R = SMALL[Cty; ] \ Cty1 NEAR[Cty2 = San Francsco] In R; the frst consttuent s the fuzzy set of small ctes; the second consttuent s the fuzzy set of ctes whch are near San Francsco; and \ denotes the ntersecton of these sets. So far we have confned our attenton to constrants of the form s R: In fact, constrants can have a varety of forms. In partcular, a constrant expressed as a canoncal form may be condtonal, that s, of the form whch may also be wrtten as f s R then Y s S Y s S f s R: The constrants n queston wll be referred to as basc. For purposes of meanng representaton, the rchness of natural languages necesstates a wde varety of constrants n relaton to whch the basc constrants form an mportant though specal class. The so-called generalzed constrants [49] contan the basc constrants as a specal case and are defned as follows. The need for generalzed constrants becomes obvous when one attempts to represent the meanng of smple propostons such as Robert loves women John s very honest checkout tme s 11 am slmness s attractve n the language of standard logcal systems. A generalzed constrant s represented as sr R where sr, pronounced ezar, s a varable copula whch defnes the way n whch R constrans : More specfcally, the role of R n relaton to s defned by the value of the dscrete varable r: The values of r and ther nterpretatons are defned below: e equal (abbrevated to =); d dsjunctve (possblstc) (abbrevated to blank); verstc; p probablstc; probablty value; u usualty; rs random set; rfs random fuzzy set; fg fuzzy graph; ps rough set; As an llustraton, when r = e; the constrant s an equalty constrant and s abbrevated to = : When r takes the value d; the constrant s dsjunctve (possblstc) and sd abbrevated to s, leadng to the expresson s R n whch R s a fuzzy relaton whch constrans by playng the role of the possblty dstrbuton of : More specfcally,

9 ZADEH: FROM COMPUTING WITH NUMBERS TO COMPUTING WITH WORDS 113 f takes values n a unverse of dscourse, U = fug; then Possf = ug = R(u); where R s the membershp functon of R; and 5 s the possblty dstrbuton of ; that s, the fuzzy set of ts possble values [42]. In schematc form s R 0! 5 = R n! Possf = ug = R (u): Smlarly, when r takes the value ; the constrant s verstc. In the case, sv R means that f the grade of membershp of u n R s ; then = u has truth value : For example, a canoncal form of the proposton p = John s procent n Englsh; French; and German may be expressed as Procency(John) sv (1jEnglsh + 0:7jFrench + 0:6jGerman) n whch 1.0, 0.7, and 0.6 represent, respectvely, the truth values of the propostons John s profcent n Englsh, John s profcent n French and John s profcent n German. Ina smlar ven, the verstc constrant Ethncty(John) sv (0:5jGerman + 0:25jFrench + 0:25jItalan) represents the meanng of the proposton John s half German, quarter French and quarter Italan. When r = p; the constrant s probablstc. In ths case, sp R means that R s the probablty dstrbuton of : For example sp N (m; 2 ) means that s normally dstrbuted wth mean m and varance 2 : Smlarly, sp (0:2na +0:5nb +0:3nc) means that s a random varable whch takes the values a; b; and c wth respectve probabltes 0.2, 0.5, and 0.3. The constrant s an abbrevaton for whch n turn means that su R usually( s R) Probf s Rg s usually: In ths expresson s R s a fuzzy event and usually s ts fuzzy probablty, that s, the possblty dstrbuton of ts crsp probablty. Fg. 14. Representaton of meanng n test-score semantcs. The constrant srs P s a random set constrant. Ths constrant s a combnaton of probablstc and possblstc constrants. More specfcally, n a schematc form, t s expressed as sp P (; Y )sq Y srs R where Q s a jont possbltstc constrant on and Y; and R s a random set. It s of nterest to note that the Dempster Shafer theory of evdence [29] s, n essence, a