soft comp fuzzy logic

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0 Soft computing is a collectionof methodologies that aim toexploit the tolerance for impreci-sion and uncertainty to achievewactability, robustness, and lowsolution cost. It s principal con-stituents are f izzy logic, neuro-computing, and probabilistic rea-soning. Softcomputing is likelyto play an increasingly impor-tant role in many applicationareas, including sof2ware engi-neering. The role modelforSOBcomputing is the human mind.

Lomputing1 ;rand Fuzzy

LogicLOTFIA. ZADEH,University of California at Berkeley0e of the deepesttraditions in science is that of accord-ing respectability to what is quantita-tive, precise, rigorous, and categorical-ly true. It is a fact, however, that welive in a world that is pervasivelyimprecise, uncertain, and hard to becategorical about. It is also a fact thatprecision and certainty carry a cost.Driven by our quest for respectability,we tend to close our eyes to these factsand thereby lose sight of the steepprice we must pay for high precisionand low uncertainty. Another visibleconcomitant of the quest forrespectability is that in much of thescientific literature elegance takesprecedence over relevance.

A case in point is the travelingsalesman problem, which is frequentlyused as a testbed for assessing theeffectiveness of various methods ofsolution. What is striking about thisproblem is the steep rise in computing

time as a function of precision of solu-tion. As the data in Table 1show, low-ering the accuracy to 3 .50 percentreduces the computing time by anorder of magnitude for a ten-foldincrease in the number of cities.A more familiar example that illus-trates the point is the problem ofparking a car. We find it relativelyeasy to park a car because the finalposition of the car is not specified pre-cisely. If it were, the difficulty of park-ing would increase geometrically withthe increase in precision, and eventu-ally parking would become impossible.

Guiding principle. These and manysimilar examples lead to the basicpremises and the guiding principle ofsoft computing.The basic premises of soft comput-ing are+ The real world is pervasivelyimprecise and uncertain.

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+ Precision and certainty carry a soft computing is aimed at accommo-cost. dating the pervasive imprecision of theThe guiding principle of soft com- real world. Although soft computingputing is has not as yet had a visible impact on+ Exploit the tolerance for impre- software engineering, it is likely to docision, uncertainty, and partial truth to so in the years ahead. Among the areasachieve tractabi lity , robustness , and in which it is likely to be applied arelow solution cost. programming languages, computerThe label soft computing is grow- security, database management, user-ing in use. What does it mean? Where 1 friendly interfaces, automated p ro-does it stand today and where is it ~ gramming, fault diagnosis, and net-headed? And what is the role of fuzzylogic in soft computing? In this article, I will focus on some

i of the basic ideas that underlie softSample application. Some of the most 1 computing and relate them to its guid-

working.

striking examples of the application of ~ ing principle.the guiding principle of soft comput-ing are the data-compression tech-niques that play a key role in high-definition television and audio record- ~ing and reproduction. ~ Basically, soft computing is not aFor example, in N H K s Muse sys- homogeneous body of concepts andtem, a motion-compensating tech- techniques. Rather, it is a partnershipnique determines the outline, direc- of distinct methods that in one way ortion, and speed of the moving body, another conform to its guiding princi-then shifts the moving image without ple.L4this juncture, the dominant aimwaiting to receive all th e pixel data. of soft computing is to exploit the tol-Th e resulting moving image does not erance for imprecision and uncertaintyhave the resolution of the still picture. to achieve tractability, robusmess, andMuse exploits the fact that the human ~ low solution cost. The principal con-eye cannot grasp the details of moving stituents of soft computing are fuzzyobjects with the same precision as still logic , neurocompu ting, and proba-objects. Even more impressive is what bilistic reasoning, with the latter sub-is achieved in the recently developed suming genetic algorithms, belief net-digital HDTV systems. For example, ~ works, chaotic systems, and parts ofthe Gen era l Ins tr ume nt syst em, learning theory. In the partnership ofinstead of transmitting data for every fuzzy logic, ne uro com put in g, andcol or d ot i n a blu e sky, se nds an probabilistic reasoning, fuzzy logic isinstruction to paint the sky. Th e com- mainly concerned with imprecisionpression ratio this system achieves is and approximate reasoning; neuro-on the order of 60 to 1. In aud io computing with learning and curve-recording and reproduction, similar fitti ng; and probabilistic reasoningideas are embodied in Sonys R/ID-l with uncertainty and belief propaga-system and PhilipsDCC. tion.In its current incarnation, the con- In large measure, fuzzy logic, neu-cept of soft computing has links to rocomputing, and probabilistic rea-many earlier influences, among them , soning are complementary, not com-my 1965 paper on fuzzy sets; 1973 ~ petitive. It is becoming increasinglypaper on the use of linguistic variables clear that in many cases it is advanta-in the analysis and control of complex geou s to comb ine t he m. A case insystems; and 1979 report (1981 paper) point is the growing number of neu-on possibili ty theo ry and soft data rofuzzy consumer products and sys-analysis.3 tems that use a combination of fuzzyUnlike traditional hard computing, logic and neural-network techniques.

