The Paradoxical Success of Fuzzy Logic Charles Elkan, University of California, San Diego

Home , Degree of truth, Fuzzy logic, Fuzzy rule

Fuzzy logic methods have been used suc- Definition 1: Let A and B be arbitrary as- itively, and it is natural to apply it in rea- cessfully in many real-world applications, sertions. Then soning about a set of fuzzy rules, since but the foundations of fuzzy logic remain 7(A A 4)and B v (4A 4)are both t(A A B) = min [ t(A),t(B)) under attack. Taken together, these two reexpressions of the classical implication t(A v B) = max { t(A),t(B)] facts constitute a paradox. A second para- 4 4 B. It was chosen for this reason, but t(4)= 1 - t(A) dox is that almost all of the successful the same result can also be proved using t(A)= t(B)if A and B are logically fuzzy logic applications are embedded con- many other ostensibly reasonable logical equivalent. trollers, while most of the theoretical pa- aquivalences. pers on fuzzy methods deal with knowl- Depending how the phrase “logically equiv- It is important to be clear on what ex- edge representation and reasoning. I hope alent” is understood, Definition 1 yields actly Theorem 1 says, and what it does not here to resolve these paradoxes by identify- different formal systems. A fuzzy logic sys- say. On the one hand, the theorem applies ing which aspects of fuzzy logic render it tem is intended to allow an indefinite variety to any more general formal system that useful in practice, and which aspects are of numerical truth values. However, for includes the four postulates listed in Defin- inessential. My conclusions are based on a many notions of logical equivalence, only ition 1. Any extension of fuzzy logic to mathematical result, on a survey of litera- two different truth values are possible given accommodate first-order sentences, for ture on the use of fuzzy logic in heuristic the postulates of Definition 1. example, collapses to two truth values if it control and in expert systems, and on prac- admits the propositional fuzzy logic of Theorem 1: Given the formal system of Def- tical experience developing expert systems. Definition 1 and the equivalence used in inition 1, if l(A A 4)and B v (4A 4) the statement of Theorem 1 as a special are logically equivalent, then for any two An apparent paradox case. The theorem also applies to fuzzy set assertions A and B, either t(B)= t(A)or As is natural in a research area as active theory given the equation (A fl B‘)‘ = t(B) = 1-t(A). W as fuzzy logic, theoreticians have investi- B U (A‘ n BC),because Definition 1 can be gated many formal systems, and a variety A direct proof of Theorem 1 appears in the understood as axiomatizing degrees of of systems have been used in applications. sidebar, but it can also be proved using membership for fuzzy set intersections, Nevertheless, the basic intuitions have re- similar results couched in more abstract unions, and complements. mained relatively constant. At its simplest, On the other hand, the theorem does not fuzzy logic is a generalization of standard necessarily apply to versions of fuzzy logic Proposition: Let P be a finite Boolean al- propositional logic from two truth values, that modify or reject any of the postulates of gebra of propositions and let z be a truth- false and true, to degrees of truth between Definition 1 or the equivalence used in The- assignment function P + [0,1], supposedly 0 and 1. orem 1. However, it is possible to carry truth-functional via continuous connec- Formally, let A denote an assertion. In through the proof of the theorem in many tives. Then for all p E P, Q) E { 0, 1 ] W fuzzy logic, A is assigned a numerical value variant fuzzy logic systems. In particular, t(A),called the degree of truth of A, such The link between Theorem 1 and this propo- the theorem remains true when negation is that 0 5 t(A)I 1. For a sentence composed sition is that l(A A 4)= B v (4A -IB) is modeled by any operator in the Sugeno from simple assertions and the logical con- a valid equivalence of Boolean algebra. class,’ and when disjunction or conjunction

nectives “and” (A), “or” (v), and “not” (1) Theorem 1 is stronger in that it relies on are modeled by operators in the Yager degree of truth is defined as follows: only one particular equivalence, while the classes! The theorem also does not depend proposition is stronger because it applies to on any particular definition of implication in any connectives that are truth-functional fuzzy logic. New definitions of fuzzy impli- and continuous (as defined in its authors’ cation are still being proposed as new appli- paper). cations of fuzzy logic are investigated.’ ~ The equivalence used in Theorem 1 is Of course, the last postulate of Definition An earlier version with the same title rather complicated, but it is plausible intu- 1 is the most controversial one. To preserve appeared in Proceedings of the Eleventh Na trona1 Conference on Artificial Intelligence (AAA1 ’93), MIT Press, 1993, pp 698-703

AUGUST 1994 3 Proof of Theorem 1

Theorem I; Given the formal system t(B)< 1 - r(B) < 1 - r(A), By the same reasoning as before, none of of Definition 1, if l(A A 4l) and B v the following can be true: that is if t(B) < 1 - t(B) and t(A) < r(B), (4A lB) are logically equivalent, then which happens if t(A) < t(B) < 0.5. So it 1 - r(A) < [(E)< 0.5 for any two assertions A and E, either t(B)= cannot be true that r(A) < t(B) < 0.5. !(A) < 1 - t(B)< 0.5 t(A) or r(B) = 1-t(A). Now note that the sentences -(A A 4) 1 - t(A)< 1 - t(B)< 0.5 Prmj Given the assumed equivalence, and E v (-A A 4)are both reexpressions of r(B) < t(A)< 0.5 (,(A A 4))= t(B v (-A A TB)). Now the material implication A 4B. One by one, 1 - t(B) < t(A) < 0.5 tf7(A A 7B)) = 1 - min [ r(A), 1 - t(B)] consider the seven other material implication t(B) < 1 - [(A)< 0.5 = 1 + max {-r(A), -1 + t(B)) sentences involving A and B, which are 1 - t(B) < I - t(A)< 0.5 = max [ 1 - (A), t(B)) 44B Now let x = min { r(A), 1 - r(A))and let A+yB and y = min [ t(B), 1 - t(B)].Clearly x I 0.5 and 434 y < 0.5 so if x # y. then one of the eight V A B+A f(B (4 iB)) = inequalities derived must he satisfied. Thus max {t(B),min { 1 - t(A), 1 - t(B))1. iB+A t(B)= t(A) or r(B) = 1 - r(A). B-4 The numerical expressions above are dif- -lB 44 ferent if

a continuum of degrees of truth, one natu- Fuzzy logic in expert systems view of the extent of fuzzy logic applica- rally wants to restrict the notion of logical The basic motivation for fuzzy logic is tion in current commercial and industrial equivalence. In intuitive descriptions, fuzzy clear: While many ideas resemble tradi- knowledge-based systems. All the systems logic is often characterized as arising from tional assertions, they are not naturally in actual use described at the 1992 IEEE the rejection of the law of excluded middle: either true or false; uncertainty of some Intemational Conference on Fuzzy Sys- the assertion A v 4. Unfortunately, reject- sort is attached to them. Fuzzy logic is an tems are controllers, as opposed to reasoning this law is not sufficient to avoid col- attempt to capture valid reasoning pattems ing systems. At the 1993 IEEE Conference lapse to just two truth values. Intuitionistic about uncertainty. The notion is now well on AI for Applications, no applications of logic rejects the law of excluded middle, accepted that there are many different types fuzzy logic in knowledge-based systems but the formal system of Definition 1 still of uncertainty, vagueness, and ign~rance.~ were reported. Of the 16 deployed systems collapses when logical equivalence means However, there is still debate as to what described at the 1993 AAA1 Conference on intuitionistic equivalence? (The Godel types of uncertainty are captured by fuzzy Innovative Applications of AI, three - the translations of classically equivalent sen- logic. Many papers have discussed (at a CAPE,* D~dger,~and DYCE'" systems - tences are intuitionistically equivalenL6 For high level of mathematical abstraction) the used fuzzy logic in some way. However, any sentence, the first three postulates of question of whether fuzzy logic provides none of these systems uses fuzzy logic op- Definition 1 make its degree of truth and suitable laws of thought for reasoning erators for reasoning about uncertainty. the degree of truth of its Godel translation about uncertainty - and if so, which vari- Input observations are assigned degrees of equal. Thus the proof in the sidebar can be eties of uncertainty. The question of inter- membership in fuzzy sets, but inference carried over directly.) Dubois and Prade est here is more empirical: whether or not with these degrees of membership uses note that if all the properties of a Boolean fuzzy logic is in practice an adequate for- other formalisms. algebra are preserved except for the law of malism for uncertain reasoning in knowl- In addition to DYCE, a team at IBM has excluded middle, their proposition no edge-based systems. developed and fielded several knowledge- longer holds? This observation is compati- I conducted a thorough search of the tech- based systems over the past five years. ble with a collapse assuming only the nical literature using the Inspec and Com- Some of these systems are used for software equivalences of intuitionistic logic, because puter Articles databases of more than 1.3 and hardware diagnosis, for data analysis, although intuitionistic logic rejects the law million papers published since 1988. Using and for operator The systems of excluded middle, it admits a doubly the abstracts as a guide, I found no have varying architectures and cope with negated version of the law, namely published report of a deployed expert sys- different varieties of uncertainty. Experience 7(7 4 v -A). Of course, collapse to tem that uses fuzzy logic as its primary for- with them suggests that fuzzy logic is rarely two truth values is avoided if we admit only malism for reasoning under uncertainty. suitable in practice for reasoning about un- the equivalences generated by the operators While many theoretical papers on fuzzy certainty. The basic problem is that items of minimum, maximum, and complement to logic in expert systems have been published, uncertain knowledge must be combined one. However, these equivalences are es- and several prototype systems have been carefully to avoid incorrect inferences. sentially the axioms of de Morgan, which described, it is hard to find reports of fielded Fixed domain-independent operators like allow only restricted reasoning about col- systems doing knowledge-intensive tasks those of fuzzy logic do not work. lections of fuzzy assertions. such as diagnosis, scheduling, or design. The correct propagation of certainty Recent conferences give a representative degrees must account for the content of the uncertain propositions being combined. I