theory of random set constrants. In computng wth words, the startng pont s a collecton of propostons whch play the role of premses. In many cases, the canoncal forms of these propostons are constrants of the basc, possblstc type. In a more general settng, the constrants are of the generalzed type, mplyng that explctaton of a proposton, p; may be represented as p! sr R where sr R s the canoncal form of p (Fg. 14). As n the case of basc constrants, the canoncal form of a proposton may be derved through the use of test-score semantcs. In ths context, the depth of p s, roughly, a measure of the effort that s needed to explctate p; that s, to translate p nto ts canoncal form. In ths sense, the proposton sr R s a surface constrant (depth = zero), wth the depth of explctaton ncreasng n the downward drecton (Fg. 15). Thus a proposton such as Mary s young s shallow, whereas t s unlkely that there wll be a substantal ncrease n the prce of ol n the near future, s not. Once the propostons n the ntal data set are expressed n ther canoncal forms, the groundwork s lad for fuzzy

10 114 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 45, NO. 1, JANUARY 1999 Conjunctve Rule for sv: sv A sv B Fg. 15. Depth of explctaton. constrant propagaton. Ths s a basc part of CW whch s dscussed n the followng secton. IV. FUZZY CONSTRAINT PROPAGATION AND THE RULES OF INFERENCE IN FUZZY LOGIC The rules governng fuzzy constrant propagaton are, n effect, the rules of nference n fuzzy logc. In addton to these rules, t s helpful to have rules governng fuzzy constrant modfcaton, The latter rules wll be dscussed at a later pont n ths secton. In a summarzed form, the rules governng fuzzy constrant propagaton are the followng [51]. (A and B are fuzzy relatons. Dsjuncton and conjuncton are defned, respectvely, as max and mn, wth the understandng that, more generally, they could be defned va t-norms and s-norms [15], [24]. The antecedent and consequent constrants are separated by a horzontal lne.) Conjunctve Rule 1: s A s B s A \ B Conjunctve Rule 2: ( 2 U; Y 2 B; A U; B V ) Dsjunctve Rule 1: s A Y s B (; Y )sa 2 B or s A s B s A [ B Dsjunctve Rule 2: (A U; B V ) A s A Y s B (; Y )sa 2 V [ U 2 B where A 2 V and U 2 B are cylndrcal extensons of A and B; respectvely. Projectve Rule: where proj V A = sup u A Surjectve Rule: Derved Rules: Compostonal Rule: sv A [ B (; Y )sa Y s proj V s A A (; Y )sa 2 V s A (; Y )sb Y s A B where A B denotes the composton of A and B: Extenson Prncple (mappng rule) [34], [40]: s A f() sf(a) where f: U! V; and f(a) s defned by f(a)() = Inverse Mappng Rule: sup uj=f(u) A(u) f() sa s f 01 (A) where f 01(A)(u) = A (f(u)) Generalzed modus ponens: s A f s B then Y s C Y s A ((:B) 8 C) where the bounded sum :B 8 C represents Lukasewcz s defnton of mplcaton. Generalzed Extenson Prncple: where f() sa q() sq(f 01 (A)) q () = sup uj=f(u) A(q(u)): The generalzed extenson prncple plays a pvotal role n fuzzy constrant propagaton. However, what s used most

11 ZADEH: FROM COMPUTING WITH NUMBERS TO COMPUTING WITH WORDS 115 frequently n practcal applcatons of fuzzy logc s the basc nterpolatve rule, whch s a specal case of the compostonal rule of nference appled to a functon whch s defned by a fuzzy graph [38], [51]. More specfcally, f f s defned by a fuzzy rule set f : f s A then s B ; =1; 111;n or equvalently, by a fuzzy graph f s A 2 B and ts argument, ; s defned by the antecedent constrant s A then the consequent constrant on Y may be expressed as Y s m ^ B where m s a matchng coeffcent, m = sup (A \ A) whch serves as a measure of the degree to whch A matches A : Syllogstc Rule: [48] Q 1 A s are B s Q 2 (A and B) s are C s (Q 1 Q 2 ) A s are (B and C) s where Q 1 and Q 2 are fuzzy quantfers; A; B; and C are fuzzy relatons; and Q 1 Q 2 s the product of Q 1 and