SOFT COMPUTING AND FUZZY LOGIC

~~~ .

Number Accuracy Computingof cities time_ _ _ _ _ _ ~

Soztrce h$u. 07l Ti?iiu\ \1/19ib 12, 1991

In this article, I focus on fuzzylogic.

FUZZY LOGIC CONCEPTSAs one of the principal constituentsof soft computing, fuzzy logic is play-ing a key role in what might be calledhigh MIQ (machine intelligence quo-tient) systems.Two concepts within fuzzy logicplay a central role in its applications.+ Th e first is a linguistic variable;that is, a variable whose values arewords or sentences in a natural or syn-thetic language.?+ The other is afizzy $-then d e ,in which the antecedent and conse-

quents are propositions containing lin-guistic variables.!The essential function of linguisticvariables is that of granulation of vari-ables and their dependencies. In effect,the use of linguistic variables and fuzzyif-then rules results- hrough granu-lation- n lossy data compression. Inthis respect, fuzzy logic mimics theremarkable ability of the human mindto summarize data and focus on deci-sion-relevant information.With regard to fuzzy logic, there isan issue of semantics that is in need ofclarification. Specifically, it is fre-quently not recognized that the termfuzzy logic is actually used in two dif-ferent senses. In a narrow sense, fuzzylogic ( F L n ) is a logical system- nextension of multivalued logic that isintended to serve as a logic of approxi-mate reasoning. In a wider sense, fuzzyI E E E S O F T W A R E 49


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Figure 1. Interpretation of' middle-aged as a linguistic value.

logic (FLm) s more o r less synony-mous with fuzzy set theory; that is, thetheory of classes with unsharp bound-aries. In this perspective, FL = FLU ,and FLn is merely a branch of FL.\%%at is important to recognize is thattoday the term fuzzy logic is used pre-dominantly in its wider sense. It is inthis sense that any field X can be"fuzzified"- nd hence generalized- y replacing the concept of a crispset in X by a fuzzy set. In application tobasic fields such as set theory, arith-metic, topology, graph theory, proba-bility theory, and logic, fuzzificationleads to fuzzy set theory, fuzzy arith-metic, fuzzy topology, fuzzy graph the-ory, and fuzzy logic in its narrow sense.Similarly, in application to appliedfields like neurocomputing, stabilitytheory, pattern recognition and mathe-matical programming, fuzzificationleads to fuzzy neurocomputing, fuzzystability theory, fuzzy pattern recogni-tion, and fuzzy mathematical program-ming. What is gained through fuzzifi-cation is greater generality, higherexpressive power, an enhanced abilityto model real-world problems, and -most important- methodology forexploiting the tolerance for impreci-sion, a methodology that fits the guid-ing principle of soft computing andthus serves to achieve tractability,