This is necessary whether the uncertain The fundamental issue here is that a con- be implemented by embedded specialized propositions constitute deep or shallow junction’s degree of uncertainty is not in microprocessors. l9 knowledge. In the case of shallow knowl- general determined uniquely by the degree Despite industry interest, and consumer edge, which may be defined as knowledge of uncertainty of the assertions entering into interest in Japan, fuzzy logic technology that is valid only in a limited context (for the conjunction. There does not exist a :ontinues to meet resistance. For example, example, a correlation between a symptom functionfsuch that the rule t(A A B) = at IJCAI ’9 1, Takeo Kanade gave a talk on and a fault), how degrees of uncertainty are flt(A),t(B)) is always valid, whatever the computer vision, describing at length Mat- combined must be adjusted to account for type of uncertainty represented by t(.). For sushita’s camcorder image stabilizing sys- unstated background knowledge. example, in the case of probabilistic uncer- tem without mentioning its use of fuzzy A simple example illustrates the diffi- tainty, the rule t(A A B) = t(A) . t(B)is valid logic. Also, while a fuzzy logic controller culty. Consider a system that reasons in a if and only if A and B represent independent is embedded in the 1994 Honda Accord’s shallow way using a notion of “strength of events. In general, for probabilistic uncer- automatic transmission, the advertising evidence,” and assume that, as in many tainty all one knows is that max [ 0, t(A)+ brochures describe it as “grade logic.” expert systems, this notion is left primitive t(B)- 1 ] 5 t(A A B) 5 min (t(A),t(B)]. Almost all currently deployed heuristic and not analyzed more deeply. (Certainly Methods for reasoning about uncertain controllers using fuzzy logic are similar in “strength of evidence” is an intuitively evidence are an active research area in AI, five important aspects (a good example of meaningful concept that may or may not be and the conclusions here are not new. How- this standard architecture appears in a probabilistic, but it is definitely different ever, our practical experience independently paper by Sugeno and his colleagues2’): from “degree of truth.”) For concreteness, confirms previous arguments about the in- (1) The typical fuzzy controller knowl- suppose the context of discourse is a col- adequacy of systems for reasoning about edge base consists of fewer than 100 lection of melons, and in this context by uncertainty that propagate numerical factors rules; often fewer than 20 rules are definition wnfermelon(x)e redinside(x) A according only to which connectives appear used. Fuzzy controllers are orders of greenoutside(x). For some melon m, sup- in assertions.I3 magnitude smaller than systems built pose that t(redinside(m))= 0.5 and t(green- using traditional AI formalisms. oufside(m))= 0.8, meaning that the evi- Fuzzy logic in heuristic tontrol (2) The knowledge entering into fuzzy dence that m is red internally has strength The application of fuzzy logic has been controllers is structurally shallow, 0.5, and that m is green externally with most successful in heuristic control, where both statically and dynamically. Con- strength of evidence 0.8. Are the rules of there is wide consensus that traditional clusions produced by rules are not fuzzy logic adequate for reasoning about techniques of mathematical control theory used as premises in other rules; stati- this particular type of uncertainty? They are often inadequate. The reasons for this cally rules are organized in a flat list, say that the strength of evidence that m is a include the reliance of traditional methods and dynamically there is no runtime watermelon is t(watermelon(m))= min on linear models of systems to be chaining of inferences. (0.5,0.8]= 0.5. However, implicit back- controlled, their propensity to produce (3) The knowledge recorded in a fuzzy ground knowledge in this context says that “bang-bang” control regimes, and their controller typically reflects immediate being red inside and green outside are mu- focus on worst-case convergence and sta- correlations between the inputs and tually reinforcing pieces of evidence to- bility rather than typical-case efficiency. outputs to be controlled, as opposed to ward being a watermelon, so m is a water- Heuristic control techniques give up math- a deep, causal model of the system. melon with strength of evidence over 0.5. ematical simplicity and performance guar- The premises of rules refer to sensor Deep knowledge can be defined as antees in exchange for increased realism observations, and rule conclusions knowledge that is detailed and explicit and better performance in practice. For refer to actuator settings. (Rule enough to be valid in multiple contexts. example, a heuristic controller using fuzzy premises refer to qualitative or “lin- Deep knowledge is general purpose and logic has been shown to have less over- guistic” sensor observations, and rule usable in complex chains reasoning. shoot and quicker settling.’4 of conclusions refer to qualitative actua- However, Theorem 1 says that if more than The first demonstrations that fuzzy logic tor settings, whereas outputs and in- two different truth values are assigned to could be used in heuristic controllers were puts of sensors and actuators are typi- the input propositions of long inference published in the 1970s.15*16Work continued cally real-valued. This means that chains using fuzzy logic rules and one through the 1980s, and recently there has normally two controller components plausible equivalence, then it is possible to been an explosion of industrial interest in map between numerical values and arrive at inconsistent conclusions. Fuzzy the area.17,18One reason for this recent qualitative values. In fuzzy logic ter- logic cannot be used for general reasoning interest in fuzzy controllers is that they can minology, these components are said under uncertainty with deep knowledge. to defuzzify outputs and implement membership functions.)

AUGUST 1994 5 (4) In deployed fuzzy controllers, the nu- he controller’s parameters allows small properties has been ~tudied,~’as have merical parameters of their rules and pantities of performance information to be methodological problems in constructing of their qualitative input and output ised to make small system changes. models of complex systems based on deep modules are tuned in a learning Thus, what makes fuzzy controllers use- knowledge.** Controllers with intermediate process. The tuning can be done by [ul in practice is the combination of a rule- variables, thus with chaining of inferences, human engineers or by leaming algo- Jased formalism with numerical factors have also been in~estigated.~~ rithms; neural network methods have palifying rules and the premises entering However, the designers of larger systems been especially successful.22What the into rules. The principal advantage of rule- based on fuzzy logic are encountering all the tuning algorithms themselves have in xsed formalisms is that knowledge can be problems of scale already identified in tradi- common is that they are gradient-de- acquired from experts or from experience tional knowledge-based systems. It appears scent “hill-climbing” algorithms that incrementally. Individual rules and that the research history of fuzzy logic is learn by local 0ptimi~ation.l~ premises can be refined independently, or recapitulating that of other areas in AI as (5) By definition, fuzzy controllers use at least more independently than items of well, particularly those dealing with knowl- fuzzy logic operators. Typically, mini- knowledge in other formalisms. Numerical edge engineering and state information. mum and maximum are used, as are factors have two main advantages. They The rules in the knowledge bases of cur- explicit possibility distributions (usu- allow a heuristic control system to inter- rent fuzzy controllers are obtained directly ally trapezoidal) and some fuzzy im- Face smoothly with the continuous outside by interviewing experts. Indeed, the origi- plication operator. world, and they allow it to be tuned gradu- nal motivation for using fuzzy logic in ally - small changes in numerical factor building heuristic controllers was that fuzzy The question that naturally arises is, Which values cause small changes in behavior. logic is designed to capture human state- of these five features are essential to the None of the features contributing to the ments involving vague quantifiers such as success of fuzzy controllers? It appears that success of systems based on fuzzy logic is “considerable.” More recently, consensus the first four are vital to practical success, unique to fuzzy logic. It seems that most has developed around the idea that research because they make the celebrated credit current fuzzy logic applications could use must focus on obtaining “procedures for assignment problem solvable, while the use other numerical rule-based formalisms fuzzy controller design based on fuzzy of fuzzy logic is not essential. instead - if a human or a learning algo- models of the process.”30Mainstream work In a nutshell, the credit assignment prob- rithm tuned numerical values for those on knowledge engineering, however, has lem is to improve a complex system by formalisms, as is customary when using already transcended the dichotomy between modifying a part of it, given only an evalua- fuzzy logic. A quote from the originator of rule-based and model-based reasoning. tion of its overall performance. In general, fuzzy heuristic control is relevant here: Expert systems with knowledge consisting of $-then rules have at least two disad- solving the credit assignment problem is ... it should be remarked that the work on impossible: the task is tantamount to gener- process control using fuzzy logic was inspired vantages. First, maintenance of a rule base ating many bits of information (a change to as much by Waterman and his approach to becomes complex and time-consuming as the internals of the system) from just a few rule-based decision making as by Zadeh ... the system size increases. Second, rule- and his novel theory of fuzzy subsets.23 bits of information (the system’s inputlout- based systems tend to be brittle: If an item put performance). However, the first four of knowledge is missing from a rule, the shared features of fuzzy controllers can Several knowledge representation for- system may fail to find a solution, or solve this problem for the following reasons. malisms that are rule-based and numerical worse, may draw an incorrect conclusion. First, since it consists of only a few rules, have been proposed besides fuzzy 10gic.~~,*~The main disadvantage of model-based the knowledge base of a fuzzy controller is To the extent that numerical factors can be approaches, on the other hand, is that it is a small system to modify. Second, the short tuned in these formalisms, they should be very difficult to construct sufficiently de- paths between the fuzzy controller’s inputs equally useful for constructing heuristic tailed and accurate models of complex sys- and outputs localize the effect of a change, controllers. Indeed, at least one has already tems. Moreover, the models constructed making it easier to discover a change with a been so used.26 tend to be highly application-specific and desired effect without producing undesired not generali~able.~’ consequences. Third, because of the itera- Retapitulatingmainstream AI Many recent expert systems, therefore, tive way in which fuzzy controllers are re- Several research groups are attempting are neither rule-based nor model-based in fined, many observations of inputloutput to scale up systems based on fuzzy logic the standard way.12 For these systems, the performance are available for system im- and lift the architectural limitations of cur- aim of the knowledge engineering process provement. Fourth, the continuous nature of rent fuzzy controllers. For example, a methodology for designing block-struc- tured controllers with guaranteed stability