Q 2 n fuzzy arthmetc. Constrant Modfcaton Rules: [36], [43] s ma! s f(a) where m s a modfer such as not, very, more, or less, and f(a) defnes the way n whch m modfes A: Specfcally, f m = not then f(a) =A 0 (complement) f m = very then f(a) = 2 A (left square) where 2 A(u) = ( A (u)) 2 : Ths rule s a conventon and should not be construed as a realstc approxmaton to the way n whch the modfer very functons n a natural language. Probablty Qualfcaton Rule [45]: ( s A) s3! P s 3 where s a random varable takng values n U wth probablty densty p(u); 3 s a lngustc probablty expressed n words lke lkely, not very lkely, etc.; and P s the probablty of the fuzzy event s A; expressed as P = Z U A (u)p(u) du: The prmary purpose of ths summary s to underscore the concdence of the prncpal rules governng fuzzy constrant propagaton wth the prncpal values of nference n fuzzy logc. Of necessty, the summary s not complete and there are many specalzed rules whch are not ncluded. Furthermore, most of the rules n the summary apply to constrants whch are of the basc, possblstc type. Further development of the rules governng fuzzy constrant propagaton wll requre an extenson of the rules of nference to generalzed constrants. As was alluded to n the summary, the prncpal rule governng constrant propagaton s the generalzed extenson prncple whch n a schematc form may be represented as f( 1 ; 111; n )sa : q( 1 ; 111; n )sq(f 01 (A)) In ths expresson, 1 ; 111; n are database varables; the term above the lne represents the constrant nduced by the IDS; and the term below the lne s the TDS expressed as a constrant on the query q( 1 ; 111; n ): In the latter constrant, f 01 (A) denotes the premage of the fuzzy relaton A under the mappng f: U! V; where A s a fuzzy subset of V and U s the doman of f( 1 ; 111; n ): Expressed n terms of the membershp functons of A and q(f 01 (A)); the generalzed extenson prncple reduces the dervaton of the TDS to the soluton of the constraned maxmzaton problem q(1;111;n)() = n whch u 1 ; 111;u n are constraned by sup ( A (f(u 1 ; 111;u n ))) (u 1;111;un) = q(u 1 ; 111;u n ): The generalzed extenson prncple s smpler than t appears. An llustraton of ts use s provded by the followng example. The IDS s: most Swedes are tall The query s: What s the average heght of Swedes? The explanatory database conssts of a populaton of N Swedes, Name 1 ; 111; Name N : The database varables are h 1 ; 111;h N ; where h s the heght of Name ; and the grade of membershp of Name n tall s tall (h ); =1; 111;n: The proporton of Swedes who are tall s gven by the sgma-count [43] Count(tall 1 Swedes=Swedes) = 1 tall (h ) N from whch t follows that the constrant on the database varables nduced by the IDS s 1 N tall (h )smost: In terms of the database varables h 1 ; 111;h N ; the average heght of Swedes s gven by h ave = 1 N h :

12 116 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 45, NO. 1, JANUARY 1999 Fg Cuts of a functon descrbed by a fuzzy graph. Snce the IDS s a fuzzy proposton, h ave s a fuzzy set whose determnaton reduces to the constraned maxmzaton problem have () = subject to the constrant 1 sup most h 1;111;h N N = 1 h : N tall (h )!! It s possble that approxmate solutons to problems of ths type mght be obtanable through the use of neurocomputng or evolutonary-computng-based methods. As a further example, we wll return to a problem stated n an earler secton, namely, maxmzaton of a functon, f; whch s descrbed n words by ts fuzzy graph, f 3 (Fg. 10). More specfcally, consder the standard problem of maxmzaton of an objectve functon n decson analyss. Let us assume as s frequently the case n real-world problems that the objectve functon, f; s not well-defned and that what we know about f can be expressed as a fuzzy rule set f : f s A 1 then Y ts B 1 f s A 2 then Y