robustness. and low solution cost.linguistic variables. A concept in fuzzylogic that plays a key role in exploitingthe tolerance for imprecision is thelinguistic variable. A linguistic vari-able, as its name suggests, is a variablewhose values are words or sentences ina natural or synthetic language. Forexample, age is a linguistic variable ifits linguistic values are young, old, mid-dle-aged,very old, not very young , and soon. A linguistic variable is interpretedas a label of a fuzzy set that is charac-terized by a membership function, asillustrated in Figure 1.Thus, if u is anumerical age, say 5 3 , then F , ~ ~ ~ ~ ~,A53) is the grade of membership of 53in middle-aged. Subjectively, you ma)interpret F ~ ~ ~ ~ ~ ~ . ~ ~ ~ ~ ( u )s the degree tcwhich u fits your perception of mid-dle-aged in a specified context.In a general setting, a linguisticvariable, V , can be viewed as 2microlanguage with context-free gram-mar and attributed-grammar seman-tics. T he context-free grammar define:the legal values of V . For example, irthe case of age, the legal values arcyoung, not young, not very young, quit(

o ld , middle-aged, and so on. Th e attrib.uted-grammar semantics provides ;mechanism for computing the membership function of any value of L

from the knowledge of the member-ship functions of the so-called przmaryterms -young and o l d , for example. Aprimary term plays the role of a gener-ator whose meaning (its membershipfunction) must be calibrated in context.For example, the meaning of not veryyoung might be computed asknot i.m OUnE (4= 1 - J l y o u n g ( 4 ) 2

where very plays the role of an intensi-fier and young is a primary term whosemembership function is specified incontext.Most current applications of fuzzylogic employ a simpler framework,illustrated in Figure 2 . Specifically, themembership functions are assumed tobe triangular or trapezoidal, and thenumber of linguistic values is usually inthe range of three to seven.Th e concept of a linguistic variableplays a central role in the applicationsof fuzzy logic because it goes to theheart of the way in which humans per-ceive, reason, and communicate.Quintessentially, the use of wordsmay be viewed as a form of data com-pression that exploits the tolerancefor imprecision to achieve tractabili-ty, robustness, and economy of com-munication. This fits almost preciselythe guiding principle of soft comput-ing.

Granulat ion. In a related sense, theuse of words may be viewed as a formof fuzzy quantization or more general-ly as granulation, as Figure 3 shows.Basically, granulation involves areplacement of a constraint of the formX = a

with a constraint of the formXis A

whereA is a fuzzy subset of U , he uni-verse ofX. or example,x=might be replaced withX is small

In fuzzy logic, X is a is interpreted as a5 0 N O V E M B E R 1994


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characterization of the possible valuesof X , with A representing a possibilitydistribution. Thus, the possibility thatX can take a value U is given by

Pon{X = U] M(U)It is in this sense that X is ~ ( u ) ,ithpossibility nterpreted as ease of attain-ment or assignment, may be interpret-ed as an elastic constraint on X .

not, the calculi of fuzzy rules and fuzzygraphs provide an alternative method-ology aimed at results that are in thespirit of the guiding principle of softcomputing.

X X__ __-

lSJ3oc\.o0 0Figure 3. (A) Quantization versus (B) granulation @zzy quantization).