6 IEEEEXPERT 1

1 is not simply to acquire knowledge from ’erence, and they are developed informally, 3. M. Sugeno, “Fuzzy Measures and Fuzzy human experts, but rather to develop a the- Nith no formal reasoning about their rules Integrals -A Survey,” Fuzzy Automata and Decision Processes, Elsevier/North-Hol- :hat applies equivalences such as the one ory of the experts’ situated performance land, New York, 1977, pp. 89-102. (this is true regardless of whether the de- ised in the statement of Theorem 1. Sec- sired knowledge is correlational, as in pre- md, the knowledge recorded in a fuzzy 4. R.R. Yager, “On a General Class of Fuzzy Connectives,” Fuzzy Sets and Systems, Vol. :ontroller is not a consistent causal model sent fuzzy controllers, or deep, as in 4, No. 3, Nov. 1980, pp. 235-242. model-based expert systems). Concretely, 3f the process being controlled, but rather under this view of knowledge engineering. m assemblage of visible correlations be- 5. D. Dubois and H. Prade, “Gradual Inference Rules in Approximate Reasoning,” Infor- tween sensor observations and actuator knowledge bases are constructed to model mation Sciences, Vol. 61, No. 1-2, 1992, pp. the beliefs and practices of experts and not settings. Since this knowledge is not itself 103-1 22. “objective” truths about underlying physi- general-purpose, the inadequacy of fuzzy 6. D. van Dalen, Logic and Structure, second cal processes. An important benefit of this logic for general reasoning about uncer- ed., Springer-Verlag, New York, 1983. approach is that the organization of an ex- tainty is not an issue. Moreover, the ability pert’s beliefs provides an implicit organiza- to refine the parameters of a fuzzy 7. P. Smets, “Varieties of Ignorance and the Need for Well-Founded Theories,” Informa- controller iteratively can compensate for tion of knowledge about the external tion Sciences, No. 57-58, Sept.-Dec. 1991, process with which the knowledge-based the arbitrariness of the fuzzy logic opera- pp. 135-144. system is intended to interact. tors as applied inside a limited domain. A. Cunningham and R. Smart, “Computer- The common assumption that heuristic 8. The more sophisticated view of knowl- Aided Parts Estimation,” Proc. 5th Innova- edge engineering just outlined is clearly controllers based on fuzzy logic are suc- tive Applications ofAI Con$, AAAI Press, relevant to research on constructing more cessful because they use fuzzy logic ap- Menlo Park, Calif., 1993, pp. 14-25. pears to be an instance of the post hoc, ergo intricate fuzzy controllers. For a second 9. A.J. Levy et al:, “Dodger: A Diagnostic example of relevant AI work, consider con- propter hoc fallacy. The fact that using Expert System for the Evaluation of Nonde- trollers that can carry state information fuzzy logic is correlated with success does structive Test Data,” Proc. 5th Innovative from one moment to the next (mentioned not entail that using fuzzy logic causes Applications of AI Con$, AAA1 Press, as a topic for future research by von Al- success. In the future, as fuzzy controllers Menlo Park, Calif., 1993, pp. 107-117. trock and colleague^^^). Symbolic AI for- are scaled up, the technical difficulties 10. D.D. Pierson and G.J. Gallant, “Diagnostic malisms for representing systems whose identified in this article can be expected to Yield Characterization Expert (DYCE): A behavior depends on their history have become important in practice. Diagnostic Knowledge-Based System Shell for Automated Data Analysis,” Proc. 5th been available since the 1960s. Neural net- Theorem 1 is a crisp demonstration of Innovative Applications of AI Con$, AAAI works with similar properties (called recur- one of several deep difficulties of scale in Press, Menlo Park, Calif., 1993, pp. rent networks) have been available for sev- AI: the problem of maintaining consistency 152-160. eral years, and have already been used in in long sequences of reasoning. Other diffi- 11. G. Gallant and J. Thygesen, “Digitized control application^.^^ It remains to be seen culties of scale can also be expected to be- Expert Pictures (Depict): An Intelligent whether research from a fuzzy logic per- come critical - in particular, the issue of Information Repository,” Proc. 5th Inuova- spective will provide new solutions to the designing learning mechanisms that can tive Applications of AI Con$, AAAI Press, Menlo Park, Calif., 1993, pp. 50-60. fundamental issues of AI. solve the credit assignment problem when the simplifying features of present 12. A. Hekmatpour and C. Elkan, “Categoriza- controllers are absent. tion-Based Diagnostic Problem Solving in the VLSI Design Domain,” Proc. IEEE Int’l Con$ on AI for Applications, IEEE Com- Applications of fuzzy logic in heuristic Acknowledgments puter Society Press, Los Alamitos, Calif., control have been highly successful, de- The author is grateful to many colleagues for 1993, pp. 121-127. useful comments on earlier versions of this article. spite the collapse of fuzzy logic to two- 13. J. Pearl, Probabilistic Reasoning in Intelli- valued logic under an apparently reason- References gent Systems, Morgan Kaufmann, San Fran- able condition, and despite the inadequacy 1, D. Dubois and H. Prade, “New Results cisco, Calif., 1988. of fuzzy logic for general inference with about Properties and Semantics of Fuzzy 14. D.G. Burkhardt and P.P. Bonissone, “Auto- uncertain knowledge. These difficulties Set-Theoretic Operators,” Fuzzy Sets: The- mated Fuzzy Knowledge Base Generation have not been harmful in practice because ory and Applications to Policy Analysis and and Tuning,” Proc. IEEE Int’l Con$ on Information Systems, Plenum Press, New current fuzzy controllers are far simpler Fuzzy Systems, IEEE Computer Society York, 1980, pp. 59-75. Press, Los Alamitos, Calif., 1992, pp. than other knowledge-based systems. The- 179-1 88. orem 1 is not an issue for fuzzy controllers 2. D. Dubois and H. Prade, “An Introduction to Possibilistic and Fuzzy Logics,” Non- because they do not perform chains of in- 15. L.A. Zadeh, “Outline of a New Approach to Standard Logics for Automated Reasoning, the Analysis of Complex Systems and Deci- Academic Press, San Diego, 1988.

