elk an i eee discussion

7/22/2019 Elk an i Eee Discussion

1/47

I

The Paradoxical Success ofFuzzy LogicCharles Elkan, University of California, San DiegoFuzzy logic methods have been used suc-cessfully in many real-world applications,but the foundations of fuzzy logic remainunder attack. Taken together, these twofacts constitute a paradox. A second para-dox is that almost all of the successfulfuzzy logic applications are embedded con-trollers, while most of the theoretical pa-pers on fuzzy methods deal with knowl-edge representation and reasoning. I hopehere to resolve these paradoxes by identify-ing which aspects of fuzzy logic render ituseful in practice, and which aspects areinessential. My conclusions are based on amathematical result, on a survey of litera-ture on the use of fuzzy logic in heuristiccontrol and in expert systems, and on prac-tical experience developing expert systems.An apparent paradox

As is natural in a research area as activeas fuzzy logic, theoreticians have investi-gated many formal systems, and a varietyof systems have been used in applications.Nevertheless, the basic intuitions have re-mained relatively constant. At its simplest,fuzzy logic is a generalization of standardpropositional logic from two truth values,f a l s e and true, to degrees of truth between0 and 1.Formally, let A denote an assertion. Infuzzy logic, A is assigned a numerical valuet ( A ) ,called the d e g r e e of t ru th of A , suchthat 0 5 t ( A ) 1. For a sentence composedfrom simple assertions and the logical con-nectives and A), or (v), and not 1)degree of truth is defined as follows:

~

An earlier version with the same titleappeared in Proceedings o the Eleventh Natrona1 Conference o n Artificial Intelligence(AAA1 93),M I T Press, 1993,pp 698-703

Def in i t ion 1: Let A and B be arbitrary as-sertions. Then

t ( A A B ) = min [ ( A ) , ( B ) )t ( A v B ) = max { t ( A ) , ( B ) ]t ( A )= t ( B ) f A and B are logicallyt ( 4 )= 1 - ( A )equivalent.

Depending how the phrase logically equiv-alent is understood, Definition 1yieldsdifferent formal systems. A fuzzy logic sys-tem is intended to allow an ndefinite varietyof numerical truth values. However, formany notions of logical equivalence, onlytwo different truth values are possible giventhe postulates of Definition 1.Th e o r e m 1: Given the formal system of Def-inition 1, if l ( A A 4 nd B v 4 4are logically equivalent, then for any twoassertions A and B , either t ( B )= t ( A )ort ( B ) = 1 - t ( A ) . WA direct proof of Theorem 1 appears in thesidebar, but it can also be proved usingsimilar results couched in more abstract

Propos i t ion: Let P be a finite Boolean al-gebra of propositions and let z be a truth-assignment function P + 0,1], supposedlytruth-functional via continuous connec-tives. Then for all p E P , Q) E { 0, 1] WThe link between Theorem 1 and this propo-sition is that l ( A A 4 B v 4 -IB) isa valid equivalence of Boolean algebra.Theorem 1 s stronger in that it relies ononly one particular equivalence, while theproposition is stronger because it applies toany connectives that are truth-functionaland continuous (as defined in its authors

The equivalence used in Theorem 1 israther complicated, but it is plausible intu-paper).

itively, and it is natural to apply it in rea-soning about a set of fuzzy rules, since7 ( A A4 nd B v 44 re bothreexpressions of the classical implication44 B . It was chosen for this reason, butthe same result can also be proved usingmany other ostensibly reasonable logicalaquivalences.

It is important to be clear on what ex-actly Theorem 1 says, and what it does notsay. On the one hand, the theorem appliesto any more general formal system thatincludes the four postulates listed in Defin-ition 1.Any extension of fuzzy logic toaccommodate first-order sentences, forexample, collapses to two truth values if itadmits the propositional fuzzy logic ofDefinition 1 and the equivalence used inthe statement of Theorem 1 as a specialcase. The theorem also applies to fuzzy settheory given the equation ( A fl B) =B U (A n BC),ecause Definition 1 can beunderstood as axiomatizing degrees ofmembership for fuzzy set intersections,unions, and complements.

On the other hand, the theorem does notnecessarily apply to versions of fuzzy logicthat modify or reject any of the postula tes ofDefinition 1 or the equivalence used in The-orem 1. However, it is possible to carrythrough the proof of the theorem in manyvariant fuzzy logic systems. In particular,the theorem remains true when negation ismodeled by any operator in the Sugenoclass, and when disjunction or conjunctionare modeled by operators in the Yagerclasses The theorem also does not dependon any particular definition of implication infuzzy logic. New definitions of fuzzy impli-cation are still being proposed as new appli-cations of fuzzy logic are investigated.

Of course, the last postulate of Definition1 is the most controversial one. To preserve

AUGUST 1994 3


2/47

Proof of Theorem 1Theorem I Given the formal system t B)< 1 - (B) < 1- (A) ,

of Definition 1, if l(A A 4l nd B v4 lB) are logically equivalent, thenfor any two assertions A and E , either t ( B )=

t ( A )or r(B)= 1-t(A).P r m j Given the assumed equivalence,

( ,(A A 4 t (B v (-A A T B ) ) . Nowtf7(A 7B))= 1 -min [ r(A) , 1 - ( B ) ]= 1 + max {-r(A), -1 + t ( B ) )

= max [ 1 - A), t ( B ) )and

f ( BV 4 i B ) ) =max { t ( B ) ,min { 1 - A) , 1 - ( B ) ) .The numerical expressions above are dif-ferent if

that is if t (B) < 1 - ( B ) and t ( A ) < r(B) ,which happens if t ( A ) < t ( B ) < 0.5. So itcannot be true that r(A)< t ( B )< 0.5.Now note that the sentences -(A 4and E v (-A A 4 re both reexpressions ofthe material implication A . One by one,consider the seven other material implicationsentences involvingA and B, which are

4 4 BA + y B4 4

B + Ai B + AB - 4

-lB 4

a continuum of degrees of truth, one natu-rally wants to restrict the notion of logicalequivalence. In intuitive descriptions, fuzzylogic is often characterized as arising fromthe rejection of the law of excluded middle:the assertion A v 4.Unfortunately, reject-ing this law is not sufficient to avoid col-lapse to just two truth values. Intuitionisticlogic rejects the law of excluded middle,but the formal system of Definition 1 stillcollapses when logical equivalence meansintuitionistic equivalence? (The Godeltranslations of classically equivalent sen-tences are intuitionistically equivalenL6 Forany sentence, the first three postulates ofDefinition 1 make its degree of truth andthe degree of truth of its Godel translationequal. Thus the proof in the sidebar can becarried over directly.) Dubois and Pradenote that if all the properties of a Booleanalgebra are preserved except for the law ofexcluded middle, their proposition nolonger holds? This observation is compati-ble with a collapse assuming only theequivalences of intuitionistic logic, becausealthough intuitionistic logic rejects the lawof excluded middle, it admits a doublynegated version of the law, namely

7(7 4 -A). Of course, collapse totwo truth values is avoided if we admit onlythe equivalences generated by the operatorsminimum, maximum, and complement toone. However, these equivalences are es-sentially the axioms of de Morgan, whichallow only restricted reasoning about col-lections of fuzzy assertions.

Fuzzy logic in expert systemsThe basic motivation for fuzzy logic is

clear: While many ideas resemble tradi-tional assertions, they are not naturallyeither true or false; uncertainty of somesort is attached to them. Fuzzy logic is anattempt to capture valid reasoning pattemsabout uncertainty. The notion is now wellaccepted that there are many different typesof uncertainty, vagueness, and ign~rance.~However, there is still debate as to whattypes of uncertainty are captured by fuzzylogic. Many papers have discussed (at ahigh level of mathematical abstraction) thequestion of whether fuzzy logic providessuitable laws of thought for reasoningabout uncertainty nd if so, which vari-eties of uncertainty. The question of inter-est here is more empirical: whether or notfuzzy logic is in practice an adequate for-malism for uncertain reasoning in knowl-edge-based systems.nical literature using the Inspec and Com-puter Articles databases of more than 1.3million papers published since 1988. Usingthe abstracts as a guide, I found nopublished report of a deployed expert sys-tem that uses fuzzy logic as its primary for-malism for reasoning under uncertainty.While many theoretical papers on fuzzylogic in expert systems have been published,and several prototype systems have beendescribed, it is hard to find reports of fieldedsystems doing knowledge-intensive taskssuch as diagnosis, scheduling, or design.

I conducted a thorough search of the tech-

Recent conferences give a representative

By the same reasoning as before, none ofthe following can be true:1 - (A)< [ ( E )< 0.5(A) < 1 - ( B ) < 0.5

1- A)< 1 - ( B )< 0.5r(B)< ( A )< 0.51 - ( B ) < t ( A )< 0.5t ( B )< 1 - ( A )< 0.5

1 - ( B ) < - ( A )< 0.5Now let x = min { r(A), 1 - (A) )and let

y = min [ B), 1 - ( B ) ] .Clearly x 0.5 andy < 0.5 so if x y. then one of the eightinequalities derived must he satisfied. Thust ( B ) = t ( A ) or r(B)= 1 - (A).

view of the extent of fuzzy logic applica-tion in current commercial and industrialknowledge-based systems. All the systemsin actual use described at the 1992 IEEEIntemational Conference on Fuzzy Sys-tems are controllers, as opposed to reason-ing systems. At the 1993 IEEE Conferenceon AI for Applications, no applications offuzzy logic in knowledge-based systemswere reported. Of the 6 deployed systemsdescribed at the 1993 AAA1 Conference onInnovative Applications of AI, three heCAPE,* D~ dg er ,~nd DYCE'" systemsused fuzzy logic in some way. However,none of these systems uses fuzzy logic op-erators for reasoning about uncertainty.Input observations are assigned degrees ofmembership in fuzzy sets, but inferencewith these degrees of membership usesother formalisms.

In addition to DYCE, a team at IBM hasdeveloped and fielded several knowledge-based systems over the past five years.Some of these systems are used for softwareand hardware diagnosis, for data analysis,and for operator The systemshave varying architectures and cope withdifferent varieties of uncertainty. Experiencewith them suggests that fuzzy logic is rarelysuitable in practice for reasoning about un-certainty. The basic problem is that items ofuncertain knowledge must be combinedcarefully to avoid incorrect inferences.Fixed domain-independent operators likethose of fuzzy logic do not work.The correct propagation of certaintydegrees must account for the content of theuncertain propositions being combined.