achieving a higherMIQ.Interestingly, the development offuzzy-set theory was motivated by thefirst situation, but today most applica-tions of fuzzy logic in the realm ofconsumer products are motivated bythe second.FDCL has many facets. Here, Ishall sketch some of the basic ideasthat underlie FDCL and the calculi offuzzy rules and fuzzy graphs.Like any language, FDCL is char-acterized by its syntax and semantics.T he syntax of FD CL is concernedwith the form of admissible fuzzyrules; the semantics is concerned withtheir meaning. It is important to notethat FDCL is not a fuzzifiedversionof a standard programming language,as is true of Fuzzy Prolog.6Fuzzy rules. FDCL allows the use of

a wide variety of fuzzy if-then rules, orsimply fuzzy rules. A typical fuzzyrule relates m antecedent variablesX I , .., , to n consequent variables,

Y,...,Y,.and has the form:if X , s A, a n d ...X , s A,t h e n Y,s B , a n d ...Y, s B,

where X = ( X , ...,X,) and Y = ( Y ,,...,Y J are linguistic variables and (A ,,...4,) nd ( B , ,..., ,) their respectivelinguistic values. For example:if Pressure is high a n d Tmperatureis high t h e n Volume s small

For simplicity,I will discuss only rulesin which m= = 1.A rule can have a rurface stmctzlre ora deep stmcture. The surface stmcture isthe rule in its symbolic form:i f X i s A t h e n Y i s B

Such a rule is said to be uncalibrated,which means that the membershipfunctions of A and B are not specified.T h e deep structure is the surfacestructure together with a characteriza-tion of the membership functions oflinguistic values of variables. In thiscase, the rule is said to be calibrated.-

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. -

Low High

WnsP 30 130 180

Speed[A)- -

Control rulesi-. If (speed 1s l o w ) ond (shi f t IS high) lhen (-3)1 2 ll speed 15 high) and (shi f t is low) then(4)3 If (throt IS ow ) and (speed s high) then (t3)4 If (throt IS low) and (speed s l ow) then (tl) 5 If (throt is high) ond (speed s high) then (-1)6 If (throt is high) and (speed IS l ow ) then (-3)4

._ - - - _ ._ -- __ ___- -- - - _.__ --Figure 4. Fuz zy rules used in Honda i izzy-logic transmission. Here, the meaning o the numeric values associated with thed e s s not important; they only illustrate how d e s are calibrated

Figure 4 shows an example of the cali- later, one of the central problems inbrated fuzzy rules used in Hondas ~ the applications of fuzzy logic is that of expressed asfiiz7y-logic transmission. .As I explain deriving the deep structure of a set of 1 the proposition ;l/lary isyoung might be,wdF3, svozrng -$ is young1 fuzzy ules from U 0 data. I where Age(Mary) is the focal variableand young is a fuzzy constraint o nApplying this concept of meaningi f x i S A then y i s we can express the meaning of the rule

in question as a fuzzy constraint on thejoint variable ( X , Y ) .More specifically,

Age(lMd?y).-71______

, DERIVING RULES AND GRAPHS~ In the semantics of FDCL, the1 basic questions are, what is the mean-ing of a single rule and what is themeaning of a collection of rules?

. representation to the fuzzy rule

iI A 1 i f Xi sA then Iis B + (X,qs A x B ,U I I type of rule:Deriving rules. Consider the simplestn 1 , if X sA then Y is B I where A x B is the Cartesian productofA and B Th e membership function

IFig41-e 5. .I x E ~ n t c i p r tr d 1i r n f i i z : ) where A and B are linguistlc values ofX and Y , respectlvely. The questlon is, ofA x B is given by

the membership functlons ofA and B? 1In fuzzy logic, the meaning of apropositlon p is expressed as a ranonzcalfilm

what 1s the meaning of this rule P e n p , xR ( = p I (U) \ p&)1 where A is the conjunction operator,usually defined as min. A x B may be1 interpreted as afizzy point or a grunule, p + Z 1 s C as shown in Figure 5 .

I

f

II where + means translates into, Z is Deriving graphs. In the case of a col-lection of rules expressed asthe constrained variable, and C is anfuzzy relation that plays the role of afuzzy constrain t on 2.What th isimplies is that the meaning of p isexpressed as a fuzzy- r, equivalent-variable. To illustrate, the meaning ofFigzlre 6. InteVretation o a colkction ly, elastic- onstraint on a designatedof&zzy rules as a@z zy graph.