AUGUST 1994

I Programming Languages for Parallel Processing

edited by David B. Skillicorn sion Processes,” IEEE Trans. Systems, Man, 28. W. Pedrycz, “Fuzzy Modeling: Fundamen- and Domenico Talia and Cybernetics,Vol. 3, No. 1, Jan. 1973, tals, Construction, and Evaluation,” Fuzzy pp. 28-44. Sets and Systems, Vol. 41, NO. 1, 199 1, pp. Discusses programming lan- 1-15. guages for parallel processing ar- 16. E.H. Mamdani, “Application of Fuzzy Al- gorithms for Control of Simple Dynamic 29. C. von Altrock, B. Krause, and H.J. Zim- chitectures and describes the Plant,” Proc. Institution of Electrical Engi- mermann, “Advanced Fuzzy Logic Control implementation of various para- neers, Vol. 121, No. 12, 1974, pp. 1585-8. of a Model Car in Extreme Situations,” digms to support different models Fuzzy Sets and Systems, Vol. 48, No. 1, of parallelism. The book provides 17. T. Yamakawa and K. Hirota, eds., special 1992, pp. 41-52. issue on applications of fuzzy logic control an overview of the most important to industry, Fuzzy Sets and Systems, Vol. 32, 30. D. Driankov and P. Eklund, “Workshop parallel programming languages No. 2, 1989. Goals,” distributed by the authors at IJCAI designed over the last decade and ’91 Workshop on Fuzzy Control. 18. C.C. Lee, “Fuzzy Logic in Control Systems introduces issues and concepts -Parts 1 and 2,” IEEE Trans. Systems, 3 1. J.R. Bourne et al., “Organizing and Under- related to the development of par- Man, and Cybernetics, Vol. 20, No. 2, Mar. standing Beliefs in Advice-Giving Diagnos- allel software. The text covers par- 1990, pp. 404435. tic Systems. IEEE Trans. Knowledge and allel languages currently used to Data Engineering, Vol. 3, No. 3, Sept. 1991, 19. T. Yamakawa, “Stabilization of an Inverted develop parallel applications in pp. 269-280. Pendulum by a High-speed Fuzzy Logic many areas from numerical to sym- Controller Hardware System,” Fuzzy Sets 32. J. McCarthy and P.J. Hayes. “Some Philo- bolic computing. and Systems, Vol. 32, No. 2, 1989, pp. sophical Problems from the Standpoint of 161-180. Artificial Intelligence,” Machine Intelli- 384 pages. 1994. 0-8 186-6502-5. gence, Vol. 4, Edinburgh Univ. Press, Edin- # 6502-01 20. K. Uomori et al., “Automatic Image Stabi- burgh, Scotland, 1969, pp. 463-502. lizing System by Full-Digital Signal Pro- $45.00 Members $34.00 cessing,’’ IEEE Trans. Consumer Electron- 33. R.L. Watrous and L. Shastri, “Learning ics,Vol. 36, No. 3,Aug. 1990, pp. 510-519. Phonetic Features Using Connectionist Advances in Networks,” Proc. 10th Int’l Joint Con$ on 21. M. Sugeno et al., “Fuzzy Algorithmic Con- AI, Morgan Kaufmann, San Francisco, Ultra-Dependable trol of a Model Car by Oral Instructions,” Calif., 1987, pp. 851-854. Fuzzy Sers and Systems, Vol. 32, No. 2, Distributed Systems 1989, pp. 135-156. 34. J.L. Elman, “Finding Structure in Time,” Cognitive Science, Vol. 14, No. 2, 1990, pp edited by N.j Suri, C.J. Walter, 22. J.M. Keller and H. Tahani, “Backpropaga- 179-21 1. and M.M. Hugue tion Neural Networks for Fuzzy Logic,” Information Sciences, Vol. 62, No. 3, 1992, 35. C.C. Ku, K.Y. Lee, and R.M. Edwards, Focuses on fault tolerance pp. 205-221. “Improved Nuclear Reactor Temperature concepts and hard real-time Control Using Diagonal Recurrent Neural 23. E.H. Mamdani and B.S. Sembi, “Process Networks,” IEEE Trans. Nuclear Science, perspectives that apply jointly to Control Using Fuzzy Logic,” Fuzzy Sets: Vol. 39, NO. 6, 1992, pp. 2298-2308. ultra-dependable systems. The Theory and Applications to Policy Analysis hook emphasizes the theoretical and Information Systems, Plenum Press, basis for achieving dependability New York, 1980, pp. 249-265. Charles Elkan is an assistant professor in the of objectives and presents a variety 24. E. Sandewall, “Combining Logic and Dif- Department of Computer Science and Engineer- of system models built on these ferential Equations for Describing Real- ing at the University of Califomia, San Diego. principles. It is written for system World Systems,’’ Proc. First Int’l Con$ on His main research interests are in artificial intel- designers and researchers in the Principles of Knowledge Representation ligence. With students and colleagues, he has and Reasoning, R.J. Brachman, H.J. worked recently on leaming algorithms for DNA fields of fault tolerance and real- Levesque, and R. Reiter, eds., Morgan time systems and presents system- and protein sequence analysis, algorithms for Kaufmann, San Francisco, Calif., 1989, pp. reasoning about database queries and updates, level perspectives on a multitude 4 12420. methods of formalizing commonsense knowl- of issues related to this topic. 25. A. Collins and R. Michalski, “The Logic of edge, and other topics. In the field of knowl- Plausible Reasoning: A Core Theory,” Cog- edge-based systems, his paper with A. Hekmat- 480 pages. 1994. 0-8 186-6287-5. nitive Science, Vol. 13, No. 1, 1989, pp. pour, “Categorization-Based Diagnostic cJi%/Og# 6287-0 1 149. Problem Solving in the VLSI Design Domain,” $48.00 Members $36.00 won a best paper award at the 1993 IEEE Con- 26. C. Sammut and D. Michie, “Controlling a ference on Artificial Intelligence for Applica- ‘Black Box’ Simulation of a Space Craft,” tions. Before joining UCSD in 1990, Dr. Elkan IEEE COMPUTER AIMagazine, Vol. 12, No. 1, 1991,pp. was a postdoctoral fellow at the University of SOCIETY PRESS 56-63. Toronto. He earned his PhD and MS at Come11 University in computer science, and his BA in 27. K. Tanaka and M. Sugeno, “Stability Analy- mathematics at Cambridge University. Charles Call toll-free: sis and Design of Fuzzy Control Systems,” Elkan can be contacted at the Dept. of Computer Fuzzy Sets and Systems, Vol. 45, No. 2, Science and Engineering, UCSD, La Jolla, CA 1 -800-CS-BOOKS 1992, pp. 135-156. 92093-01 14; Intemet: [email protected] , Fax:(714)821-4641 IEEE EXPERT I

The Unique Strength of Fuzzy Logic Control Hamid R. Berenji, Intelligent Inference SystemdNASA Ames Research Center

I am pleased to see that Elkan has revised as these, the result of the first level of con- a complex task that would have been very his paper based on comments from fuzzy trol is used in deriving control rules for the difficult, if not impossible, if other sym- logic experts. His reference to Dubois and second set, and so on. These examples bolic control techniques had been used. Prade indicates that he has realized, finally, prove that fuzzy-logic control systems can In summary, I see two major misunder- that his alleged “new discovery” has long be developed to reason with considerable standings in Elkan’s paper. First, it relies on been known by specialists in fuzzy and depth of complexity. Similarly, the control a theorem that is irrelevant to fuzzy logic to multivalued logics. mechanisms for the local-motion control of argue that the methodology is paradoxical. Unfortunately, the new version still con- SRI’s autonomous robot2 rely on several Second, it fails to note that the advantages tains many misunderstandings and errors. I deliberation levels to determine the rele- provided by fuzzy-set constructs give fuzzy will briefly respond to some of them, vance level of each control rule (by evalu- control a unique methodological strength avoiding a discussion of the supposedly ating the operational environment charac- - a fact Elkan mistakenly interprets as startling proof about the purported incon- teristics); to identify current goals and their technological immaturity. sistency of fuzzy logic, which is covered in state of achievement; to activate control responses by Enrique Ruspini and others. I rules according to the current context; and References H.R. Berenji et al., “A Hierarchical Approach will confine my comments primarily to a to blend their control recommendations. 1. to Designing Approximate Reasoning-Based fundamental misunderstanding that is the At any rate, the “depth” of a reasoning Controllers for Dynamic Physical Systems,” source of many of Elkan’s mistaken asser- process as Elkan seems to understand it is in Uncertainty in Arti$cial Intelligence, P.P. tions about the use of fuzzy logic in heuris- not even a well-defined measure of infer- Bonissone et al, eds. North-Holland, Amster- tic control and expert systems. ential system complexity. This is seen in dam, 1991, pp. 331-343. 2. A. Saffiotti, E. Ruspini, and K. Konolige, the fact that the two-level forward chain Elkan lists a number of powerful fea- “Blending Reactivity and Goal-Directedness tures of fuzzy-logic control, but then erro- A -+ (B -+ c)is often “compiled” in real- in a Fuzzy Controller, Proc. Fuzzy Logic, in neously concludes that none is unique to time applications (such as control systems) Proc. Second IEEE Int’l Con$ Fuzzy Sys- fuzzy logic. He fails to realize that the into the single-level rule A A B -+ C to sim- tem, IEEE Computer Society Press, Los unique strength of fuzzy-logic control is its plify and speed computation. This simplifi- Alamitos, Calif., 1993, pp. 134-139. 3. H.R. Berenji et al., “Space Shuttle Attitude dependence on fuzzy-set theory and its cation mechanism, which turns what Elkan Control by Fuzzy Logic and Reinforcement representational capabilities. The small would consider “complex” into an equiva- Learning,” in Proc. Second IEEE Int ’1 Con$ number of rules typical in these systems is lent “simple” version, is used to introduce Fuzzy Systems, IEEE Press, Pistcataway, not the result of mere luck, but the direct contextual and goal-dependence considera- N.J., 1993, pp. 1396-1401. 4. H.R. Berenji and P. Khedkar, “Learning and consequence of the fuzzy predicates that tions into the reasoning chain both in the Tuning Fuzzy Logic Controllers through appear in the rules. Each of these predi- SRI’s mobile robot controller and in our Reinforcements,” IEEE Trans. on Neural cates covers a wide range of state variable own two-goal inverted pendulum. Networks, Vol. 3, No. 5, 1992, pp. 724-740. values while facilitating interpolation of Using fuzzy sets to describe a general 5. H.R. Berenji, “An Architecture for Design- rule consequents. Fuzzy sets provide for a linguistic variable also significantly ing Fuzzy Controllers using Neural Net- works,” in Int ’1 J. Approximute Reasoning, general, yet compact characterization of reduces the complexity of the search Vol. 6., No. 2, Feb. 1992, pp. 267-292. system state that requires fewer rules. process in fuzzy systems that learn from Elkan’s assertion about the shallowness experience. Elkan correctly points out that Hamid R. Berenji is a senior research scientist of fuzzy controller knowledge is simply using fewer rules simplifies the credit as- and principal investigator on intelligent control in the AI branch of the NASA Ames Research signment problem, but he fails to realize wrong. Recent fuzzy-logic controllers, Center. He was a program chair for the IEEE developed for more challenging tasks, use that this is a consequence of using fuzzy International Conference on Neural Networks, hierarchical fuzzy control methods.’ Exam- logic rather than an indicator of its current and was a program cochair of the 1994 IEEE ples include the helicopter control devel- or future applicability. This feature is desir- Conference on Fuzzy Systems. He serves on the oped by Sugeno and his collaborators at the able in any control system, as is seen in the editorial board of several technical publications, and is an associate editor of IEEE Transactions Tokyo Institute of Technology (a system fuzzy-logic controller developed at NASA on Fuzzy Systems and IEEE Transactions on that can appear trivial only to those unfa- Ames for the Space Shuttle’s rendezvous Neural Networks. He is a member of IEEE, and miliar with control theory), and the con- and docking operation^.^ This controller chairs the Neural Networks Council’s Technical troller for a three-linked inverted pendulum learns to improve itself from experience Committee on Fuzzy Systems. Hamid Berenji can be reached at [email protected] developed at Aptronix. In applications such using reinforcement learning technique^:,^