3/47

I

This is necessary whether the uncertainpropositions constitute deep or shallowknowledge. In the case of shallow knowl-edge, which may be defined as knowledgethat is valid only in a limited context (forexample, a correlation between a symptomand a fault), how degrees of uncertainty arecombined must be adjusted to account forunstated background knowledge.

A simple example illustrates the diffi-culty. Consider a system that reasons in ashallow way using a notion of strength ofevidence, and assume that, as in manyexpert systems, this notion is left primitiveand not analyzed more deeply. (Certainlystrength of evidence is an intuitivelymeaningful concept that may or may not beprobabilistic, but it is definitely differentfrom degree of truth.) For concreteness,suppose the context of discourse is a col-lection of melons, and in this context bydefinition wnfermelon(x)e edinside(x) Agreenoutside(x). For some melon m, sup-pose that t(redinside(m))= 0.5 and t(green-oufside(m))= 0.8,meaning that the evi-dence that m is red internally has strength0.5, and that m is green externally withstrength of evidence 0.8. Are the rules offuzzy logic adequate for reasoning aboutthis particular type of uncertainty? Theysay that the strength of evidence that m is awatermelon is t (watermelon(m))= min(0.5,0.8]= 0.5.However, implicit back-ground knowledge in this context say s thatbeing red inside and green outside are mu-tually reinforcing pieces of evidence to-ward being a watermelon, so m is a water-melon with strength of evidence over 0.5.

Deep knowledge can be defined asknowledge that is detailed and explicitenough to be valid in multiple contexts.Deep knowledge is general purpose andusable in complex chains of reasoning.However, Theorem 1 says that if more thantwo different truth values are assigned tothe input propositions of long inferencechains using fuzzy logic rules and oneplausible equivalence, then it is possible toarrive at inconsistent conclusions. Fuzzylogic cannot be used for general reasoningunder uncertainty with deep knowledge.

The fundamental issue here is that a con-junctions degree of uncertainty is not ingeneral determined uniquely by the degreeof uncertainty of the assertions entering intothe conjunction. There does not exist afunctionfsuc h that the rule t(AA B ) =f l t (A) , ( B ) ) s always valid, whatever thetype of uncertainty represented by t(.). Forexample, in the case of probabilistic uncer-tainty, the rule t(AA B ) = t (A) . ( B ) s validif and only if A and B represent independentevents. In general, for probabilistic uncer-tainty all one knows is that max [0, t A)+t ( B )- 1 ] 5 t A A B ) 5 min t A), ( B ) ] .

Methods for reasoning about uncertainevidence are an active research area in AI,and the conclusions here are not new. How-ever, our practical experience independentlyconfirms previous arguments about the in-adequacy of systems for reasoning aboutuncertainty that propagate numerical factorsaccording only to which connectives appearin assertions.I3Fuzzy logic i n heuristic tont rol

The application of fuzzy logic has beenmost successful in heuristic control, wherethere is wide consensus that traditionaltechniques of mathematical control theoryare often inadequate. The reasons for thisinclude the reliance of traditional methodson linear models of systems to becontrolled, their propensity to producebang-bang control regimes, and theirfocus on worst-case convergence and sta-bility rather than typical-case efficiency.Heuristic control techniques give up math-ematical simplicity and performance guar-antees in exchange for increased realismand better performance in practice. Forexample, a heuristic controller using fuzzylogic has been shown to have less over-shoot and quicker settling.4

The first demonstrations that fuzzy logiccould be used in heuristic controllers werepublished in the 1970s.15*16ork continuedthrough the 1980s,and recently there hasbeen an explosion of industrial interest inthe area.17, 18 ne reason for this recentinterest in fuzzy controllers is that they can

be implemented by embedded specializedmicroprocessors. 9Despite industry interest, and consumerinterest in Japan, fuzzy logic technology:ontinues to meet resistance. For example,at IJCAI 91, Takeo Kanade gave a talk oncomputer vision, describing at length Mat-sushitas camcorder image stabilizing sys-tem without mentioning its use of fuzzylogic. Also, while a fuzzy logic controlleris embedded in the 1994 Honda Accordsautomatic transmission, the advertisingbrochures describe it as grade logic.

Almost all currently deployed heuristiccontrollers using fuzzy logic are similar infive important aspects (a good example ofthis standard architecture appears in apaper by Sugeno and his collea gues2):(1) The typical fuzzy controller knowl-edge base consists of fewer than 100

rules; often fewer than 20 rules areused. Fuzzy controllers are orders ofmagnitude smaller than systems builtusing traditional AI formalisms.

( 2 ) The knowledge entering into fuzzycontrollers is structurally shallow,both statically and dynamically. Con-clusions produced by rules are notused as premises in other rules; stati-cally rules are organized in a flat list,and dynamically there is no runtimechaining of inferences.

(3) The knowledge recorded in a fuzzycontroller typically reflects immediatecorrelations between the inputs andoutputs to be controlled, as opposed toa deep, causal model of the system.The premises of rules refer to sensorobservations, and rule conclusionsrefer to actuator settings. (Rulepremises refer to qualitative or lin-guistic sensor observations, and ruleconclusions refer to qualitative actua-tor settings, whereas outputs and in-puts of sensors and actuators are typi-cally real-valued. This means thatnormally two controller componentsmap between numerical values andqualitative values. In fuzzy logic ter-minology, these components are saidto defuzzify outputs and implementmembership functions.)

AUGUST 1994 5


4/47

(4) In deployed fuzzy controllers, the nu-merical parameters of their rules andof their qualitative input and outputmodules are tuned in a learningprocess. The tuning can be done byhuman engineers or by leaming algo-rithms; neural network methods havebeen especially successful.22What thetuning algorithms themselves have incommon is that they are gradient-de-scent hill-climbing algorithms thatlearn by local 0ptimi~ atio n.l~

( 5 ) By definition, fuzzy controllers usefuzzy logic operators. Typically, mini-mum and maximum are used, as areexplicit possibility distributions (usu-ally trapezoidal) and some fuzzy im-plication operator.

The question that naturally arises is, Whichof these five features are essential to thesuccess of fuzzy controllers? It appears thatthe first four are vital to practical success,because they make the celebrated creditassignment problem solvable, while the useof fuzzy logic is not essential.

In a nutshell, the credit assignment prob-lem is to improve a complex system bymodifying a part of it, given only an evalua-tion of its overall performance. In general,solving the credit assignment problem isimpossible: the task is tantamount to gener-ating many bits of information (a change tothe internals of the system) from just a fewbits of information (the systems inputlout-put performance). However, the first fourshared features of fuzzy controllers cansolve this problem for the following reasons.

First, since it consists of only a few rules,the knowledge base of a fuzzy controller isa small system to modify. Second, the shortpaths between the fuzzy controllers inputsand outputs localize the effect of a change,making it easier to discover a change with adesired effect without producing undesiredconsequences. Third, because of the itera-tive way in which fuzzy controllers are re-fined, many observations of inputloutputperformance are available for system im-provement. Fourth, the continuous nature of

he controllers parameters allows smallpant itie s of performance information to beised to make small system changes.

Thus, what makes fuzzy controllers use-[ul in practice is the combination of a rule-Jased formalism with numerical factorspal ify ing rules and the premises enteringinto rules. The principal advantage of rule-xs ed formalisms is that knowledge can beacquired from experts or from experienceincrementally. Individual rules andpremises can be refined independently, orat least more independently than items ofknowledge in other formalisms. Numericalfactors have two main advantages. Theyallow a heuristic control system to inter-Face smoothly with the continuous outsideworld, and they allow it to be tuned gradu-ally mall changes in numerical factorvalues cause small changes in behavior.

None of the features contributing to thesuccess of systems based on fuzzy logic isunique to fuzzy logic. It seems that mostcurrent fuzzy logic applications could useother numerical rule-based formalismsinstead f a human or a learning algo-rithm tuned numerical values for thoseformalisms, as is customary when usingfuzzy logic. A quote from the originator offuzzy heuristic control is relevant here:

...it should be remarked that the work onprocess control using fuzzy logic was inspiredas much by Waterman and his approach torule-based decision making as by Zadeh ...and his novel theory of fuzzy subsets.23Several knowledge representation for-

malisms that are rule-based and numericalhave been proposed besides fuzzy 10 gi c. ~~ ,* ~To the extent that numerical factors can betuned in these formalisms, they should beequally useful for constructing heuristiccontrollers. Indeed, at least one has alreadybeen so used.26RetapitulatingmainstreamAI

Several research groups are attemptingto scale up systems based on fuzzy logicand lift the architectural limitations of cur-rent fuzzy controllers. For example, amethodology for designing block-struc-tured controllers with guaranteed stability

properties has been ~tu di ed, ~s havemethodological problems in constructingmodels of complex systems based on deepknowledge.** Controllers with intermediatevariables, thus with chaining of inferences,have also been in~ est iga ted .~~However, the designers of larger systemsbased on fuzzy logic are encountering all theproblems of scale already identified in tradi-tional knowledge-based systems. It appearsthat the research history of fuzzy logic isrecapitulating that of other areas in AI aswell, particularly those dealing with knowl-edge engineering and state information.