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i f X i s , 4 , t h e n I i s B , , t = l , ...,nand the meaning of the collection isdefined asIfXisA, then 1-1sB,,(2 = 1,...,n )+ (XY)s

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(A, B , + ...+A,>x Jwhere + is used in place of v to denotethe disjunction operator, which is usu-ally defined as max. For simplicity, theright-hand member of the collection 1may be written as

~

( X , .? is (E,4 x BJTh e expression 1, , x B, may beviewed as a superposition of fuzzypoints o r granules, as illustrated inFigure 7 . In effect, it represents acoarse- r, equivalently, compressed- haracterization of the dependencyand for this reason it is called a&zzygraph.5 Thus, a collection of fuzzy

rules is represented as a fuzzy graph.For example:if X is smal l then Y is smal lif X is medium then Y is largeif X is large then Y is small I

is a coarse characterization of thedependency illustrated in Figure 6.

INTERPOLATIONIf we interpret a collection of rulesas a coarse representation of the func-tional dependence of Yon X , the prob-lem o interpolation may be defined asthat of computing the value of Y givena value of X that may not be a perfectmatch with any of the antecedent vari-ables in the collection. More specifi-cally, this problem can be expressed asthe inference schema( X , ? is (1,, x B,)X i s AY s?B

in which ? B signifies that B is theobject of computation. In graphicalterms, as shown in Figure 8, theproblem may be viewed as that ofassigning a linguistic value to X andcomputing the corresponding linguis-tic value of Y.In fuzzy logic, computation of B iscarried out through the basic rule ofinference, called the compositional ruleof inference.? T h e rule in questionreads

Figure 7. R c p m ~ t i t i n g collection of fiizzy rr1le.r ns n f i izzy g m p h ,f: whichnppl~orlltI"rer to f .

Figure 8. Interpolation o f a@zzy gvaph. The value of Y may be interpreted as theprojection of the intersection o f th eh zz y graph with the cylindrical extension ofA .

(A kJ is KX i s AY is R* A

in which the composition operation isdefined byp R e.3 (v) rLip&R(zi,z') Ap4('))

in which pR(u,u)and pz4(u) re, respec-tively, the membership functions of Rand A.In the example considered earlier, Ris given byR = (E, , B, )

or, equivalently,B = Z,p,A B ,

in whichP2= (F,4,(.) A 1 (U)Th e sequence of computations thatleads to B is standard in most fuzzylogic applications and is usually imple-mented in software or hardware. Insome implementations, called max-product implementations, the conjunc-tion A is interpreted as the arithmeticproduct.Interpolation lies at the heart of theutility of fuzzy rule-based systemsbecause it makes it possible to employa relatively small number of fuzzy rulesto characterize a complex relationshipbetween two or more variables. In atypical application in a consumer

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11 Figure 9. Interpolationof t w o mles and defizzq5cation.product or industrial-control system,the number of rules is on the order ofIn applications in which B plays therole of a control variable (an input to amotor, for example), B must be"defuzzified"- onverted to a single-ton - efore it is applied. In currentpractice, the center-of-gravity methodis generally used to achieve defuzzifi-cation. Figure 9 shows a simple exam-ple of interpolation and defuzzifica-tion.

io to 20.73

In this figure, the rules are

i f X i s A , t h e n Yi s B ,if X isA, h e n Y s B,and the input is

X i s Am l an d m2 are, respectively, thedegrees to which A matches A1 andA2. Th e expression for the output is

P B ( U ) = 2,P/ A P d 4which upon center-of-gravity dehzzi-fication leads to a numerical value ofE .

1 Grodient pragromming 14 Genetic:olgorithms 11 Reinforcement earning f I I

Figure 10. Summary o alternative methods t o deduce the deep stmctures of a set11 oj-rmles.