s B 2 :::::::::::::::::::::::: f s A n then Y s B n or, equvalently, as a fuzzy graph f s A 2 B : The queston s: What s the pont or, more generally, the maxmzng set [54] at whch f s maxmzed, and what s the maxmum value of f? The problem can be solved by employng the technque of -cuts [34], [40]. Wth reference to Fg. 16, f A and B are -cuts of A and B ; respectvely, then the correspondng -cut of f 3 s gven by f 3 = A 2 B : From ths expresson, the maxmzng fuzzy set, the maxmum fuzzy set and maxmum value fuzzy set can readly be derved, as shown n Fgs. 16 and 17. A key pont whch s brought out by these examples and the precedng dscusson s that explctaton and constrant propagaton play pvotal roles n CW. Ths role can be concretzed Fg. 17. set. Fg. 18. Computaton of maxmzng set, maxmum set, and maxmum value Conceptual structure of computng wth words. by vewng explctaton and constrant propagaton as translaton of propostons expressed n a natural language nto what mght be called the generalzed constrant language (GCL) and applyng rules of constrant propagaton to expressons n ths language expressons whch are typcally canoncal forms of propostons expressed n a natural language. Ths process s schematzed n Fg. 18. The conceptual framework of GCL s substantvely dfferently from that of conventonal logcal systems, e.g., predcate logc. But what matters most s that the expressve power of GCL whch s based on fuzzy logc s much greater than that of standard logcal calcul. As an llustraton of ths pont, consder the followng problem. A box contans ten balls of varous szes of whch several are large and a few are small. What s the probablty that a ball drawn at random s nether large nor small? To be able to answer ths queston t s necessary to be able to defne the meanngs of large, small, several large balls, few small balls, and nether large nor small. Ths s a problem n semantcs whch falls outsde of probablty theory, neurocomputng and other methodologes. An mportant applcaton area for computng wth words and manpulaton of perceptons s decson analyss snce n most realstc settngs the underlyng probabltes and utltes are not known wth suffcent precson to justfy the use of numercal valuatons. There exsts an extensve lterature on the use of fuzzy probabltes and fuzzy utltes n decson analyss. In what follows, we shall restrct our dscusson

13 ZADEH: FROM COMPUTING WITH NUMBERS TO COMPUTING WITH WORDS 117 Fg. 19. A box wth black and whte balls. Fg. 21. Computaton of expectaton through use of the extenson prncple. Fg. 20. Membershp functon of most. to two very smple examples whch llustrate the use of perceptons. Frst, consder a box whch contans black balls and whte balls (Fg. 19). If we could count the number of black balls and whte balls, the probablty of pckng a black ball at random would be equal to the proporton, r; of black balls n the box. Now suppose that we cannot count the number of black balls n the box but our percepton s that most of the balls are black. What, then, s the probablty, p; that a ball drawn at random s black? Assume that most s characterzed by ts possblty dstrbuton (Fg. 20). In ths case, p s a fuzzy number whose possblty dstrbuton s most, that s, p s most: Fg. 22. Fg. 23. Expectaton of gan. A box wth balls of varous szes and a defnton of large ball. Next, assume that there s a reward of a dollars f the ball drawn at random s black and a penalty of b dollars f the ball s whte. In ths case, f p were known as a number, the expected value of the gan would be e = ap 0 b(1 0 p): Snce we know not p but ts possblty dstrbuton, the problem s to compute the value of e when p s most. For ths purpose, we can employ the extenson prncple [34], [40], whch mples that the possblty dstrbuton, E; of e s a fuzzy number whch may be expressed as E = a most 0 b(1 0 most): For