The rules in the knowledge bases of cur-rent fuzzy controllers are obtained directlyby interviewing experts. Indeed, the origi-nal motivation for using fuzzy logic inbuilding heuristic controllers was that fuzzylogic is designed to capture human state-ments involving vague quantifiers such asconsiderable. More recently, consensushas developed around the idea that researchmust focus on obtaining procedures forfuzzy controller design based on fuzzymodels of the process.30Mainstream workon knowledge engineering, however, hasalready transcended the dichotomy betweenrule-based and model-based reasoning.Expert systems with knowledge consist-ing of $-then rules have at least two disad-vantages. First, maintenance of a rule basebecomes complex and time-consuming asthe system size increases. Second, rule-based systems tend to be brittle: If an itemof knowledge is missing from a rule, thesystem may fail to find a solution, orworse, may draw an incorrect conclusion.The main disadvantage of model-basedapproaches, on the other hand, is that it isvery difficult to construct sufficiently de-tailed and accurate models of complex sys-tems. Moreover, the models constructedtend to be highly application-specific andnot gen erali~able.~

Many recent expert systems, therefore,are neither rule-based nor model-based inthe standard way.12 For these systems, theaim of the knowledge engineering process

6 IEEEEXPERT


5/47

is not simply to acquire knowledge fromhuman experts, but rather to develop a the-ory of the experts situated performance(this is true regardless of whether the de-sired knowledge is correlational, as in pre-sent fuzzy controllers, or deep, as inmodel-based expert systems). Concretely,under this view of knowledge engineering.knowledge bases are constructed to modelthe beliefs and practices of experts and notobjective truths about underlying physi-cal processes. An important benefit of thisapproach is that the organization of an ex-perts beliefs provides an implicit organiza-tion of knowledge about the externalprocess with which the knowledge-basedsystem is intended to interact.The more sophisticated view of knowl-edge engineering just outlined is clearlyrelevant to research on constructing moreintricate fuzzy controllers. For a secondexample of relevant AI work, consider con-trollers that can carry state informationfrom one moment to the next (mentionedas a topic for future research by von Al-trock and colleague^^^). Symbolic AI for-malisms for representing systems whosebehavior depends on their history havebeen available since the 1960s.Neural net-works with similar properties (called recur-rent networks) have been available fo r sev-eral years, and have already been used incontrol application^.^^ It remains to be seenwhether research from a fuzzy logic per-spective will provide new solutions to thefundamental issues of AI.

Applications of fuzzy logic in heuristiccontrol have been highly successful, de-spite the collapse of fuzzy logic to two-valued logic under an apparently reason-able condition, and despite the inadequacyof fuzzy logic for general inference withuncertain knowledge. These difficultieshave not been harmful in practice becausecurrent fuzzy controllers are far simplerthan other knowledge-based systems. The-orem 1 is not an issue for fuzzy controllersbecause they do not perform chains of in -

erence, and they are developed informally,Nith no formal reasoning about their rules:hat applies equivalences such as the oneised in the statement of Theorem 1. Sec-md, the knowledge recorded in a fuzzy:ontroller is not a consistent causal model3f the process being controlled, but ratherm assemblage of visible correlations be-tween sensor observations and actuatorsettings. Since this knowledge is not itselfgeneral-purpose, the inadequacy of fuzzylogic for general reasoning about uncer-tainty is not an issue. Moreover, the abilityto refine the parameters of a fuzzycontroller iteratively can compensate forthe arbitrariness of the fuzzy logic opera-tors as applied inside a limited domain.The common assumption that heuristiccontrollers based on fuzzy logic are suc-cessful because they use fuzzy logic ap-pears to be an instance of the p o s t h o c , e r g oprop ter hoc fallacy. The fact that usingfuzzy logic is correlated with success doesnot entail that using fuzzy logic causessuccess. In the future, as fuzzy controllersare scaled up, the technical difficultiesidentified in this article can be expected tobecome important in practice.

Theorem 1 is a crisp demonstration ofone of several deep difficulties of scale inAI: the problem of maintaining consistencyin long sequences of reasoning. Other diffi-culties of scale can also be expected to be-come critical n particular, the issue ofdesigning learning mechanisms that cansolve the credit assignment problem whenthe simplifying features of presentcontrollers are absent.AcknowledgmentsThe author is grateful to many colleagues foruseful comments on earlier versions of this article.References1, D. Dubois and H. Prade, New Resultsabout Properties and Semantics of FuzzySet-Theoretic Operators, Fuzzy Sets: The-

ory and Applications to Policy Analysis andInformation Systems, Plenum Press,NewYork, 1980, pp. 59-75.2. D. Dubois and H . Prade, An Introductionto Possibilistic and Fuzzy Logics, Non-Standard Logics for Automated Reasoning,Academic Press, San Diego, 1988.

3. M. Sugeno, Fuzzy Measures and FuzzyIntegrals -A Survey, Fuzzy Automata andDecision P rocesses, Elsevier/North-Hol-land, New York, 1977, pp. 89-102.R.R. Yager, On a General Class of FuzzyConnectives, Fuzzy Sets and Systems, Vol.4, No. 3, Nov. 1980, pp. 235-242.D. Dubois and H. Prade, Gradual InferenceRules in Approximate Reasoning, Infor-mation Sciences, Vol. 61, No. 1-2, 1992, pp.103-1 22.D. van Dalen, Logic and Structure, seconded., Springer-Verlag, New York, 1983.P. Smets, Varieties of Ignorance and theNeed for Well-Founded Theories, Informa-tion Sciences, No. 57-58, Sept.-Dec. 1991,pp. 135-144.A. Cunningham and R. Smart, Computer-Aided Parts Estimation, Proc. 5th Innova-tive Applications ofA I Con , AAAI Press,Menlo Park, Calif., 1993, pp. 14-25.

9. A.J. Levy et al:, Dodger: A DiagnosticExpert System for the Evaluation of Nonde-structive Test Data, Proc. 5th InnovativeApplications of AI Con , AAA1Press,Menlo Park, Calif., 1993, pp. 107-117.10. D.D. Pierson and G.J. Gallant, Diagnostic

4.

5.

6.7.

8.

Yield Characterization Expert (DYCE): ADiagnostic Knowledge-Based System Shellfor Automated Data Analysis, Proc. 5thInnovative Applications of AI Con , AAAIPress, Menlo Park, Calif., 1993, pp.152-160.11. G. Gallant and J. Thygesen, Digitized

Expert Pictures (Depict): An IntelligentInformation Repository, Proc. 5th Inuova-tive Applications of AI Con , AAAI Press,Menlo Park, Calif., 1993, pp.50-60.12. A. Hekmatpour and C. Elkan, Categoriza-tion-Based Diagnostic Problem Solving inthe VLSI Design Domain, Proc. IEEE IntlCon on AI fo r Applications, IEEE Com-puter Society Press, Los Alamitos, Calif.,1993, pp. 121-127.13. J. Pearl, Probabilistic Reasoning in Intelli-gent Systems, Morgan Kaufmann, San Fran-cisco, Calif., 1988.14. D.G. Burkhardt and P.P. Bonissone, Auto-mated Fuzzy Knowledge Base Generationand Tuning, Proc. IEEE Intl Con onFuzzy Systems, IEEE Computer Society

Press, Los Alamitos, Calif., 1992, pp.179-1 88.15. L.A. Zadeh, Outline of a New Approach tothe Analysis of Complex Systems and Deci-

AUGUST 1994


6/47

ProgrammingLanguages forParal lel Processingedi ted by David B . Skil l icornand Domenico Tal iaDiscusses programming lan-gua ges for parallel processing ar-

ch i tectures and descr ibes theimplementation of various para-digms to support different modelsof parallelism. The book providesan overview of the most importantparallel programming languagesdesigned over the last decade a ndintroduces issues and conceptsrelated to the development of par-allel software. The text covers par-allel languages currently used todevelop parallel applications inmany areas from numerical t o sym-bolic computing.

384 pages. 1994. 0-8 186-6502-5 .45.00 Members 34.00

6502-01

Advances inUltra-DependableDistributed Systems

edi ted by N.j Suri, C.J. Walter,and M.M. HugueFocuses o n fau l t to le ranceconcepts and hard rea l- t ime

perspectives that apply jointly toultra-dependable systems. Thehook emphasizes the theoreticalbasis for achieving dependabilityof objectives and presen ts a varietyof system models built on theseprinciples. It is written for systemdesigners and researchers in thefields of fault tolerance and real-time systems an d presents system-level perspectives on a multitudeof issues related to this topic.

480 pages. 1994. 0-8 186-6287-5.48.00 Members 36.00

cJi /Og 6287-0

IEEE COMPUTERSOCIETY PRESS

Call toll -free:1 -800-CS-BOOKSFax: 714)821-4641

sion Processes, IEEE Trans. Systems, Man,and Cybernetics,Vol. 3,No. 1, Jan. 1973,pp. 28-44.

16. E.H. Mamdani, Application of Fuzzy Al-gorithms for Control of Simple DynamicPlant, Proc. Institution o Electrical Engi-neers, Vol. 121,No. 12,1974, pp. 1585-8.issue on applications of fuzzy logic controlto industry, Fuzzy Sets and Systems, Vol. 32,No. 2, 1989.

17. T. Yamakawa and K. Hirota, eds., special

18. C.C. Lee, Fuzzy Logic in Control Systems-Parts 1 and 2, IEEE Trans. Systems,Man, and Cybernetics,Vol. 20,No. 2, Mar.1990, pp. 404435.19. T. Yamakawa, Stabilization of an InvertedPendulum by a High-speed Fuzzy LogicController Hardware System, Fuzzy Sets

and Systems, Vol. 32,No. 2, 1989, pp.161-180.20. K. Uomori et al., Automatic Image Stabi-lizing System by Full-Digital SignalPro-cessing,IEEE Trans. Consumer Electron-ics,Vol. 36,No. 3,Aug. 1990, pp. 510-519.21. M. Sugeno et al., Fuzzy Algorithmic Con-trol of a Model Car by Oral Instructions,Fuzzy Sers and Systems, Vol. 32,No. 2,1989, pp. 135-156.22. J.M. Keller and H. Tahani, Backpropaga-tion Neural Networks for Fuzzy Logic,Information Scienc es, Vol. 62,No. 3,1992,pp. 205-221.23. E.H. Mamdani and B.S. Sembi, ProcessControl Using Fuzzy Logic,Fuzzy Sets:Theory and Applications to Policy Analysis

and Information Systems, Plenum Press,New York, 1980, pp. 249-265.24. E. Sandewall, Combining Logic and Dif-ferential Equations for Describing Real-World Systems,Proc. First Intl Con onPrinciples of Knowledge Representationand Reasoning, R.J. Brachman, H.J.Levesque, and R. Reiter, eds., MorganKaufmann, San Francisco, Calif., 1989, pp.412420.25. A. Collins and R. Michalski, TheLogic ofPlausible Reasoning: A Core Theory, Cog-nitive Science, Vol. 13,No. 1, 1989, pp.149 .26. C. Sammut and D. Michie, Controlling aBlack Box Simulation of a Space Craft,AIMagaz ine , Vol. 12, No. 1, 1991,pp.

56-63.27. K. Tanaka and M. Sugeno, Stability Analy-sis and Design of Fuzzy Control Systems,Fuzzy Sets and Systems, Vol. 45,No. 2,1992, pp. 135-156.