INDUCING RULES FROMOBSERVATIONOne of the central problems in theapplications of fuzzy logic relates tothe question, how can rules be inferredfrom observations; that is, from theknowledge of a collection of U 0 pairs?[n the context of self-organizing sys-tems, this problem was first formulat-ed and analyzed by T.J.Procyk andE.H.Mamdani. Later, a seminal paperbyT. Takagi and M. Sugeno made amajor contribution.9During the past several years,researchers have made importantadvances toward at least a partial solu-

tion to the problem by applying neur-al-network techniques or, more gener-ally, dynamic and gradient program-ming,7.10-J Other promising approach-es involve the use of genetic algo-rithmsl?$13 nd reinfo rcemen t learn-ing.'+ Figure 10 summarizes the waysto derive the deep structure of a set ofrules from the surface structure.A basic idea underlying theseapproaches involves representing afuzzy rule-based system as a multilay-ered structure, such as that shown inFigure 11 'I In a simple version of thisarchitecture11 that is rooted in theTakagi-Sugeno-Kang approach,Y therules are assumed to be of the form(ifX, sA,, a n d ... a n d X , i s A , ,then Y = J , i = I,..,,n

where b, are constants (singleton con-sequents). If the numerical values ofX I , ... X , are U,, ... U,, , respectively,and the grades of membership of u I ,... U,, , in A,, ...A,, are p I r ( u l , ...~ ~ ~ ( u ~ ~ ) ,hen the combined degree towhich the input n-tuple X ( u I , , ... u,Jmatches the antecedents is taken to bethe product

m, P I I (Ul i I,... Il rn, (U,"J ,Then, defining the normalized weightwrs

w , = m,/ i ...+ mrnthe output is expressed as

Y = Z, m,b,l!

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Note that in this architecture thereis no defuzzifier because the inputs X , ,...X , are assumed to be singletons.In the application of gradient pro-gramming to this architecture, themembership functions of A , , , ...A,,are assumed to be triangular, trape-zoidal, or Gaussian in form. Then,using backward iteration, the values ofmembership-function parameters arecomputed from right to left.10-11 Inthis way, from the knowledge of 110pairs we can compute the values ofparameters and thereby induce therules from observations.The approach sketched here is oneway the methodologies of fuzzy rule-based systems and neural networks canbe combined, leading to neurofuzzysystems. Such systems are growing innumber and visibility and are illustra-tive of the advantages derived fromcombining soft computings con-stituent methodologies. In this con-text, it is important to note that inter-polation and induction of rules fromobservations are key issues in bothfuzzy logic and neurocomputing.

approach that is model-dependent -in the sense of requiring a formulation

FUZZY BALL AND BEAM PROBLEM

Figure 11 . Representing a fu zzy Jystem as a multilayered structure. Il and Ndenote mu ltipliers and norm alizers, respectively.

of equations governing system behav-ior- an be employed because we donot know how to model a ball rollingor sliding on a rug-like surface. Thisrules out the use of classical controltheory as well as any approach thatrequires simulation.The set-interval may be viewed as adisjunctive goal. This feature makes itdifficult to employ neural-networktechniques.By contrast, the problem is easy tosolve with fuzzy logic because it is rel-atively easy for a human. In fact, thepresence of a fuzzy layer makes theball-and-beam problem easy forhumans and difficult or impossible foralternative methodologies. As in mostfuzzy-logic applications, the solution isin effect a translation of a human solu-tion into FDCL. A human solutionwould normally involve seven steps:1 . Compile uncalibrated fuzzy orcrisp rules from knowledge of naturallaws, to govern the behavior of theball and beam. For example

if 0 i s P e g a t i v et h e n Y i s posi t ive

if 0 i s ,p o s i t i v et h e n Y i s n eg a t i v eth e more n e g a t i v e 0,th e more posi t ive Ythe more posi t ive 0 ,t h e more n eg at i v e Y2 . Construct a plan of action (an

algorithm), expressed in terms ofuncalibrated fuzzy rules of the formif State is A t h e n Action is B3. Test the system without tryingto solve the problem4. Calibrate the fuzzy rules in step2 using metarules, rules that modifyother rules5. Test the algorithm constructedin step 2 .6. Refine the calibrated fuzzy rulesderived in step 5 .7 . Iterate steps 5 and 6 until theball stays in the set-interval.