AUGUST 1994

I Broader Issues At Stake A Response to Elkan B. Chandrasekaran, Ohio State University

The fuzzy set approach has clearly cap- In the second argument, Elkan asserts above terminology, rule-based languages tured the interest of many researchers that when fuzzy control systems that work would be M, Mycin and R1 would be the around the world and has been used to well are analyzed, the real source of their P’s, and simple diagnosis and configura- build applications of various sorts, of success seems to be not the inferential cation would be the corresponding tasks, T. which fuzzy control applications are cur- pabilities of fuzzy set theory (derived from The success of the two programs led to rently the most prominent. The approach, the theory’s composition axioms) but claims about the power of the rule-based however, remains controversial. While this rather a combination of things exclusive of mechanism. Similar examples involving controversy has many sources, there are fuzzy set axioms. Among these are the abil- other mechanisms, such as belief nets and relatively few places where the arguments ity to represent certain things as continuous truth maintenance systems, can be are set out in a fashion that allows debate. quantities rather than all-or-nothing quanti- constructed. It is thus useful to have both Charles ties; certain heuristic techniques - that are In a series of articles (such as one from Elkan’s analysis of the fuzzy set approach themselves outside fuzzy set theory -to 1986,’ for example), I made the following to representing uncertainty, and his exami- get the right parameters for the problems; points regarding rule-based systems as a nation of which features of fuzzy set theory and the fact that there is little complex rule- mechanism. The specifics of the mecha- are responsible for the success of fuzzy chaining going on. A number of alterna- nism were incidental in accounting for control systems. In particular, I commend tives and rivals to fuzzy set theory would many aspects of why the programs worked. Elkan for making his arguments about work as well in those applications. The mechanism was computation-univer- these techniques in a nonpolemical way, Part of Elkan’s point - that the success sal, and of course could be used to imple- letting technical arguments and results do of fuzzy control systems thus far is not ment any other mechanism or strategy. A most of the talking. really a full test or proof of the axioms and higher order strategy - classification in In Elkan’s first argument, he claims that claims of fuzzy set theory - is actually an the case of Mycin, or linear sequencing of the axioms of fuzzy set theory, in conjunc- instance of a larger phenomenon in AI. I subtasks in the case of R1 -was the prob- tion with what appear to be a number of think that Elkan’s point can be made lem-level strategy that was responsible for reasonable versions of logical equivalence against the claims of not only fuzzy control the programs’ performance. Not only was between sentences, lead to a collapse of proposals, but also against a number of the rule-based mechanism not the direct truth functions into just two values - a other proposals in AI, including the rivals cause of the good performance, but they fate that fuzzy set theory was expressly of fuzzy sets, such as belief nets. actually hid the reasons for success: The meant to avoid. The general problem is a kind of credit higher level strategies were programmed in As Elkan points out, a result similar to allocation problem and can be stated as the language of the lower level mechanism. his collapse theorem was already known to follows. Given some mechanism M, and The strategies had to be brought out by researchers within the fuzzy set community some specific task T, suppose I write a pro- analysis, rather than seen by a direct in- (Dubois and Prade). My understanding is gram P, using M as the basis for the pro- spection of the mechanism. The limitations that they weren’t too worried by this result, gram. And, let us say that P does well in and success of Mycin and R1 could be since they think that the traditional notion the task T. What conclusions can we draw more insightfully analyzed by examining of logical equivalence or any of its variants about mechanism M from the success of P the adequacy of classification for diagnosis should be abandoned for fuzzy sets. This in tackling T? How much credit should M and linear subtasking for configuration response seems to me to be formally rea- get for the success of P? design. Clancey also analyzed Mycin as a sonable, but I think in practice it would be heuristic classifier2 and pointed out the hard to work with a system in which logi- A historical perspective. In the late power such high-level analysis brought to cal equivalence itself is a fuzzy relation. 1970’s, rule-based expert systems were building diagnostic systems. In the last Ultimately, we will have to see how much capturing the imagination of many people. decade or so, there has been a decisive shift really interesting work is possible with this Mycin and R1 were great successes. In the in emphasis in the field of knowledge- notion of fuzzy equivalence. based systems from mechanisms at the rule level to phenomena at the task level.

10 IEEE EXPERT Thus, given an M-T-P triad, it is not al- much was made of the uncertainty-factor A psychological theory? At the heart of ways easy to decide exactly what the role of formalism. Debates raged about this for- fuzzy set theory is an ambiguity about the M was in the success of P in achieving that malism versus Bayesian formalism versus nature of the theory, and how one goes version of T. This is not to say that M’s fuzzy set formalisms as an appropriate cal- about validating it. If it is a psychological properties are irrelevant. There are several culus. Cooper and Clancey got the idea of theory - that is, a theory of how humans ways a given mechanism might play less of doing an experiment in which they coars- deal with certain types of uncertainty - a role than is readily apparent, among them: ened the uncertainty factors in Mycin’s we would need certain kinds of evidence knowledge base rules and examined how about human behavior in uncertainty han- M might simply be one among many well the modified Mycin did in the same dling. I am unconvinced that fuzzy set the- perfectly reasonable lower level mecha- cases3 The modified Mycin solved the ory is a psychological theory. I have not nisms to implement the causally more problems as well as the original Mycin. done an extensive literature survey, but the relevant higher level mechanism. How could this be? Clearly the calculus work of Kempt0n~3~raises doubts that M might have features which actually as such didn’t play as fundamental a role in human behavior in uncertainty handling impede good performance for the class the ability of Mycin to solve the problems. follows the axioms of fuzzy set theory. of problems in T. This might not be The fine structure of uncertainty didn’t Even if it turns out that the theory does evident from the specific instance of T really matter. The knowledge base had correspond to human behavior in this area, for which P was written. In this enough knowledge to establish or reject the we must then decide what kinds of scaling instance, the troublesome features of M conclusions in a near-definitive way. None and rationality properties the relevant might not have been used or their effect of the conclusions were based on even human behavior has before it is used to might be minimal. Fuzzy set theory has moderate distinctions in uncertainty be- make machines that make decisions. been successfully applied to simple tween the candidates. There were multiple Two relevant analogies are found in versions of the control problem. As ways to get to or reject conclusions, and commonsense physical reasoning and rea- Elkan argues, however, the problematic even moderate changes in the uncertainties soning about probabilistic uncertainty. We features of the theory might start show- didn’t matter. The correct conclusions were all have approximate rules about how the ing up as more complex versions of the very strongly established, and the incorrect physical world behaves: “If we push this a control problem are encountered. conclusions were very strongly rejected. little, this will move a moderate distance, In some cases, M has many more fea- Mycin did well, not because of the fine while the other object would hardly move.” tures than needed for capturing the points of its uncertainty calculus - it We use such rules when we have to predict essence of T. Hence, using M to build P would have done just as well with any of a behavior in the physical world, but these for solving T calls for making commit- number of alternative calculi -but be- rules are typically chained over a few steps. ments to details that are either irrelevant cause of the robustness of its knowledge When a problem calls for many steps, these or that detract from building good P’s. base. This is another instance of the alloca- rules start accumulating large errors (to be However, when such a program is built, tion of credit problem. expected), but curiously, they also start it takes quite a bit of analysis to tell accumulating ambiguities of another sort. which features of M are necessary. The nature of fuzzy theory So many alternative possibilities are gener- There is often a tendency, especially I have followed fuzzy set theory almost ated that we adopt all kinds of goal- and among those who are enthusiasts of M from its inception. The theory’s claim that context-specific strategies to select a “fu- for other reasons, to ascribe the success all senses of uncertainty in human knowl- ture history” over other alternatives. Or, if of P to those features of M that were edge cannot be reduced to some version of we are physicists, we resort to a pencil and actually incidental to P’s success. Even probability has always struck me as right. paper for more exact calculations even if more seriously, success with M might One of the most useful consequences of the what we really want are approximate an- lead to its use for more complex ver- fuzzy set movement has been the identifi- swers. Clearly such approximate reasoning sions of T,where these additional fea- cation of different types of uncertainty. In by humans does not scale up very well. tures actually make building successful particular, the theory suggests that many In the case of probability assessment P’s more difficult. Elkan makes a good predicates such as “bald,” “most,” and behavior, human behavior is not always case for this possibility as fuzzy control “large” are neither binary predicates, nor what an outside observer might regard as approaches are applied to more com- are they simply probabilistic. This also rationaL6 Thus, in addition to the scalabil- plex control problems. seems to me to be true. However, the spe- ity problem, there is the problem of ratio- The history of Mycin is another source of cific solutions offered and claims made by nality of human behavior as well. wisdom about the role of uncertainty-han- fuzzy set theory, and the way they have dling mechanisms. When Mycin came out, often been applied to problems like control, are problematic for me.