28. W. Pedrycz, Fuzzy Modeling: Fundamen-tals, Construction, and Evaluation,FuzzySets and Systems, Vol. 41,NO. 1, 1991 pp.1-15.

29. C. von Altrock, B . Krause, and H.J. Zim-mermann, Advanced Fuzzy Logic Controlof a Model Car in Extreme Situations,Fuzzy Sets and Sy stems, Vol. 48,No. 11992,pp. 41-52.30. D. Driankov and P. Eklund, WorkshopGoals, distributed by the authors at IJCAI91 Workshop on Fuzzy Control.31. J.R. Bourne et al., Organizing and Under-standing Beliefs in Advice-Giving Diagnos-tic Systems. IEEE Trans. Knowledge andData Engineering, Vol. 3, No. 3,Sept. 1991,pp. 269-280.32. J. McCarthy and P.J. Hayes. Some Philo-sophical Problems from the Standpoint ofArtificial Intelligence, Machine Intelli-gence, Vol. 4, Edinburgh Univ. Press, Edin-burgh, Scotland, 1969, pp. 463-502.33. R.L. Watrous and L. Shastri, LearningPhonetic Features Using ConnectionistNetworks, Proc. 1 0th Intl Joint Con onAI, Morgan Kaufmann, San Francisco,Calif., 1987, pp. 851-854.34. J.L. Elman, Finding Structure in Time,Cognitive Science , Vol. 14,No. 2, 1990, pp179-21 1.35. C.C. Ku, K.Y. Lee, and R.M. Edwards,Improved Nuclear Reactor TemperatureControl Using Diagonal Recurrent NeuralNetworks, IEEE Trans. Nuclear Science,Vol. 39,NO. 6,1992, pp. 2298-2308.

CharlesElkan is an assistant professor in theDepartment of Computer Science and Engineer-ing at the University of Califomia, San Diego.His main research interests are in artificial intel-ligence. With students and colleagues, he hasworked recently on leaming algorithms for DNAand protein sequence analysis, algorithms forreasoning about database queries and updates,methods of formalizing commonsense knowl-edge, and other topics. In the field of knowl-edge-based systems, his paper with A. Hekmat-pour, Categorization-Based DiagnosticProblem Solving in the VLSI Design Domain,won a best paper award at the 1993 IEEE Con-ference on Artificial Intelligence for Applica-tions. Before joining UCSD in 1990, Dr. Elkanwas a postdoctoral fellow at the University ofToronto. He earned his PhD and MS at Come11University in computer science, and his BA inmathematics at Cambridge University. CharlesElkan can be contacted at the Dept. of ComputerScience and Engineering, UCSD, La Jolla, CA92093-0114;Intemet: [email protected]

I EXPERT


7/47

I

T h e Unique Strength ofFuzzy Logic ControlHamid R. Berenji, Intelligent Inference SystemdNASA Ames Research CenterI am pleased to see that Elkan has revisedhis paper based on comments from fuzzylogic experts. His reference to Dubois andPrade indicates that he has realized, finally,that his alleged new discovery has longbeen known by specialists in fuzzy andmultivalued logics.

Unfortunately, the new version still con-tains many misunderstandings and errors. Iwill briefly respond to some of them,avoiding a discussion of the supposedlystartling proof about the purported incon-sistency of fuzzy logic, which is covered inresponses by Enrique Ruspini and others. Iwill confine my comments primarily to afundamental misunderstanding that is thesource of many of Elkans mistaken asser-tions about the use of fuzzy logic in heuris-tic control and expert systems.

Elkan lists a number of powerful fea-tures of fuzzy-logic control, but then erro-neously concludes that none is unique tofuzzy logic. He fails to realize that theunique strength of fuzzy-logic control is itsdependence on fuzzy-set theory and itsrepresentational capabilities. The smallnumber of rules typical in these systems isnot the result of mere luck, but the directconsequence of the fuzzy predicates thatappear in the rules. Each of these predi-cates covers a wide range of state variablevalues while facilitating interpolation ofrule consequents. Fuzzy sets provide for ageneral, yet compact characterization ofsystem state that requires fewer rules.of fuzzy controller knowledge is simplywrong. Recent fuzzy-logic controllers,developed for more challenging tasks, usehierarchical fuzzy control methods. Exam-ples include the helicopter control devel-oped by Sugeno and his collaborators at theTokyo Institute of Technology (a systemthat can appear trivial only to those unfa-miliar with control theory), and the con-troller for a three-linked inverted pendulumdeveloped at Aptronix. In applications such

Elkans assertion about the shallowness

AUGUST 1994

as these, the result of the first level of con-trol is used in deriving control rules for thesecond set, and so on. These examplesprove that fuzzy-logic control systems canbe developed to reason with considerabledepth of complexity. Similarly, the controlmechanisms for the local-motion control ofSRIs autonomous robot2 rely on severaldeliberation levels to determine the rele-vance level of each control rule (by evalu-ating the operational environment charac-teristics); to identify current goals and theirstate of achievement; to activate controlrules according to the current context; andto blend their control recommendations.

At any rate, the depth of a reasoningprocess as Elkan seems to understand it isnot even a well-defined measure of infer-ential system complexity. This is seen inthe fact that the two-level forward chainA -+ ( B -+c s often compiled in real-time applications (such as control systems)into the single-level rule A A B -+ C to sim-plify and speed computation. This simplifi-cation mechanism, which turns what Elkanwould consider complex into an equiva-lent simple version, is used to introducecontextual and goal-dependence considera-tions into the reasoning chain both in theSRIs mobile robot controller and in ourown two-goal inverted pendulum.

Using fuzzy sets to describe a generallinguistic variable a lso significantlyreduces the complexity of the searchprocess in fuzzy systems that learn fromexperience. Elkan correctly points out thatusing fewer rules simplifies the credit as-signment problem, but he fails to realizethat this is a consequence of using fuzzylogic rather than an indicator of its currentor future applicability. This feature is desir-able in any control system, as is seen in thefuzzy-logic controller developed at NASAAmes for the Sp ace Shuttles rendezvousand docking operation^.^ This controllerlearns to improve itself from experienceusing reinforcement learning technique^:,^

a complex task that would have been verydifficult, if not impossible, if other sym-bolic control techniques had been used.

In summary, I see two major misunder-standings in Elkans paper. First, it relies ona theorem that is irrelevant to fuzzy logic toargue that the methodology is paradoxical.Second, it fails to note that the advantagesprovided by fuzzy-set constructs give fuzzycontrol a unique methodological strength

fact Elkan mistakenly interprets astechnological immaturity.References1. H.R. Berenji etal., A Hierarchical Approachto Designing Approximate Reasoning-BasedControllers for Dynamic Physical Systems,in Uncertainty in Arti$cial Intelligence, P.P.Bonissone etal, eds. North-Holland, Amster-dam, 1991, pp. 331-343.2. A. Saffiotti, E. Ruspini, and K. Konolige,Blending Reactivity and Goal-Directednessin a Fuzzy Controller, Proc. Fuzzy Logic, inProc. Second IEEE Intl Con Fuzzy Sys-t e m , IEEE Computer Society Press, LosAlamitos, Calif., 1993,pp. 134-139.H.R. Berenji et al., Space Shuttle AttitudeControl by Fuzzy Logic and ReinforcementLearning, in Proc. Second IEEE Int Con

Fuzzy Systems, IEEE Press, Pistcataway,H.R. Berenji and P. Khedkar, Learning andTuning Fuzzy Logic Controllers throughReinforcements, IEEE Trans. on NeuralNetworks, Vol. 3, No. 5 , 1992, pp. 724-740.5 . H.R. Berenji, An Architecture for Design-ing Fuzzy Controllers using Neural Net-works, in Int 1J. Approximute Reasoning,Vol. 6., No. 2, Feb. 1992,pp. 267-292.Hamid R.Berenji is a senior research scientistand principal investigator on intelligent controlin the AI branch of the NASA Ames ResearchCenter. He was a program chair for the IEEEInternational Conference on Neural Networks,and was a program cochair of the 1994IEEEConference on Fuzzy Systems. He serves on theeditorial board of several technical publications,and is an associate editor of IEEE Transactionson Fuzzy Systems and IEEE Transactions onNeural Networks. He is a member of IEEE, andchairs the Neural Networks Councils TechnicalCommittee on Fuzzy Systems. Hamid Berenjican be reached at [email protected]

3.

N.J., 1993, pp. 1396-1401.4.

I
mailto:[email protected]:[email protected]


8/47

Broader Issues At StakeA Response to E lk a nB. Chandrasekaran, Ohio State University

The fuzzy set approach has clearly cap-tured the interest of many researchersaround the world and has been used tobuild applications of various sorts, ofwhich fuzzy control applications are cur-rently the most prominent. The approach,however, remains controversial. While thiscontroversy has many sources, there arerelatively few places where the argumentsare set out in a fashion that allows debate.It is thus useful to have both CharlesElkans analysis of the fuzzy set approachto representing uncertainty, and his exami-nation of which features of fuzzy set theoryare responsible for the success of fuzzycontrol systems. In particular, I commendElkan for making his arguments aboutthese techniques in a nonpolemical way,letting technical arguments and results domost of the talking.

In Elkans first argument, he claims thatthe axioms of fuzzy set theory, in conjunc-tion with what appear to be a number ofreasonable versions of logical equivalencebetween sentences, lead to a collapse oftruth functions into just two valuesfate that fuzzy set theory was expresslymeant to avoid.

As Elkan points out, a result similar tohis collapse theorem was already known toresearchers within the fuzzy set community(Dubois and Prade). My understanding isthat they werent too worried by this result,since they think that the traditional notionof logical equivalence or any of its variantsshould be abandoned fo r fuzzy sets. Thisresponse seems to me to be formally rea-sonable, but I think in practice it would behard to work with a system in which logi-cal equivalence itself is a fuzzy relation.Ultimately, we will have to see how muchreally interesting work is possible with thisnotion of fuzzy equivalence.

In the second argument, Elkan assertsthat when fuzzy control systems that workwell are analyzed, the real source of theirsuccess seems to be not the inferential ca-pabilities of fuzzy set theory (derived fromthe theorys composition axioms) butrather a combination of things exclusive offuzzy set axioms. Among these are the abil-ity to represent certain things as continuousquantities rather than all-or-nothing quanti-ties; certain heuristic techniques hat arethemselves outside fuzzy set theory -toget the right parameters for the problems;and the fact that there is little complex rule-chaining going on. A number of alterna-tives and rivals to fuzzy set theory wouldwork as well in those applications.