smplcty, assume that most has a trapezodal possblty dstrbuton (Fg. 20). In ths case, the trapezodal possblty dstrbuton of E can be computed as shown n Fg. 21. It s of nterest to observe that f the support of E s an nterval [; ] whch straddles O (Fg. 22), then there s no noncontroversal decson prncple whch can be employed to answer the queston: Would t be advantageous to play a game n whch a ball s pcked at random from a box n whch most balls are black, and a and b are such that the support of E contans O. Next, consder a box n whch the balls b 1 ; 111;b n have the same color but vary n sze, wth b ; =1; 111;n havng the grade of membershp n the fuzzy set of large balls (Fg. 23). The queston s: What s the probablty that a ball drawn at random s large, gven the percepton that most balls are large? The dfference between ths example and the precedng one s that the event the ball drawn at random s large s a fuzzy event, n contrast to the crsp event the ball drawn at random s black. The probablty of drawng b s 1=n: Snce the grade of membershp of b n the fuzzy set of large balls s ; the probablty of the fuzzy event the ball drawn at random s large s gven by [35] P = 1 n : On the other hand, the proporton of large balls n the box s gven by the relatve sgma-count [40], [43] Count(large 1 balls=balls 1 n 1 box )= 1 n : Consequently, the canoncal form of the percepton most balls are large may be expressed as 1 n s most

14 118 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 45, NO. 1, JANUARY 1999 whch leads to the concluson that P s most: It s of nterest to observe that the possblty dstrbuton of P s the same as n the precedng example. If the queston were: What s the probablty that a ball drawn at random s small, the answer would be P s 1 n where ; = 1; 111;n; s the grade of membershp of b n the fuzzy set of small balls, gven that 1 n s most: What s nvolved n ths case s constrant propagaton from the antecedent constrant on the to a consequent constrant on the : Ths problem reduces to the soluton of a nonlnear program. What ths example ponts to s that n usng fuzzy constrant propagaton rules, applcaton of the extenson prncple reduces, n general, to the soluton of a nonlnear program. What we need and do not have at present are approxmate methods of solvng such programs whch are capable of explotng the tolerance for mprecson. Wthout such methods, the cost of solutons may be excessve n relaton to the mprecson whch s ntrnsc n the use of words. In ths connecton, an ntrgung possblty s to use neurocomputng and evolutonary computng technques to arrve at approxmate solutons to constraned maxmzaton problems. The use of such technques may provde a closer approxmaton to the ways n whch human manpulate perceptons. V. CONCLUDING REMARKS In our quest for machnes whch have a hgh degree of machne ntellgence (hgh MIQ), we are developng a better understandng of the fundamental mportance of the remarkable human capacty to perform a wde varety of physcal and mental tasks wthout any measurements and any computatons. Underlyng ths remarkable capablty s the bran s crucal ablty to manpulate perceptons perceptons of dstance, sze, weght, force, color, numbers, lkelhood, truth and other characterstcs of physcal and mental objects. A basc dfference between percepton and measurements s that, n general, measurements are crsp whereas perceptons are fuzzy. In a fundamental way, ths s the reason why to deal wth perceptons t s necessary to employ a logcal system that s fuzzy rather than crsp. Humans employ words to descrbe perceptons. It s ths obvous observaton that s the pont of departure for the theory outlned n the precedng sectons. When perceptons are descrbed n words, manpulaton of perceptons s reduced to computng wth words (CW). In CW, the objects of computaton are words or, more