AUGUST 1994 11

I The point that I want to make with these What do I mean by “such an abstract ties through human common sense. What if two examples is that, in many domains, world may not exist”? Again, the analogy human behavior, in combining everyday automated decision systems should not be of qualitative physics is relevant. We know uncertainties, is really governed by a com- designed to emulate human behavior. Thus, there is a real physics, whose laws relate bination of goal- and context-dependent even if fuzzy set theory turns out to be a values of some state variables to the values strategies that make use of a rich body of model of how humans handle a certain type of other state variables. If we have an exact domain-specific knowledge? What if this of uncertainty, we need additional argu- value for the independent variables, we can cannot be captured by a calculus of the ments to make the theory the basis of auto- calculate, using these laws, the exact val- type that fuzzy set or other theorists are matic control. ues of the dependent variables. looking for? If human conclusions are ro- The equations of physics are not a psy- bust with respect to moderate changes in A mathematical theory? On the other chological theory. However, consider the the uncertainty values of the constituents hand, fuzzy set could be a theory of an ab- ordinary commonsense reasoning about the - as in the Mycin experiment by Cooper stract mathematical system whose proper- physical world that I discussed earlier. Peo- and Clancey - then the real explanation of ties model some domain of human interest. ple do make qualitative predictions about human behavior is not given by a calculus, Examples of such systems are arithmetic the physical world in response to qualita- fuzzy or otherwise, but by the complex and deductive logic. The formalization of tive changes in some state of the world. As collection of situation- and goal-specific arithmetic starts with our intuitive notions I said, the qualitative rules people have knowledge that people bring to bear on about numbers, but it is not a psychological cannot be chained into long inferences: The instances of the problem. theory. It posits a world of numbers and ambiguities multiply, resulting in too many Like the case in qualitative reasoning operations on them, and the formalization possible future histories. Which one of the mentioned earlier, people might in fact is an attempt to capture the properties of histories will be realized often depends on avoid anything like a chain of uncertainty this world. We can in fact construct the a more exact value for some variables than combination. If the conclusion seems ro- abstract world, recognize its objects as the we can get from qualitative rules alone. I bust with respect to moderate changes in familiar numbers and perform operations have described elsewhere a number of the uncertainty values of its constituents, on them, and then verify those operations strategies people use to handle such an people feel comfortable with the conclu- against the predictions of the axiomatiza- explosion of possibilities, but almost all of sion. If not, they might get additional data tion. For example, we can multiply 2 and 3, the strategies depend on the problem-solv- so that a robust conclusion can be reached, and check if the axiom system in fact gen- ing goal and ~ontext.~The conclusion is not postpone making a decision, or make deci- erates the number 6 for the answer. the result of applying an abstract, context- sions that may not in general be considered If fuzzy set theory is a theory of an ab- independent calculus. In short, there is no the best, but that are fine for the specific stract world whose constituents are uncer- qualitative physics that is a homomorphism goal at hand. In other words, the same val- tainties of certain types, and whose opera- of the quantitative physics such that the ues of uncertainties for two constituent tions are the sort of things we do when we qualitative physics gives answers that are beliefs would lead to a conclusion with an combine uncertainties, then the theory has to just qualitative versions of the answers uncertainty value A in one situation, an give two kinds of evidence. First, there must given by the quantitative physics. uncertainty value B in another, additional be evidence that such an abstract world in- With respect to uncertainty handling, information gathering in a third, explicit deed exists. Many abstract worlds that can many researchers seem to be looking for a use of probability models in a fourth, and be postulated fail to exist because their ax- similar abstract system that may not exist. simply a shrugging of shoulders and no ioms lack a certain internal coherence. Sec- They are looking for a calculus of uncer- decision at all in a fifth. If this is the case, ond, it must give evidence that the fuzzy set tainty handling which has the following then the search for a calculus of the type axioms capture the operations of this world. features: fuzzy set theorists (and many others in the Establishing that such an abstract world research community concerned with mod- The semantics of its uncertainty terms exists is actually quite hard. In fact, I think it eling uncertainty in reasoning) are looking capture the intuitive meaning of uncer- is quite possible that there is no abstract for is likely to be futile. The issue is illus- tainty terms that people use in their world of uncertainty combination of the trated well in Elkan’s example of his expert commonsense behavior. type that fuzzy set theory attempts to cap- system, for which neither the probability The operations of combination in the ture. In any case, fuzzy set theory has to scheme nor the fuzzy set approach was calculus capture human behavior when worry about validation of its assumptions appropriate. their uncertainties are combined. and about the existence of an abstract calcu- The problem with fuzzy set theory, in lus for combining this kind of uncertainty. This assumes that there is in fact a calculus that underlies the combining of uncertain-

12 IEEE EXPERT I

my view, is not in the mathematics of the to resist the mathematical attractions of an 3. B.G. Buchanan and E.H. Shortliffe, Rule- formal system. It is clearly a mathematical abstract calculus. Instead, we developed a Based Expert Systems: The Mycin Experi- system of some interest. However, a theory formalism in which we could incorporate ments of the Stanford Heuristic Program- ming Project, Addison-Wesley, Reading, of this type has to be judged either as a the uncertainty-combining behavior of Mass.. 1984. psychological theory or as a theory that has experts,8 who were compiling a complex of captured an abstract calculus that underlies background knowledge in such context- 4. W. Kempton, “Category Grading and Taxo- some type of human reasoning. As I have specific rules. It was also important to note nomic Relations: A Mug is a Sort of a Cup,” American Ethnologist Vol. 5, No. 1, 1978, just argued, an abstract calculus of this type that the chaining length was relatively pp. 44-65; revised version reprinted in Lan- may not exist. small: Two or three steps were all that were guage, Culture, and Cognition: Anthropo- used. If the problem called for much longer logical Perspectives, R.W. Casson, ed., The problem of context. In the 1980’s, my chaining, we took it as a sign that we were Macmillan, New York, 1981. colleagues and I were faced with a similar modeling the expert knowledge inaccu- 5. W. Kempton, The Folk Classification of problem with uncertainty in medical diag- rately, and sought additional pieces of Ceramics: A Study of Cognitive Prototypes, nosis. Physicians have to come up with an knowledge that would shorten the chain. Academic Press, San Diego, 1981. assessment of the “likelihood” of some disease for which a number of data were 6. A. Tversky and D. Kahneman, “Judgment Under Uncertainty: Heuristics and Biases,’’ potentially relevant. The relation between Science, Vol. 185, 1974, pp. 300-306. the data and the strength of belief in the Fuzzy set theory has done quite well as a disease was of course a classic example of formal mathematical system. Whether its 7. B. Chandrasekaran, “QP is More than uncertainty. For various reasons - not the theorems are interesting is a subjective SPQR and Dynamical System Theory,” least of which was that we didn’t have the opinion among mathematicians, but a large Computational Intelligence, Vol. 8, No. 2, 1992, pp. 216-222. data needed to use the frequency version of body of mathematical work exists. Where the probabilities for this relationship - we more work needs to be done is in establish- 8. B. Chandrasekaran and S. Mittal, “Concep- needed a technique to model human exper- ing that fuzzy set theory actually captures tual Representation of Medical Knowledge tise in this area. Bayesian approaches, something real and can make a pragmatic by Computer: MDX and Related Systems,” Advances in Computers, Vol. 22, Academic fuzzy set theory, Dempster-Shafer theory, difference, for the right reasons. Press, 1983, pp. 217-293. and uncertainty factor calculus were all I think Elkan has performed a service by available to us. All these calculi shared one initiating a debate about the properties of important property or assumption about fuzzy set theory. I have argued that the B. Chandrasekaranis director of the Labora- human expertise -that there was a situa- points Elkan makes about fuzzy sets are tory for AI Research and a professor of computer and information science at Ohio State Univer- tion- and goal-independent way of combin- really an instance of problems that apply to sity. His research interests include knowledge- ing uncertainties. a number of other AI mechanisms and based systems, using images in problem solving, For example, if two symptoms, sl and s2, ideas, and specifically to many other pro- and the foundations of cognitive science and AI. were relevant to making a decision about posals for subjective calculi for handling Chandrasekaran received his PhD from the Uni- versity of Pennsylvania in 1967. He is editor-in- disease d, such calculi would provide ways uncertainty. The issues raised are large in chief of IEEE Expert, a fellow of the IEEE and in which evidence for sl and s2 would be scope, and not only the fuzzy set commu- AAAI, and a member of the IEEE Computer combined to give evidence about d, and nity, but the AI community as a whole Society. B. Chandrasekaran can be reached at additionally, that the rule of combination could benefit from giving them thought. the Department of Computer and Information Science, Ohio State Univ., 591 Dreese Labs, itself is independent of the specific labels for 2015 Neil Ave., Columbus, OH 43210-1277. s 1, s2, and d. If the evidence for sl is large, Referentes and s2 is medium, the rule would specify 1. B. Chandrasekaran, “Generic Tasks in what the evidence for d would be. But the Knowledge-Based Reasoning: High-Level rule cannot be one thing where sl is “biliru- Building Blocks for Expert System bin,” s2 is “alkaline phosphatase,” and d is Design,” IEEE Expert, Vol. 1, No. 3, Fall “liver disease,” while another rule is used 1986, pp. 23-30. where sl is “cholesterol level,” s2 is “alka- 2. W.J. Clancey, “Heuristic Classification,” line phosphatase,” and d is “heart disease.” Artificial Intelligence Vol. 27, No. 3, 1985, We found, however, that expert behavior pp. 289-350. in uncertainty combination in fact differed from context to context, and problem-solving goal to problem-solving goal. We had