Part of Elkans point hat the successof fuzzy control systems thus far is notreally a full test or proof of the axioms andclaims of fuzzy set theory s actually aninstance of a larger phenomenon in AI. Ithink that Elkans point can be madeagainst the claims of not only fuzzy controlproposals, but also against a number ofother proposals in AI, including the rivalsof fuzzy sets, such as belief nets.

The general problem is a kind of creditallocation problem and can be stated asfollows. Given some mechanism M , andsome specific task T , suppose I write a pro-gram P , using M as the basis for the pro-gram. And, let us say that P does well inthe task T . What conclusions can we drawabout mechanism M from the success of Pin tackling T ow much credit should Mget for the success of P?A historical perspective.In the late1970s, rule-based expert systems werecapturing the imagination of many people.Mycin and R1 were great successes. In the

above terminology, rule-based languageswould be M , Mycin and R1 would be thePs, and simple diagnosis and configura-tion would be the corresponding tasks, T.The success of the two programs led toclaims about the power of the rule-basedmechanism. Similar examples involvingother mechanisms, such as belief nets andtruth maintenance systems, can beconstructed.In a series of articles (such as one from1986, for example), I made the followingpoints regarding rule-based systems as amechanism. The specifics of the mecha-nism were incidental in accounting formany aspects of why the programs worked.The mechanism was computation-univer-sal, and of course could be used to imple-ment any other mechanism or strategy. Ahigher order strategy lassification inthe case of Mycin, or linear sequencing ofsubtasks in the case of R1 -was the prob-lem-level strategy that was responsible forthe programs performance. Not only wasthe rule-based mechanism not the directcause of the good performance, but theyactually hid the reasons for success: Thehigher level strategies were programmed inthe language of the lower level mechanism.The strategies had to be brought out byanalysis, rather than seen by a direct in-spection of the mechanism. The limitationsand success of Mycin and R1 could bemore insightfully analyzed by examiningthe adequacy of classification for diagnosisand linear subtasking for configurationdesign. Clancey also analyzed Mycin as aheuristic classifier2 and pointed out thepower such high-level analysis brought tobuilding diagnostic systems. In the lastdecade or so, there has been a decisive shiftin emphasis in the field of knowledge-based systems from mechanisms at the rulelevel to phenomena at the task level.

10 IEEE EXPERT


9/47

Thus, given an M-T-P triad, it is not al-ways easy to decide exactly what the role ofM was in the success of P in achieving thatversion of T . This is not to say that Msproperties are irrelevant. There are severalways a given mechanism might play less ofa role than is readily apparent, among them:

M might simply be one among manyperfectly reasonable lower l evel mecha-nisms to implement the causally morerelevant higher level mechanism.M might have features which actuallyimpede good performance f or the classof problems in T. This might not beevident from the specific instance of Tfor which P was written. In thisinstance, the troublesome features of Mmight not have been used or their effectmight be minimal. Fuzzy set theory hasbeen successfully applied to simpleversions of the control problem. AsElkan argues, however, the problematicfeatures of the theory might start show-ing up as more complex versions of thecontrol problem are encountered.In some cases, M has many more fea-tures than needed for capturing theessence of T. Hence, using M to build Pfor solving T calls for making commit-ments to details that are either irrelevantor that detract from building good Ps.However, when such a program is built,it takes quite a bit of analysis to tellwhich features of M are necessary.There is often a tendency, especiallyamong those who are enthusiasts of Mfor other reasons, to ascribe the successof P to those features of M that wereactually incidental to Ps success. Evenmore seriously, success with M mightlead to its use for more complex ver-sions of T ,where these additional fea-tures actually make building successfulPs more difficult. Elkan makes a goodcase for this possibility as fuzzy controlapproaches are applied to more com-plex control problems.

The history of Mycin is another source ofwisdom about the role of uncertainty-han-dling mechanisms. When Mycin came out,

much was made of the uncertainty-factorformalism. Debates raged about this for-malism versus Bayesian formalism versusfuzzy set formalisms as an appropriate cal-culus. Cooper and Clancey got the idea ofdoing an experiment in which they coars-ened the uncertainty factors in Mycinsknowledge base rules and examined howwell the modified Mycin did in the samecas es3 The modified Mycin solved theproblems as well as the original Mycin.

How could this be? Clearly the calculusas such didnt play as fundamental a role inthe ability of Mycin to solve the problems.The fine structure of uncertainty didntreally matter. The knowledge base hadenough knowledge to establish or reject theconclusions in a near-definitive way. Noneof the conclusions were based on evenmoderate distinctions in uncertainty be-tween the candidates. There were multipleways to get to or reject conclusions, andeven moderate changes in the uncertaintiesdidnt matter. The correct conclusions werevery strongly established, and the incorrectconclusions were very strongly rejected.Mycin did well, not because of the finepoints of its uncertainty calculus twould have done just as well with any of anumber of alternative calculi -but be-cause of the robustness of its knowledgebase. This is another instance of the alloca-tion of credit problem.The nature of fuzzy theory

I have followed fuzzy set theory almostfrom its inception. The theorys claim thatall senses of uncertainty in human knowl -edge cannot be reduced to some version ofprobability has always struck me as right.One of the most useful consequences of thefuzzy set movement has been the identifi-cation of different types of uncertainty. Inparticular, the theory suggests that manypredicates such as bald, most, andlarge are neither binary predicates, norare they simply probabilistic. This alsoseems to me to be true. However, the spe-cific solutions offered and claims made byfuzzy set theory, and the way they haveoften been applied to problems like control,are problematic for me.

A psychological theory?At the heart offuzzy set theory is an ambiguity about thenature of the theory, and how one goesabout validating it. If it is a psychologicaltheory hat is, a theory of how humansdeal with certain types of uncertaintywe would need certain kinds of evidenceabout human behavior in uncertainty han-dling. I am unconvinced that fuzzy set the-ory is a psychological theory. I have notdone an extensive literature survey, but thework of Kempt0n~3~aises doubts thathuman behavior in uncertainty handlingfollows the axioms of fuzzy set theory.Even if it turns out that the theory doescorrespond to human behavior in this area,we must then decide what kinds of scalingand rationality properties the relevanthuman behavior has before it is used tomake machines that make decisions.

Two relevant analogies are found incommonsense physical reasoning and rea-soning about probabilistic uncertainty. Weall have approximate rules about how thephysical world behaves: If we push this alittle, this will move a moderate distance,while the other object would hardly move.We use such rules when we have to predictbehavior in the physical world, but theserules are typically chained over a few steps.When a problem calls for many steps, theserules start accumulating large errors (to beexpected), but curiously, they also startaccumulating ambiguities of another sort.So many alternative possibilities are gener-ated that we adopt all kinds of goal- andcontext-specific strategies to select a fu-ture history over other alternatives. Or, ifwe are physicists, we resort to a pencil andpaper for more exact calculations even ifwhat we really want are approximate an-swers. Clearly such approximate reasoningby humans does not scale up very well.

In the case of probability assessmentbehavior, human behavior is not alwayswhat an outside observer might regard asrationaL6 Thus, in addition to the scalabil-ity problem, there is the problem of ratio-nality of human behavior as well.

AUGUST 1994 1 1


10/47

The point that I want to make with thesetwo examples is that, i n many domains,automated decision systems should not bedesigned to emulate human behavior. Thus,even if fuzzy set theory turns out to be amodel of how humans handle a certain typeof uncertainty, we need additional argu-ments to make th e theory the basis of auto-matic control.A mathematical theory?On the otherhand, fuzzy set could be a theory of an ab-stract mathematical system whose proper-ties model some domain of human interest.Examples of such systems are arithmeticand deductive logic. The formalization ofarithmetic starts with our intuitive notionsabout numbers, but it is not a psychologicaltheory. It posits a world of numbers andoperations on them, and the formalizationis an attempt to capture the properties ofthis world. We can in fact construct theabstract world, recognize its objects as thefamiliar numbers and perform operationson them, and then verify those operationsagainst the predictions of the axiomatiza-tion. For example, we can multiply 2 and 3,and check if the axiom system in fact gen-erates the number 6 for the answer.

If fuzzy set theory is a theory of an ab-stract world whose constituents are uncer-tainties of certain types, and whose opera-tions are the sort of things we do when wecombine uncertainties, then the theory has togive two kinds of evidence. First, there mustbe evidence that such an abstract world in-deed exists. Many abstract worlds that canbe postulated fail to exist because their ax-ioms lack a certain internal coherence. Sec-ond, it must give evidence that the fuzzy setaxioms capture the operations of this world.Establishing that such an abstract worldexists is actually quite hard. In fact, I think itis quite possible that there is no abstractworld of uncertainty combination of thetype that fuzzy set theory attempts to cap-ture. In any case, fuzzy set theory has toworry about validation of its assumptionsand about the existence of an abstract calcu-lus for combining this kind of uncertainty.

What do I mean by such an abstractworld may not exist? Again, the analogyof qualitative physics is relevant. We knowthere is a real physics, whose laws relatevalues of some state variables to the valuesof other state variables. If we have an exactvalue for the independent variables, we cancalculate, using these laws, the exact val-ues of the dependent variables.

The equations of physics are not a psy-chological theory. However, consider theordinary commonsense reasoning about thephysical world that I discussed earlier. Peo-ple do make qualitative predictions aboutthe physical world in response to qualita-tive changes in some state of the world. AsI said, the qualitative rules people havecannot be chained into long inferences: Theambiguities multiply, resulting in too manypossible future histories. Which one of thehistories will be realized often depends ona more exact value for some variables thanwe can get from qualitative rules alone. Ihave described elsewhere a number ofstrategies people use to handle such anexplosion of possibilities, but almost all ofthe strategies depend on the problem-solv-ing goal and ~o nt ex t. ~he conclusion is notthe result of applying an abstract, context-independent calculus. In short, there is noqualitative physics that is a homomorphismof the quantitative physics such that thequalitative physics gives answers that arejust qualitative versions of the answersgiven by the quantitative physics.