generally, propostons drawn from a natural language. A basc premse n CW s that the meanng of a proposton, p, may be expressed as a generalzed constrant n whch the constraned varable and the constranng relaton are, n general, mplct n p: In comng years, computng wth words and perceptons s lkely to emerge as an mportant drecton n scence and technology. In a reversal of long-standng atttudes, manpulaton of perceptons and words whch descrbe them s destned to gan n respectablty. Ths s certan to happen because t s becomng ncreasngly clear that n dealng wth realworld problems there s much to be ganed by explotng the tolerance for mprecson, uncertanty and partal truth. Ths s the prmary motvaton for the methodology of computng wth words (CW) and the computatonal theory of perceptons (CTP) whch are outlned n ths paper. ACKNOWLEDGMENT The author acknowledges Prof. Mcho Sugeno, who has contrbuted so much and n so many ways to the development of fuzzy logc and ts applcatons. REFERENCES [1] H. R. Berenj, Fuzzy renforcement learnng and dynamc programmng, n n Fuzzy Logc n Artfcal Intellgence, Proc. IJCAI 93 Workshop, A. L. Ralescu, Ed. Berln, Germany: Sprnger-Verlag, 1994, pp [2] M. Black, Reasonng wth loose concepts, Dalog 2, pp. 1 12, [3] P. Bosch, Vagueness, ambguty and all the rest, n Sprachstruktur, Indvduum und Gesselschaft, M. Van de Velde and W. Vandeweghe, Eds. Tubngen, Germany: Nemeyer, [4] J. Bowen, R. La, and D. Bahler, Fuzzy semantcs and fuzzy constrant networks, n Proc. 1st IEEE Conf. on Fuzzy Systems, San Francsco, 1992, pp [5], Lexcal mprecson n fuzzy constrant networks, n Proc. Nat. Conf. on Artfcal Intellgence, 1992, pp [6] M. J. Cresswell, Logc and Languages. London, U.K.: Methuen, [7] D. Dubos, H. Farger, and H. Prade, Propagaton and satsfacton of flexble constrants, n Fuzzy Sets, Neural Networks, and Soft Computng, R. R. Yager, L. A. Zadeh, Eds. New York: Von Nostrand Renhold, 1994, pp [8], Possblty theory n constrant satsfacton problems: Handlng prorty, preference, and uncertanty, J. Appl. Intell., to be publshed. [9], The calculus of fuzzy restrctons as a bass for flexble constrant satsfacton, n Proc. 2nd IEEE Int. Conf. on Fuzzy Systems, San Francsco, CA, 1993, pp [10] E. C. Freuder and P. Snow, Improved relaxaton and search methods for approxmate constrant satsfacton wth a maxmn crteron, n Proc. 8th Bennal Conf. on the Canadan Socety for Computatonal Studes of Intellgence, Ont., Canada, 1990, pp [11] J. A. Goguen, The logc of nexact concepts, Synthese, vol. 19, pp , [12] J. R. Hobbs, Makng computatonal sense of Montague s ntensonal logc, Artf. Intell., vol. 9, pp , [13] O. Kata, S. Matsubara, H. Masuch, M. Ida, et al., Synergetc computaton for constrant satsfacton problems nvolvng contnuous and fuzzy varables by usng Occam, n Transputer/Occam, Proc. 4th Transputer/Occam Int. Conf., S. Noguch and H. Umeo, Eds. Amsterdam, The Netherlands: IOS Press, 1992, pp [14] A. Kaufmann and M. M. Gupta, Introducton to Fuzzy Arthmetc: Theory and Applcatons. New York: Von Nostrand, [15] G. Klr and B. Yuan, Fuzzy Sets and Fuzzy Logc. Englewood Clffs, NJ: Prentce Hall, [16] K. Lano, A constrant-based fuzzy nference system, n EPIA 91, 5th Portuguese Conf. on Artfcal Intellgence, P. Barahona, L. M. Perera, and A. Porto, Eds. Berln, Germany: Sprnger-Verlag, 1991, pp [17] W. A. Lodwck, Analyss of structure n fuzzy lnear programs, Fuzzy Sets and Systems, vol. 38, no. 1, pp , [18] E. H. Mamdan and B. R. Ganes, Eds., Fuzzy Reasonng and Its Applcatons, London, U.K [19] M. Mares, Computaton Over Fuzzy Quanttes. Boca Raton, FL: CRC, [20] V. Novak, Fuzzy logc, fuzzy sets, and natural languages, Int. J. Gen. Syst., vol. 20, no. 1, pp , [21] V. Novak, M. Ramk, M. Cerny, and J. Nekola, Eds., Fuzzy Approach to Reasonng and Decson-Makng. Boston, MA: Kluwer, 1992.