AUGUST 1994 13 A Better Path to Duplicating Human Reasoning ChristopherJS. desilva and Yianni Attikiouzel, University of WesternAustralia

The paradox that arises from Elkan’s Theo- Interrogator: Is John Doe both dead and Perhaps the real paradox of fuzzy logic’s rem 1 is mild in comparison to some of the alive? success is that proponents hail it as a suc- logical problems that lurk behind the ap- Respondent 1: It is half-true that John Doe cessful technology despite the fact that it is parently innocent equations in Definition 1. is both dead and alive. incapable of performing as they claim it In fact, although fuzzy logic has been pro- Respondent 2: It is impossible for John Doe can and does. moted as a way of writing programs that to be both dead and alive. carry out inference in the same way a per- References son might, the equations of Definition 1 While there is an element of caricature 1. R.T. Cox, The Algebra ofprobable Infer- ence, Johns Hopkins Press, Baltimore, 1961. can lead inescapably to conclusions that no in this dialogue, it serves to highlight the human being would accept. problem. It is clear that if A is any proposi- 2. E.T. Jaynes, “How Does the Brain do Plau- sible Reasoning?’ Tech. Report 42 I, Mi- Consider a simple example: You know tion with a non-zero truth value, the equa- crowave Laboratory, Stanford Univ., 1957. that the airplane on which John Doe was tions of Definition l will lead to the con- 3. M. Tribus, Rational Descriptions, Deci- traveling has crashed in some remote loca- clusion that the truth value of the compound sions, and Designs, Pergamon Press, New tion, but you have no information about statement (A and (not A)) is also non-zero. York, 1969. whether anyone on board has survived. In This is a very simple example of how fuzzy 4. P. Cheeseman, “An Inquiry into Computer this situation, you might make the follow- logic diverges from human logic. It is to be Understanding,” Computer Znrelligence, ing assignment: t(“John Doe is alive”) = expected that this divergence will increase Vol. 4, No. 2, Feb. 1988, pp. 58-66. 0.5. The equations of Definition 1 would with the complexity of the inference process. lead you immediately to t(“John Doe is Of course, people have been assigning Yianni Attikiouzel is a professor of electrical and electronic engineering at the University of dead”) = 0.5. While this is a reasonable truth values between zero and one to make Western Australia where he is director of the assignment, it would in tum lead you to inferences since the time of Laplace, on the Centre for Intelligent Information Processing t(“John Doe is both dead and alive”) = 0.5. basis of probability theory. As Cox has Systems. His work has been published in and Thus, there is an element of truth in the shown,’ using the axioms of probability presented at more than 120 international journals statement “John Doe is both dead and and conferences, and he is the author of two theory is essentially the only way to carry books. He is a member of the Industry Research alive.” However, any rational person will out this form of inference and remain con- and Development Board of the Commonwealth argue that it is impossible for John Doe to sistent with human reasoning - any other Department of Science and Technology, and sits be both dead and alive, so that the state- way will lead to contradictions and incon- on its Services and Consumer Products Commit- ment “John Doe is both dead and alive” sistencies. However, proponents of fuzzy tee. Yianni Attikiouzel can be contacted at the Centre for Intelligent Information Processing must always be false, and have a truth logic appear to be unaware of Cox’s work Systems. Department of Electrical and Elec- value of zero. and that of Jaynes2 and Trib~s,~where the tronic Engineering. University of Western Aus- We can imagine putting a fuzzy logic question of how to write programs that tralia, Nedlands, WA 6009 Australia; phone: 61 9 system to the Turing test on the matter of make inference based on incomplete 380 3134; fax: 61 9 380 1101; Internet: [email protected] John Doe’s well-being: knowledge is discussed. Christopher desilva is a research fellow at the Interrogator: Is John Doe alive ? Centre for Intelligent Information Processing Respondent 1: It is half-true that John Doe As Cheeseman4 pointed out for AI in gen- Systems at the University of Western Australia. is alive. eral, the bottom line is that if you want to He is currently working on the theory and application of artificial neural networks. His other Respondent 2: I don’t know. write a program or build a machine that research interests include syntactic pattern will perform inference in the same way as recognition and Bayesian inference. He can be Interrogator: Is John Doe dead ? people, then you must build the basic equa- reached at the Centre for Intelligent Information Respondent 1 : It is half-true that John Doe tions of probability theory into it, or face Processing Systems, Department of Electrical and Electronic Engineering, University of West- is dead. the inevitable outcome that it will not per- em Australia, Nedlands, WA 6009 Australia; Respondent 2: I don’t know. form as required. phone: 61 9 380 1765; fax: 61 9 380 1101; Inter- net: [email protected]

I 14 IEEE EXPERT 1 Partial Truth is not Uncertainty I Fuzzy Logic versus Possibilistic Logic Didier Dubois and Henri Prade, UniversitkPaul Sabatiw de Toulouse Philippe Smets, Universit6Libre de Bruxelles

Charles Elkan has questioned fuzzy logic Fuzzy logic equivalence is not classical. where t(l) = 0 and t(T)= 1. Indeed, as and cast serious doubts on the reasons for Elkan claims that in fuzzy logic, four re- many authors have emphasized, the failure its success, arguing that “fuzzy logic col- quirements hold for any assertions A and B, of contradiction and excluded-middle laws lapses mathematically to two-valued t being a truth assignment function such is typical of fuzzy logic. This is natural logic.” We completely disagree, and we that VA, t(A) E [0,1]: with gradual properties like “tall.” For ex- especially object to two points: ample, in a given context, somebody who t(A A B) = min(t(A), t(B)) (1) is 1.75 meters high might be considered t(A v B) = max(t(A), t(B)) (2) (1) Elkan’s proof uses too strong a notion neither as completely tall (tall with degree of logical equivalence. The particular t(1A) = 1 - t(A) (3) 1) nor as completely not tall (tall with de- equivalence he considers, while valid t(A) = t(B) if A and B are logically gree 0). In this case, we might have, for in Boolean algebra, has nothing to do equivalent. (4) example, ptall(1.75) = 0.5 = pYtal1(1.75). with fuzzy logic. While Equations 1-3 are indeed the To establish the collapse of fuzzy logic (2) Elkan claims that De Morgan’s alge- basic relations governing degrees of truth to binary logic, Elkan uses the logical bra “allows very little reasoning about in fuzzy logic (as well as fuzzy set mem- equivalence collections of fuzzy assertions,” al- bership degrees) as proposed by Zadeh? though he correctly states that when -(A A iB)= B v (’A A iB) (5) Equation 4 (where “logically equivalent” is logical equivalence is restricted to De understood in a stronger sense than the postulated as being “plausible intuitively.” Morgan algebra equalities’ “collapse equivalences induced by 1-3) has never If Equations 1-3 hold, the left-hand part of ~ to two truth values is avoided.” been seriously considered by any author in Equation 5 can be equivalently written in Furthermore, Elkan fails to understand the the fuzzy-set literature. (There are, as can fuzzy logic as important distinction between two totally be expected, a few erroneous attempts at ’(A A iB) -A V B different problems that fuzzy-set-based the subject in a corpus of more than 10,000 methods address. These are the handling published papers). Obviously, some classi- while the right-hand part can be equiva- of gradual (thus non-Boolean) properties cal logic equivalences still hold with fuzzy lently written as whose satisfaction is a matter of degree assertions obeying Equations 1-3, namely, B v (TA A iB) (’A v B) A (Bv iB), (even when information is complete) on those allowed by the De Morgan structure the one hand, and the handling of uncer- induced by 1-3, such as which clearly relates to the excluded-middle law. Thus, it is expected that Equation 5 fails tainty pervading Boolean propositions, the A AA =A ; A v A =A (idempotency) uncertainty being induced by incomplete to hold in fuzzy logic -and indeed it can states of knowledge that are represented by A A (B v C)= (A AB) v (A A C) ; be checked, using Equations 1-3, that a means of fuzzy sets, on the other hand.’ A v (BAC)=(A vB)A (A v C) counterexample to Equation 5 is provided The first problem requires the plain use of (distributivity) by t(A) = 0, t(B) = 0.5, for instance. Thus, fuzzy sets, while the second is the realm of But other Boolean equivalences do not Elkan’s claim of “a paradox in fuzzy logic” possibility the0ry~9~and possibilistic logic5. hold, for instance: relies only on faulty assumptions, or at best We now discuss in greater detail the points on a logical equivalence, the rationale of AA-A+L above and the distinction between truth which is far from natural in the scope of functional fuzzy (multivalued) logic and since Equations 1 and 3 entail only fuzzy logic. non-fully compositional possibilistic logic. t(A A -A) = min(t(A), 1 - t(A)) < 1/2; Gradual and interpolative reasoning. and Fuzzy logic is concerned with the handling AviA+T of assertions like “John is tall” - assertions whose truth is a matter of degree due since Equations 2 and 3 entail o~ly t(A v ’A) = max(t(A), 1 - t(A)) Z 1/2