With respect to uncertainty handling,many researchers seem to be looking for asimilar abstract system that may not exist.They are looking for a calculus of uncer-tainty handling which has the followingfeatures:

The semantics of its uncertainty termscapture the intuitive meaning of uncer-tainty terms that people use in theircommonsense behavior.The operations of combination in thecalculus capture human behavior whentheir uncertainties are combined.

This assumes that there is in fact a calculusthat underlies the combining of uncertain-

ties through human common sense. What ifhuman behavior, in combining everydayuncertainties, is really governed by a com-bination of goal- and context-dependentstrategies that make use of a rich body ofdomain-specific knowledge? What if thiscannot be captured by a calculus of thetype that fuzzy set or other theorists arelooking for? If human conclusions are ro-bust with respect to moderate changes inthe uncertainty values of the constituentss in the Mycin experiment by Cooperand Clancey hen the real explanation ofhuman behavior is not given by a calculus,fuzzy or otherwise, but by the complexcollection of situation- and goal-specificknowledge that people bring to bear oninstances of the problem.

Like the case in qualitative reasoningmentioned earlier, people might in factavoid anything like a chain of uncertaintycombination. If the conclusion seems ro-bust with respect to moderate changes inthe uncertainty values of its constituents,people feel comfortable with the conclu-sion. If not, they might get additional dataso that a robust conclusion can be reached,postpone making a decision, or make deci-sions that may not in general be consideredthe best, but that are fine for the specificgoal at hand. In other words, the same val-ues of uncertainties fo r two constituentbeliefs would lead to a conclusion with anuncertainty value A in one situation, anuncertainty value B in another, additionalinformation gathering in a third, explicituse of probability models in a fourth, andsimply a shrugging of shoulders and nodecision at all in a fifth. If this is the case,then the search for a calculus of the typefuzzy set theorists (and many others in theresearch community concerned with mod-eling uncertainty in reasoning) are lookingfor is likely to be futile. The issue is illus-trated well in Elkans example of his expertsystem, for which neither the probabilityscheme nor the fuzzy set approach wasappropriate.

The problem with fuzzy set theory, in

12 I EXPERT


11/47

I

my view, is not in the mathemati cs of theformal system. It is clearly a mathematicalsystem of some interest. However, a theoryof this type has to be judged either as apsychological theory or as a theory that hascaptured an abstract calculus that underliessome type of human reasoning. As I havejust argued, an abstract calculus of this typemay not exist.The problem of context. In the 1980s,mycolleagues and I were faced with a similarproblem with uncertainty in medical diag-nosis. Physicians have to come up with anassessment of the likelihood of somedisease for which a number of data werepotentially relevant. The relation betweenthe data and the strength of belief in thedisease was of course a classic example ofuncertainty. For various reasons ot theleast of which was that we didnt have thedata needed to use the frequency version ofthe probabilities for this relationship eneeded a technique to model human exper-tise in this area. Bayesian approaches,fuzzy set theory, Dempster-Shafer theory,and uncertainty factor calculus were allavailable to us. All these calculi shared oneimportant property or assumption abouthuman expertise -that there was a situa-tion- and goal-independent way of combin-ing uncertainties.

For example, if two symptoms, s l and s2,were relevant to making a decision aboutdisease d, such calculi would provide waysin which evidence for sl and s2 would becombined to give evidence about d, andadditionally, that the rule of combinationitself is independent of the specific labels fors , s2, and d. If the evidence for sl is large,and s2 is medium, the rule would specifywhat the evidence for d would be. But therule cannot be one thing where s l is biliru-bin, s2 is alkaline phosphatase, and d isliver disease, while another rule is usedwhere sl is cholesterol level, s2 is alka-line phosphatase, and d is heart disease.

We found, however, that expert behaviorin uncertainty combination in fact differedfrom context to context, and problem-solv-ing goal to problem-solving goal. We had

to resist the mathematical attractions of anabstract calculus. Instead, we developed aformalism in which we could incorporatethe uncertainty-combining behavior ofexperts,8 who were compiling a complex ofbackground knowledge in such context-specific rules. It was also important to notethat the chaining length was relativelysmall: Two or three steps were all that wereused. If the problem called for much longerchaining, we took it as a sign that we weremodeling the expert knowledge inaccu-rately, and sought additional pieces ofknowledge that would shorten the chain.

Fuzzy set theory has done quite well as aformal mathematical system. Whether itstheorems are interesting is a subjectiveopinion among mathematicians, but a largebody of mathematical work exists. Wheremore work needs to be done is in establish-ing that fuzzy set theory actually capturessomething real and can make a pragmaticdifference, for the right reasons.initiating a debate about the properties offuzzy set theory. I have argued that thepoints Elkan makes about fuzzy sets arereally an instance of problems that apply toa number of other AI mechanisms andideas, and specifically to many other pro-posals for subjective calculi for handlinguncertainty. The issues raised are large inscope, and not only the fuzzy set commu-nity, but the AI community as a wholecould benefit from giving them thought.

I think Elkan has performed a service by

Referentes1. B. Chandrasekaran, Generic Tasks inKnowledge-Based Reasoning: High-LevelBuilding Blocks for Expert SystemDesign, IEEE Expert, Vol. 1,No. 3, Fall1986, pp. 23-30.2. W.J. Clancey, Heuristic Classification,Artificial Intelligence Vol. 27, No. 3, 1985,pp. 289-350.

3.

4.

5.

6.

7.

8.

B.G. Buchanan and E.H. Shortliffe, Rule-Based Expert Systems: The Mycin Experi-ments o the Stanford Heuristic Program-ming Project, Addison-Wesley, Reading,Mass.. 1984.W. Kempton, Category Grading and Taxo-nomic Relations: A Mug is a Sort of a Cup,American Ethnologist Vol. 5 , No. 1, 1978,pp. 44-65; revised version reprinted in Lan-guage, Culture, and Cognition: Anthropo-logical Perspectives, R.W. Casson, ed.,Macmillan, New York, 1981.W. Kempton, The Folk Classification ofCeramics: A Study of Cognitive Prototypes,Academic Press, San Diego, 1981.A. Tversky and D. Kahneman, JudgmentUnder Uncertainty: Heuristics and Biases,Science, Vol. 185, 1974, pp. 300-306.B. Chandrasekaran, QP is More thanSPQR and Dynamical System Theory,Computational Intelligence, Vol. 8, No. 2,1992, pp. 216-222.B. Chandrasekaran and S . Mittal, Concep-tual Representation of Medical Knowledgeby Computer: MDX and Related Systems,Advances in Computers, Vol. 22, AcademicPress, 1983, pp. 217-293.

B. Chandrasekaran s director of the Labora-tory for AI Research and a professor of computerand information science at Ohio State Univer-sity. His research interests include knowledge-based systems, using images in problem solving,and the foundations of cognitive science and AI.Chandrasekaran received his PhD from the Uni-versity of Pennsylvania in 1967. He is editor-in-chief of IEEE Expert, a fellow of the IEEE andAAAI, and a member of the IEEE ComputerSociety. B. Chandrasekaran can be reached atthe Department of Computer and InformationScience, Ohio State Univ., 591 Dreese Labs,2015 Neil Ave., Columbus, OH 43210-1277.

AUGUST 1994 13


12/47

A Better Path to DuplicatingHuman ReasoningChristopherJS. desilva and Yianni Attikiouzel, Universityof WesternAustraliaThe paradox that arises from Elkans Theo-rem 1 is mild in comparison to some of thelogical problems that lurk behind the ap-parently innocent equations in Definition 1.In fact, although fuzzy logic has been pro-moted as a way of writing programs thatcarry out inference in the same way a per-son might, the equations of Definition 1can lead inescapably to conclusions that nohuman being would accept.Consider a simple example: You knowthat the airplane on which John Doe wastraveling has crashed in some remote loca-tion, but you have no information aboutwhether anyone on board has survived. Inthis situation, you might make the follow-ing assignment: t(John Doe is alive) =0.5. The equations of Definition 1 wouldlead you immediately to t(John Doe isdead) = 0.5. While this is a reasonableassignment, it would in tum lead you tot(John Doe is both dead and alive) = 0.5.Thus, there is an element of truth in thestatement John Doe is both dead andalive. However, any rational person willargue that it is impossible fo r John Doe tobe both dead and alive, so that the state-ment John Doe is both dead and alivemust always be false, and have a truthvalue of zero.We can imagine putting a fuzzy logicsystem to the Turing test on the matter ofJohn Does well-being:Interrogator:Respondent 1: It is half-true that John DoeRespondent 2: I dont know.

Is John Doe alive ?is alive.

Interrogator: Is John Doe dead ?Respondent 1 : It is half-true that John Doeis dead.

Respondent 2: I dont know.

Interrogator: Is John Doe both dead andalive?

Respondent 1: It is half-true that John Doeis both dead and alive.

Respondent 2: It is impossible for John Doeto be both dead and alive.

While there is an element of caricaturein this dialogue, it serves to highlight theproblem. It is clear that if A is any proposi-tion with a non-zero truth value, the equa-tions of Definition l will lead to the con-clusion that the truth value of the compoundstatement (A and (not A)) is also non-zero.This is a very simple example of how fuzzylogic diverges from human logic. It is to beexpected that this divergence will increasewith the complexity of the inference process.

Of course, people have been assigningtruth values between zero and one to makeinferences since the time of Laplace, on thebasis of probability theory. As Cox hasshown, using the axioms of probabilitytheory is essentially the only way to carryout this form of inference and remain con-sistent with human reasoning ny otherway will lead to contradictions and incon-sistencies. However, proponents of fuzzylogic appear to be unaware of Coxs workand that of Jaynes2 and T r i b ~ s , ~here thequestion of how to write programs thatmake inference based on incompleteknowledge is discussed.

A s Cheeseman4 pointed ou t for AI in gen-eral, the bottom line is that if you want towrite a program or build a machine thatwill perform inference in the same way aspeople, then you must build the basic equa-tions of probability theory into it, or facethe inevitable outcome that it will not per-form as required.

Perhaps the real paradox of fuzzy logicssuccess is that proponents hail it as a suc-cessful technology despite the fact that it isincapable of performing as they claim itcan and does.References1. R.T. Cox, The Algebra o fproba ble Infer-ence, Johns Hopkins Press,Baltimore, 1961.2. E.T. Jaynes, How Does the Brain do Plau-sible Reasoning? Tech. Report 42 I , Mi-crowave Laboratory, Stanford Univ., 1957.