15 ZADEH: FROM COMPUTING WITH NUMBERS TO COMPUTING WITH WORDS 119 [22] M. S. Oshan, O. M. Saad, and A. G. Hassan, On the soluton of fuzzy multobjectve nteger lnear programmng problems wth a parametrc study, Adv. Modelng Anal, A, vol. 24, no. 2, pp , [23] B. Partee, Montague Grammar. New York: Academc, [24] W. Pedrycz and F. Gomde, Introducton to Fuzzy Sets. Cambrdge, MA: MIT Press, [25] G. Q and G. Fredrch, Extendng constrant satsfacton problem solvng n structural desgn, n Industral and Engneerng Applcatons of Artfcal Intellgence and Expert Systems, 5th Int. Conf., IEA/AIE-92, F. Bell and F. J. Radermacher, Eds. Berln, Germany: Sprnger- Verlag, 1992, pp [26] H. Rasowa and M. Marek, On reachng consensus by groups of ntellgent agents, n Methodologes for Intellgent Systems, Z. W. Ras, Ed. Amsterdam, The Netherlands: North-Holland, 1989, pp [27] A. Rosenfeld, R. A. Hummel, and S. W. Zucker, Scene labelng by relaxaton operatons, IEEE Trans. Syst., Man, Cybern., vol. 6, pp , [28] M. Sakawa, K. Sawada, and M. Inuguch, A fuzzy satsfcng method for large-scale lnear programmng problems wth block angular structure, European J. Oper. Res., vol. 81, no. 2, pp , [29] G. Shafer, A Mathematcal Theory of Evdence. Prnceton, NJ: Prnceton Unv. Press, [30] S. C. Tong, Interval number and fuzzy number lnear programmng, Adv. n Modelng Anal. A, vol. 20, no. 2, pp , [31] R. Vallee, Cognton et Systeme. Pars: l Interdscplnare Systeme(s), [32] R. R. Yager, Some extensons of constrant propagaton of label sets, Int. J. Approxmate Reasonng, vol. 3, pp , [33] L. A. Zadeh, From crcut theory to system theory, Proc. IRE, vol. 50, pp , [34], Fuzzy sets, Inform. Contr., vol. 8, pp , [35], Probablty measures of fuzzy events, J. Math. Anal. Appl., vol. 23, pp , [36], A fuzzy-set-theoretc nterpretaton of lngustc hedges, J. Cybern,, vol. 2, pp. 4 34, [37], Outlne of a new approach to the analyss of complex system and decson processes, IEEE Trans. Syst., Man, Cybern., vol. SMC-3, pp , [38], On the analyss of large scale systems, Systems Approaches and Envronment Problems, H. Gottnger, Ed. Gottngen, Germany: Vandenhoeck and Ruprecht, 1974, pp [39], Calculus of fuzzy restrctons, n Fuzzy Sets and Ther Applcatons to Cogntve and Decson Processes, L. A. Zadeh, K. S. Fu, and M. Shmura, Eds. New York: Academc, 1975, pp [40], The concept of a lngustc varable and ts applcaton to approxmate reasonng, Part I: Inf. Sc., vol. 8, pp ; Part II: Inf. Sc., vol. 8, pp ; Part III: Inform. Sc., vol. 9, pp , [41], A fuzzy-algorthmc approach to the defnton of complex or mprecse concepts, Int. J. Man Machne Studes, vol. 8, pp , [42], Fuzzy sets as a bass for a theory of possblty, Fuzzy Sets Syst., vol. 1, pp. 3 28, [43], PRUF A meanng representaton language for natural languages, Int. J. Man Machnes Studes, vol. 10, pp , [44], Fuzzy sets and nformaton granularty, n Advances n Fuzzy Set Theory and Applcatons, M. Gupta, R. Ragade, and R. Yager, Eds. Amsterdam, The Netherlands: North-Holland, 1979, pp [45], A theory of approxmate reasonng, Machne Intellgence, vol. 9, J. Hayes, D. Mche, and L. I. Mkulch, Eds. New York: Halstead, 1979, pp [46], Test-score semantcs for natural languages and meanng representaton va PRUF, Emprcal Semantcs, B. Reger, Ed. Germany: Brockmeyer, pp ; also, Tech. Rep. Memo. 246, AI Center, SRI Internatonal, Menlo Park, CA, [47], Test-score semantcs for natural languages, n Proc. Nnth Int. Conf. on Computatonal Lngustcs, Prague, Czech Republc, 1982, pp [48], Syllogstc reasonng n fuzzy logc and ts applcaton to reasonng wth dspostons, n Proc Int. Symp. on Multple- Valued Logc, Wnnpeg, Canada, 1984, pp [49], Outlne of a computatonal approach to meanng and knowledge representaton based on the concept of a generalzed assgnment statement, n Proc. Int. Semnar on Artfcal Intellgence and Man Machne Systems, M. Thoma and A. Wyner, Eds. Hedelberg, Germany: Sprnger-Verlag, 1986, pp [50], Fuzzy logc, neural networks, and soft computng, Commun. ACM, vol. 37, no. 3, pp , [51], Fuzzy logc and the calcul of fuzzy rules and fuzzy graphs: A precs, Multple Valued Logc 1, Gordon and Breach Scence, 1996, pp [52], Fuzzy logc = computng wth words, IEEE Trans. Fuzzy Syst., vol. 4, pp , [53], Toward a theory of fuzzy nformaton granulaton and ts centralty n human reasonng and fuzzy logc, Fuzzy Sets Syst., vol. 90, pp , [54], Maxmzng sets and fuzzy Markoff algorthms, IEEE Trans. Syst., Man, Cybern. C, vol. 28, pp. 9 15, Lotf A. Zadeh (S 45 A 47 M 47 SM 56 F 58 LF 87) s a professor n the Graduate School, Computer Scence Dvson, Department of EECS, Unversty of Calforna, Berkeley. In addton, he s servng as the Drector of BISC (Berkeley Intatve n Soft Computng). He s an alumnus of the Unversty of Teheran, MIT, and Columba Unversty. He held vstng appontments at the Insttute for Advanced Study, Prnceton, NJ; MIT, Cambrdge, MA; IBM Research Laboratory, San Jose, CA, SRI Internatonal, Menlo Park, CA; and the Center for the Study of Language and Informaton, Stanford Unversty, CA. Hs earler work was concerned manly wth systems analyss, decson analyss, and nformaton systems. Hs current research s focused on fuzzy logc, computng wth words and soft computng, whch s a coalton of fuzzy logc, neurocomputng, evolutonary computng, probablstc computng and parts of machne learnng. The gudng prncple of soft computng s that, n general, better solutons can be obtaned by employng the consttuent methodologes of soft computng n combnaton rather than solaton. He has publshed extensvely on a wde varety of subjects relatng to the concepton, desgn and analyss of nformaton/ntellgent systems, and s servng on the edtoral boards of over 50 journals. Dr. Zadeh s a Fellow of the AAAS, ACM, and AAAI. He s a member of the Natonal Academy of Engneerng and a Foregn Member of the Russan Academy of Natural Scences. He s a recpent of the IEEE Educaton Medal, the IEEE Rchard W. Hammng Medal, the IEEE Medal of Honor, the ASME Rufus Oldenburger Medal, the B. Bolzano Medal of the Czech Academy of Scences, the Kampe de Feret Medal, the AACC Rchard E. Bellmann Contral Hertage Award, the Grgore Mosl Prze, the Honda Prze, the Okawa Prze, and other awards and honorary doctorates.

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