Figure 12. Fuzzy ball and beam prob-lem.I E E E S O F T W A R E 5 5


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Translating these steps into a col-lection of fuzzy rules expressed inFDCL is by no means a trivial prob-lem. This is particularly true of the so-called gradual rulesls of the formthe moreXis A the more Y s B

because such rules describe the globalbehavior of a functional dependencyrather than i ts local propert ies.However, what is important is that,though it is not easy, it is feasible totranslate a human solution intoFDCL, whereas it is not feasible totranslate it into anal~aical echnisues.

more difficult for a human because itinvolves a conjunction of two goals:+ confine the motion of the ball tothe prescribed set-interval [a,,a,], and+ enter the set-interval a t a time twhich is constrained to lie in a pre-scribed temporal set-interval [t , , J .In this case, formulating a humansolution and translating it into FDCLis a real challenge. We do n ot yetcompletely understand how to applyfuzzy logic to problems like this. But itis evident that fuzzy logic - sedalone or in combination with neuro-computing and probabilistic reasoning- s the methodology of choice whento reach the set interval at some time t ~ analytic models are impossible or hardNow suppose you wanted the balland stay there. This is significantly to formulate.

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ACKNOWLEDGMENTSThis research was supported in part by the BISC Prograin, N .AS h (;rant NCC 2 - ? i j , EPR IAgreement RP 8010-34 and MI CR O State Program KO.92-180.

REFERENCESI. LA . Zadeh, Fuzzy Sets, Infirmation and Conti-ol, u n c 196.i, pp . 3 38-15!.2. L.A. Zadeh, O utline of a New A pproach to the Analysisof Complex Systems and Dccision

Processes, IEEE Trans. Systems, Ma n and Cybmzetics, 1973, pp. 28-44,3. L.A. Zadeh , Possibility The ory and Soft Data Analysis, .\.lathernotical Frontreis ofthe Social un d

PoliT Sciences,L . Cobb and R. M.Thral l , eds., \Vesmew Press, Boulder, Colo., 1081, pp.69-129.4. L.A. Zadeh, Fuzzy Logic, Neural Network?an d Soft Comput ing, fConrrn. :IC.%f,Mar . 1994,5. L.A. Zadeh, O n the Analysis of Large-Scalc Systems, in $.itmu ,?pproache.r an d Enmnmnrnzt6. M. M ukaidono, Z.L. Shen, and L. Ding, Fundamentals of Fuzzy Prolog, lnt/y. lpproxmrutr7. B. Kosko, Neural Networks and Fu zzy Systems: -1 L)ynnrnicalSy.itenrs. Ippi-nach o .llai.hriw8.T. Teran o , K. Asai, and M. Sugeno, Fuzzy Systems Th e o n and lts Xpplications, .Academic9. T. Takagi and M. Sugeno, Fuzzy Identification of Systems and it5 .Applications to &lodeling10. C:T. Lin and C.S. George Lee, Neural -Network-Based F u z q Logic Control an d Decision11.J.-S.R. Jang, Self-Learning Fuzzy Controller Based on Temporal BacL~ Pr(,pagation,EE E12 .C.C. Lee, Fuzz y Logic in Control Systems: Fuzz)- 1-ogicController, Part 1 and Part 11, IEEE1 3 . C. Karr, G enetic Algorithms for Fuzzy Controllers..?I E xp r i t , h - o v . 1001, pp. 26-33.14.M.A. Lee and H . Takagi, Integrating Design Stages of 1;uzzy Systems Using G eneticAlgorithms,Proc. Intl Con$ onFuzzy System. IFEE Press, S e w York, 1903,pp. 612-617.15. D. Dubois and H. Prade, Gra dual lnference Rules in Approximate Rcasoning, Z ~ r f b ~m o t i o aSciences, 1992,pp. 103-122.

pp. 77-84.Problems, H. Gottinger, ed., Landenhoeck and Ruprecht,

soft comp fuzzy logic

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