M. Tribus, Rational Descriptions, Deci-sions, and Designs, Pergamon Press, NewYork, 1969.4. P. Cheeseman, An Inquiry into ComputerUnderstanding, Computer Znrelligence,Vol. 4, No. 2, Feb. 1988, pp. 58-66.

3.

Yianni Attikiouzel is a professor of electricaland electronic engineering at the University ofWestern Australia where he is director of theCentre for Intelligent Information ProcessingSystems. His work has been published in andpresented at more than 120 international journalsand conferences, and he is the author of twobooks. He is a member of the Industry Researchand Development Board of the CommonwealthDepartment of Science and Technology, and sitson its Services and Consumer Products Commit-tee. Yianni Attikiouzel can be contacted at theCentre for Intelligent Information ProcessingSystems. Department of Electrical and Elec-tronic Engineering. University of Western Aus-tralia, Nedlands, WA 6009 Australia; phone: 61 9380 3134; fax: 61 9 380 1101; Internet:[email protected] desilv a is a research fellow at theCentre for Intelligent Information ProcessingSystems at the University of Western Australia.He is currently working on the theory and appli-cation of artificial neural networks. His otherresearch interests include syntactic patternrecognition and Bayesian inference. He can bereached at the Centre for Intelligent InformationProcessing Systems, Department of Electricaland Electronic Engineering, University of West-em Australia, Nedlands, WA6009 Australia;phone: 61 9 380 1765; fax: 61 9 380 1101; Inter-net: [email protected]

14 IEEE EXPERT


13/47

Partial Truth is not UncertaintyFuzzy Logic versus Possibilistic LogicDidierDubois and Henri Prade, UniversitkPaul Sabatiw de ToulousePhilippe Smets, Universit6Libre de BruxellesCharles Elkan has questioned fuzzy logicand cast serious doubts on the reasons forits success, arguing that fuzzy logic c ol-lapses mathematically to two-valuedlogic. We completely disagree, and weespecially object to two points:(1) Elkans proof uses too strong a notionof logical equivalence. The particular

equivalence he considers, while validin Boolean algebra, has nothing to dowith fuzzy logic.( 2 ) Elkan claims that De Morgans alge-bra allows very little reasoning about

collections of fuzzy assertions, al-though he correctly states that whenlogical equivalence is restricted to DeMorgan algebra equalities collapseto two truth values is avoided.

Furthermore, Elkan fails to understand theimportant distinction between two totallydifferent problems that fuzzy-set-basedmethods address. These are the handlingof gradual (thus non-Boolean) propertieswhose satisfaction is a matter of degree(even when information is complete) onthe one hand, and the handling of uncer-tainty pervading Boolean propositions, theuncertainty being induced by incompletestates of knowledge that are represented bymeans of fuzzy sets, on the other hand.The first problem requires the plain use offuzzy sets, while the second is the realm ofpossibility the0ry~9~nd possibilistic logic5.We now discuss in greater detail the pointsabove and the distinction between truthfunctional fuzzy (multivalued) logic andnon-fully compositional possibilistic logic.

~

Fuzzy logic equivalence is not classical.Elkan claims that i n fuzzy logic, four re-quirements hold for any assertions A and B,t being a truth assignment function suchthat VA, t(A) E [0,1]:

t(A A B ) = min(t(A), t ( B ) ) (1)t(A v B ) = max(t(A), t ( B ) ) ( 2 )t(1A) = 1- (A) (3)equivalent. 4)t(A) = t (B) f A and B are logicallyWhile Equations 1-3 are indeed the

basic relations governing degrees of truthin fuzzy logic (as well as fuzzy set mem-bership degrees) as proposed by Zadeh?Equation 4 where logically equivalent i sunderstood in a stronger sense than theequivalences induced by 1-3) has neverbeen seriously considered by any author inthe fuzzy-set literature. (There are, as canbe expected, a few erroneous attempts atthe subject in a corpus of more than 10,000published papers). Obviously, some classi-cal logic equivalences still hold with fuzzyassertions obeying Equations 1-3, namely,those allowed by the De Morgan structureinduced by 1-3, such as

A AA = A ;A v A =A (idempotency)A A ( B v C) = (A AB)v (A A C ) ;A v ( B AC)=(A v B ) A (A v C )(distributivity)

But other Boolean equivalences d o nothold, for instance:

A A - A + Lsince Equations 1 and 3 entail only

t(A A -A) = min(t(A), 1- (A)) < 1/2;and

A v i A + Tsince Equations 2 and 3 entail o ~ l y

t(A v A) = max(t(A), 1- (A)) Z 1/2

where t ( l ) = 0 and t T )= 1. Indeed, asmany authors have emphasized, the failureof contradiction and excluded-middle lawsis typical of fuzzy logic. This is naturalwith gradual properties like tall. For ex-ample, in a given context, somebody whois 1.75 meters high might be consideredneither as completely tall (tall with degree1) nor as completely not tall (tall with de-gree 0). In this case, we might have, forexample, ptall .75) = 0.5 = pYtal1 .75).to binary logic, Elkan uses the logicalequivalence

To establish the collapse of fuzzy logic

-(A A i B )= B v (A A i B ) 5 )postulated as being plausible intuitively.If Equations 1-3 hold, the left-hand part ofEquation 5 can be equivalently written infuzzy logic as

(A A i B ) -A V Bwhile the right-hand part can be equiva-lently written as

B v (TA A i B ) (A v B ) A ( Bv i B ) ,which clearly relates to the excluded-middlelaw. Thus, it is expected that Equation 5 failsto hold in fuzzy logic -and indeed it canbe checked, using Equations 1-3, that acounterexample to Equation 5 is providedby t(A) =0, t (B)=0.5, for instance. Thus,Elkans claim of a paradox in fuzzy logicrelies only on faulty assumptions, or at beston a logical equivalence, the rationale ofwhich is far from natural in the scope offuzzy logic.Gradual and interpolative reasoning.Fuzzy logic is concerned with the handlingof assertions like John is tall sser-tions whose truth is a matter of degree due

1 AUGUST1994 15

I


14/47

to gradual predicates within them. The de-gree of truth of compound expressions canbe easily computed using Equations 1-3.(Although we restrict ourselves here to theoperators minimum, maximum, and com-plement to one, there is a panoply of

that enable us to model differentkinds of AND and OR operations betweenproperties in a multicriteria aggregationperspective.)

More than 20 years ago, R.C.T. Lee9provided the basic machinery for reasoningin fuzzy logic by extending the resolutionrule in accordance with Equations 1-3. Heestablished that if all the truth values of theparent clauses are greater than 0.5, then aresolvent clause derived by the resolutionprinciple always has a truth-value betweenthe maximum and the minimum of those ofthe parent clauses.

We can also use an implication oper atorto model gradual rules,I0 which expressknowledge of the form the more Xi s A,the more Y is B, such as, the taller youare, the heavier you are. This is capturedby the implication defined by

t ( A -+ B ) = 1 if t ( A ) 5 t ( B )= 0 if t(A) > t ( B )

This implication is the natural counter-part of Zadehs fuzzy set inclusion definedby the pointwise inequality of the member-ship functions.6 It is also directly associ-ated with Equations 1-3, since A + B =Tif and only if A A B =A. Such an implica-tion expresses a purely gradual relationshipand has nothing to do with uncertainty.Besides, Takagi and Sugeno have pro-posed an interpolation mechanism betweenn rules with fuzzy condition parts and non-fuzzy conclusions of the form if X is A,and Y is B , then Z = cl,by computing thefollowing output when X =x0and Y = yo isobserved

(6)

(7)where K = min(pA,(xd, pei(yd), i = 1,n.Again, this kind of inference (which iswidely used in fuzzy control) has nothingto do with uncertainty handling, since only

an interpolation between typical conclu-sions is performed, based on degrees ofsimilarity between the input (xo, yo) and theprototypical values in the core of the fuzzyset A, x B,. This similarity is measured bythe coefficients yl which cannot be consid-ered as degrees of uncertainty in any case.In spite of its apparently ad hoc nature,Equation 7 can be justified with one-premised rules using Equation 6 and view-ing the rules as expressing the more XisA , and Y is B,, the closer Z is to cL ndusing appropriately shaped membershipfunctions.l*

As this shows, contrary to Elkans claim,some kinds of reasoning, as exemplified byTakagi and Sugenos, and Lees methods,can be handled in a De Morgan algebraframework.Possibility theory and uncertainty. Inaddition to modeling the gradual nature ofproperties, fuzzy sets can be used to repre-sent incomplete states of knowledge. Inthis second use, the fuzzy set plays the roleof a possibility distribution that provides acomplete ordering of mutually exclusivestates of the world according to their re-spective levels of possibility or plausibility.For instance, if we know only that John istall (but not his precise height), where themeaning of tall is described, in context,by the membership function of a fuzzy set(that is, ptall),hen the greater ptall(x)s, thegreater the possibility that height(John) =x ; the smaller ptall(x)s, the smaller thispossibility.

Given a [O,l]-valued possibility distribu-tion n: describing an incomplete state ofknowledge, Zadeh4 defines a so-called pos-sibility measure n such that

(8)where A is a B o o l e a n proposition (a propo-sition that can only be true or false). It canbe easily checked that for Boolean proposi-tions A and B , we have

n(A) = sup(~ ( x ) ,makes A true}

n(A v B ) = max(n(A), n ( B ) ) (9)but that we only have the inequalityn(A A B ) 5 min(n(A), n ( B ) ) (10)

n the general case (equality holds when Aind B are l og ica l l y independen t ) . Indeed if3 TA, n(A A B )= n ( l ) 0, whilenin(n(A), n(-.A)) = 0 only if the informa-.ion is sufficiently complete for having:ither n(1A) = 0 (A is true) or n(A)= 0 (Ais false). If nothing is known about A, welave n(A) = n(-A)= 1.By duality, a ne-Zessity measure N is associated to n ac-:ording to the relation (which can beviewed as a graded version of the relationbetween what is necessary and what is pos-iible in modal logic)

11)which states that A is all the more necessar-ily true as TA has a low possibility to betrue. It entails

N(A) = 1-n(-A)

N(A A B ) = min(N(A), N ( B ) ) (12:and

N(A v B ) 2 max(N(A), N ( B ) ) . (13:Equations 9, 1 I , and 12 should not be

confused with Equations 2 , 3 , and 1, respec-tively. In 9, 11, and 12 we deal withBoolean propositions pervaded with uncer-tainty due to incomplete information, while1-3 pertain to non-Boolean propositionswhose truth is a matter of degree (the infor-mation being assumed to be complete).Very often, discussions about fuzzy expertsystems or uncertain knowledge base sys-tems get confused because of a lack of dis-tinction between degrees of truth and de-gree of uncertainty. Fuzzy logic, asunderstood by Elkan, is a logic where thetruth status of propositions is multiple-val-ued; that is, there are intermediary truthvalues between true and false (like verytrue, rather true, and so on). On the con-trary, degrees of uncertainty apply to all-or-nothing propositions, and do not modeltruth values but express the fact that thetruth value (true or false) is unknown. Theuncertainty degrees then try to assess whichone of true or false is the most plausi-ble truth value. This distinction was madeby one of the founders of subjective proba-bility theory -De Finetti13 but with afew exceptions (including ourselves) it has

16 IEEE EXPERT

~ -- -


15/47

been quite forgotten by the AI communityin general and by Elkan in particular. Still,we consider this distinction a crucial pre-requisite in any discussion about fuzzy setsand possibility theory and their use in auto-mated reasoning.

Observe also that neither n nor N arefully compositional with respect to A, v ,and 7 his is not surprising, since the onlyway to map a Boolean structure on [0,1]bya fully compositional mappingfi s to havef(A) equal to 0 or to 1 for any A. Truth-functionality in Equations 1-3 is preservedonly by having A and B elements of aweaker structure, namely, a De Morganalgebra. Thus, logics of uncertainty cannotbe fully compositional with respect to un-certainty degrees. Thi s point is also recog-nized by Elkan in the case of probabilitymeasures, and dates back at least to DeFinetti in the 1930s Partial compositional-ity is possible, however; probabilities arecompositional with respect to negation,possibilities with respect to disjunction,necessities with respect to conjunction.Based on his article, however, it seems thatElkan has not heard about possibility the-ory, which is another side of fuzzy sets.

Let us consider Elkans watermelon ex-ample:

wa t e r m e l o n ( x ) =r e d i n s i d e ( x )A g r e e n o u t s i d e ( x )

It is supposed that for some melon m , evi-dence that m is red intemally has strength0.5,and m is green externally with strengthof evidence 0.8. It is not clear what Elkanmeans by strength of evidence in the lightof the above comments . We shall assumethey are indeed degrees of uncertainty,rather than degrees of red and degrees ofgreen. But then the only way to anchor thisdiscussion in the fuzzy logic debate is tointerpret these degrees in possibility theory.Elkans watermelon sentence can be under-stood as N( r e d i n s i d e ( m ) )2 0.5 and N(green-o u t s i d e ( m ) )t .8, expressing that the avail-able information makes us certain to thedegree of 0.5 that m is red inside, and to thedegree 0.8 that it is green outside. A directapplication of Equation I2 leads to

AUGUST 1994

V ( wu t e r m e l o n ( m ) )2 min(0.5,0.8) =0.5, aresult also obtained under an equality form,y Elkan by applying Equation 1 in an inap-propriate way. However, he would like tozonclude that m is a watermelon withstrength of evidence over 0.5.This seems astrange requirement, and one that a proba-bilistic model would not satisfy either (sinceProb(A A B ) 5 min(Prob(A),Prob(B)). In-deed, we are not in a data fusion situationwhere two independent sources provide thesame conclusion with various strengths,I4but in a situation where the logical con-junction of two conditions is required toZonclude that m is a watermelon (namelythe inside redness of m and its outsidegreenness). Note that in case we have bothN(A) t and N(A) t as obtained fromdistinct arguments, we shall conclude thatN(A) tmax(u,a).Reasoning with possibility theory. Inpossibilistic logic, first-order logic formu-las are weighted by lower bounds of neces-sity or possibility measures, which reflectthe uncertainty of the available informa-tion. Possibilistic has been devel-oped both at the syntactic level, wherethere is an inference machinery based onextended resolution and refutation (thelower bound of the resolvent clause neces-sity is the minimum of the lower bounds ofparent clauses necessity measures), and atthe semantic level, where a semantics interms of a possibility distribution over a setof classical interpretations has been provedto be sound and complete with respect tothe syntax. Due to the fact that a possibilitydistribution encodes a preferential orderingover a set of possible interpretations, possi-bilistic logic has been shown to capture animportant class of nonmonotonic reasoningconsequence relations and has capabili-ties for handling partial inconsistency inknowledge bas es5 Moreover, possibilisticassumption-based truth maintenance sys-te md 6 based on possibilistic logic havebeen defined for dealing with uncertainjustifications and ranking environments ina label; they have been successfully ap-plied to a data-fusion appl i~at ion .~

However, possibility theory offers moregeneral applications to reasoning with un-certain, imprecise, or fuzzy pieces of infor-mation by manipulating possibility distrib-utions explicitly. An example of thesereasoning capabilities is provided by theso-called generalized modus ponens, lxwhich from a fuzzy fact Xis A (repre-sented by a possibility distribution J I ~p A / )and a fuzzy rule if X is A then Y is B (alsorepresented by a possibility distributionJ C ~ , ~ ) ,nables us to infer the possibilitydistribution restricting the possible valuesof Y by combining xx and n y l Xnd project-ing the result on the domain of the variableY. According to the multiple-valued logicimplication + used to compute x Y l xromp A and pB,different kinds of fuzzy rulescan be modeled. In particular, we can dis-tinguish, for example, between the purelygradual rules already mentioned (of theform the more X is A, the more Y is E)and certainty rules of the form the more Xis A the more cer ta in Y is B. Thus, gradu-ality can also be encountered in the expres-sion of incomplete knowledge states per-taining to little-known relationshipsbetween variables (like the ones expressedby fuzzy rules).

Expert systems with fuzzy rules havebeen designed that are not as simple asfuzzy controllers (where no chaining ofrules is required, but only an interpolationbetween the conclusions of a parallel rulesset). These expert systems, as expected byElkan, do knowledge-intensive tasks suchas diagnosis, scheduling, or design, andinclude Cadiag-2, Taiger,*O RUM,2Milord?2 OPAL.* All these systems wereor are used in applications in one of theabove-mentioned fields. These systems usesome form of fuzzy set or possibility-the-ory-based inference mechanisms that ismuch more sophisticated than the threeformulas proposed by Zadeh in 1965(Equations 1-3) -and to which fuzzy setand possibility theory methods cannot bereduced. There are many other importantworks on fuzzy set and possibility theory-based inference systems in temporal, quali-tative, and abductive reasoning, that, for thesake of brevity, we do not mention here.

17


16/47

Fuzzy logic is not as simple as Elkan seemsto believe. In this respect, the absence ofany mention in Elkans discussion ofZadehs possibility theory and approximatereasoning approach4,18 s quite revealing.

In the literature, the express ion fuzzylogic usually refers either to multiple-valued logic (as in the first part of Elkanspaper) or to fuzzy controllers. However,the two domains have very little in com-mon, due to the fact that control engineersusually do not know about logic, and logi-cians do not know about control. In thatsense, the first part of Elkans article hasvery little relevance to his discussion onfuzzy control.cal, it is certainly not because of Elkanscollapsing property. More importantly,Zadehs view of fuzzy logic seems to go farbeyond multiple-valued logic, and is asmuch a framework for handling incompleteinformation as a methodology fo r captur-ing graduality in propositions. The conceptof fuzzy truth values refers a s much to theidea of a partially unknown truth value asto intermediate truth values. This is whywe have emphasized the crucial distinctionbetween the truth-functional handling ofgradual properties and the possibilistictreatment of uncertainty (which is not fullycompositional).

It is certainly true that the huge quantityof fuzzy set literature hose quality isunavoidably inconsistent oes not con-tribute much toward helping newcomershave a synthetic, well-informed, and bal-anced view of the domain. Fuzzy controllershave encountered great success by provid-ing an efficient way of implementing aninterpolative mechanism, not only in small,but also in very large and complex prob-lems. However, this should not obscureother existing applications, and the greatpotential of fuzzy set and possibility theoryfor AI applications in general.

If the success of fuzzy logic is paradoxi-

References1. D. Dubois and H. Prade, An Introductionto Possibilistic and Fuzzy Logics, Non-standard Logics forAu toma ted Reasoning,Academic Press, New York, 1988, pp. 287-326.2. D. Dubois,H. Prade, and J. Lang, FuzzySets in Approximate Reasoning, FuzzySets and Systems, Vol. 40,No. 1, March1991. pp. 143-244.3. D. Dubois and H. Prade, Possibility Theory:An Approach to Computerized Processingof Uncertainty. Plenum Press, New York,1988.

L.A. Zadeh, Fuzzy Sets as a Basis for aTheory of Possibility, Fuzzy Sets and Sys-tems, Vol. 1, No. 1, Jan. 1978, pp. 3-28.D. Dubois and H. Prade, Epistemic En-trenchment and Possibilistic Logic, Art$-cia1 Intelligence, Vol. 50, No. 2, July 1991,pp. 223-239.

6. L.A. Zadeh, Fuzzy Sets, Information andControl, Vol. 8, No. 4,June 1965, pp. 338-353.7. R.R. Yager, Connectives and Quantifiers nFuzzy Sets, Fuzzy Sets and Systems,Vol.8. D. Dubois, H. Prade, and R.R. Yager, eds.,Readings in Fuzzy Sets fo r Intelligent Sys-tems, Morgan Kaufmann, San Francisco,Calif., 1993.9. R.C.T. Lee, Fuzzy Logic and the Resolu-tion Principle, . ACM, Vol. 19,No. 1, Jan.

1972, pp. 109-119.10. D. Dubois and H. Prade, Gradual InferenceRules in Approximate Reasoning, Infor-mation Sciences, Vol. 61, No. 1-2, Apr.,

4.

5.

40,NO.1, Mar. 1991, pp. 143-244.

1992, pp. 103-122.11. T. Takagi and M. Sugeno, Fuzzy Identifi-cation of Systems and its Applications toModeling and Control, EEE Trans. Sys-tems, Man and Cybernetics, Vol. 15,No. 2,1985, pp. 116-132.12. D. Dubois and H. Prade, Possibility The-

ory a

elk an i eee discussion

Documents