1 AUGUST1994 15

I to gradual predicates within them. The de- an interpolation between typical conclu- n the general case (equality holds when A gree of truth of compound expressions can sions is performed, based on degrees of ind B are logically independent). Indeed if be easily computed using Equations 1-3. similarity between the input (xo, yo) and the !3 TA, n(A AB)= n(l)= 0, while (Although we restrict ourselves here to the prototypical values in the core of the fuzzy nin(n(A), n(-.A)) = 0 only if the informa- operators minimum, maximum, and com- set A, x B,. This similarity is measured by .ion is sufficiently complete for having plement to one, there is a panoply of the coefficients yl which cannot be consid- :ither n(1A) = 0 (A is true) or n(A)= 0 (A that enable us to model different ered as degrees of uncertainty in any case. is false). If nothing is known about A, we kinds of AND and OR operations between In spite of its apparently ad hoc nature, lave n(A) = n(-A) = 1. By duality, a ne- properties in a multicriteria aggregation Equation 7 can be justified with one- Zessity measure N is associated to n ac- perspective.) premised rules using Equation 6 and view- :ording to the relation (which can be More than 20 years ago, R.C.T. Lee9 ing the rules as expressing “the more Xis viewed as a graded version of the relation provided the basic machinery for reasoning A, and Y is B,, the closer Z is to cL”and between what is necessary and what is pos- in fuzzy logic by extending the resolution using appropriately shaped membership iible in modal logic) rule in accordance with Equations 1-3. He functions.l* N(A) = 1 - n(-A) (11) established that if all the truth values of the As this shows, contrary to Elkan’s claim, parent clauses are greater than 0.5, then a some kinds of reasoning, as exemplified by which states that A is all the more necessar- resolvent clause derived by the resolution Takagi and Sugeno’s, and Lee’s methods, ily true as TA has a low possibility to be principle always has a truth-value between can be handled in a De Morgan algebra true. It entails the maximum and the minimum of those of framework. N(A A B) = min(N(A), N(B)) (12: the parent clauses. We can also use an implication operator Possibility theory and uncertainty. In and to model “gradual rules,”I0 which express addition to modeling the gradual nature of N(A v B) 2 max(N(A), N(B)). (13: knowledge of the form “the more Xis A, properties, fuzzy sets can be used to repre- the more Y is B,” such as, “the taller you sent incomplete states of knowledge. In Equations 9, 1 I, and 12 should not be are, the heavier you are.” This is captured this second use, the fuzzy set plays the role confused with Equations 2,3, and 1, respec- by the implication defined by of a possibility distribution that provides a tively. In 9, 11, and 12 we deal with complete ordering of mutually exclusive Boolean propositions pervaded with uncer- t(A B) = 1 if t(A) 5 t(B) -+ states of the world according to their re- tainty due to incomplete information, while = 0 if t(A) > t(B) (6) spective levels of possibility or plausibility. 1-3 pertain to non-Boolean propositions This implication is the natural counter- For instance, if we know only that “John is whose truth is a matter of degree (the infor- part of Zadeh’s fuzzy set inclusion defined tall” (but not his precise height), where the mation being assumed to be complete). by the pointwise inequality of the member- meaning of ‘‘tall’’ is described, in context, Very often, discussions about fuzzy expert ship functions.6 It is also directly associ- by the membership function of a fuzzy set systems or uncertain knowledge base sys- ated with Equations 1-3, since A + B = T (that is, ptall),then the greater ptall(x)is, the tems get confused because of a lack of dis- if and only if A A B =A. Such an implica- greater the possibility that height(John) = tinction between degrees of truth and de- tion expresses a purely gradual relationship x; the smaller ptall(x)is, the smaller this gree of uncertainty. Fuzzy logic, as and has nothing to do with uncertainty. possibility. understood by Elkan, is a logic where the Besides, Takagi and Sugeno” have pro- Given a [O,l]-valued possibility distribu- truth status of propositions is multiple-val- posed an interpolation mechanism between tion n: describing an incomplete state of ued; that is, there are intermediary truth n rules with fuzzy condition parts and non- knowledge, Zadeh4 defines a so-called pos- values between true and false (like “very fuzzy conclusions of the form “if X is A, sibility measure n such that true,” “rather true,” and so on). On the con- and Y is B, then Z = cl”,by computing the trary, degrees of uncertainty apply to all-or- n(A) = sup(~(x), x makes A true} (8) following output when X = x0 and Y = yo is nothing propositions, and do not model observed where A is a Boolean proposition (a propo- truth values but express the fact that the sition that can only be true or false). It can truth value (true or false) is unknown. The (7) be easily checked that for Boolean proposi- uncertainty degrees then try to assess which tions A and B, we have one of “true” or “false” is the most plausi- where K = min(pA,(xd, pei(yd), = 1,n. ble truth value. This distinction was made i n(A v B) = max(n(A), n(B)) (9) Again, this kind of “inference” (which is by one of the founders of subjective proba- widely used in fuzzy control) has nothing but that we only have the inequality bility theory -De Finetti13 -but with a to do with uncertainty handling, since only few exceptions (including ourselves) it has n(A AB)5 min(n(A), n(B)) (10)

16 IEEE EXPERT

~ -- - been quite forgotten by the AI community V(wutermelon(m))2 min(0.5,0.8) = 0.5, a However, possibility theory offers more in general and by Elkan in particular. Still, result also obtained under an equality form general applications to reasoning with un- we consider this distinction a crucial pre- ,y Elkan by applying Equation 1 in an inap- certain, imprecise, or fuzzy pieces of infor- requisite in any discussion about fuzzy sets propriate way. However, he would like to mation by manipulating possibility distrib- and possibility theory and their use in auto- zonclude that “m is a watermelon with utions explicitly. An example of these mated reasoning. strength of evidence over 0.5.” This seems a reasoning capabilities is provided by the Observe also that neither n nor N are strange requirement, and one that a proba- so-called generalized modus ponens,lx fully compositional with respect to A, v , bilistic model would not satisfy either (since which from a fuzzy fact “Xis A”’ (repre- and 7.This is not surprising, since the only Prob(A AB)5 min(Prob(A),Prob(B)). In- sented by a possibility distribution JI~= pA/) way to map a Boolean structure on [0,1] by deed, we are not in a data fusion situation and a fuzzy rule “if X is A then Y is B’ (also a fully compositional mappingfis to have where two independent sources provide the represented by a possibility distribution f(A) equal to 0 or to 1 for any A.’ Truth- same conclusion with various strengths,I4 JC~,~),enables us to infer the possibility functionality in Equations 1-3 is preserved but in a situation where the logical con- distribution restricting the possible values only by having A and B elements of a junction of two conditions is required to of Y by combining xx and nylXand project- weaker structure, namely, a De Morgan Zonclude that m is a watermelon (namely ing the result on the domain of the variable algebra. Thus, logics of uncertainty cannot the inside redness of m and its outside Y. According to the multiple-valued logic be fully compositional with respect to un- greenness). Note that in case we have both implication + used to compute xYlxfrom certainty degrees. This point is also recog- N(A) t U and N(A) t a‘ as obtained from pAand pB,different kinds of fuzzy rules nized by Elkan in the case of probability distinct arguments, we shall conclude that can be modeled. In particular, we can dis- measures, and dates back at least to De N(A) t max(u,a’). tinguish, for example, between the purely Finetti in the 1930s! Partial compositional- gradual rules already mentioned (of the ity is possible, however; probabilities are Reasoning with possibility theory. In form “the more X is A, the more Y is E’) compositional with respect to negation, possibilistic logic, first-order logic formu- and certainty rules of the form “the more X possibilities with respect to disjunction, las are weighted by lower bounds of neces- is A the more certain Y is B.” Thus, gradu- necessities with respect to conjunction. sity or possibility measures, which reflect ality can also be encountered in the expres- Based on his article, however, it seems that the uncertainty of the available informa- sion of incomplete knowledge states per- Elkan has not heard about possibility the- tion. Possibilistic has been devel- taining to little-known relationships ory, which is another side of fuzzy sets. oped both at the syntactic level, where between variables (like the ones expressed Let us consider Elkan’s watermelon ex- there is an inference machinery based on by fuzzy rules).’ ample: extended resolution and refutation (the Expert systems with fuzzy rules have lower bound of the resolvent clause neces- been designed that are not as simple as watermelon(x) = sity is the minimum of the lower bounds of fuzzy controllers (where no chaining of redinside(x)A greenoutside(x) parent clauses necessity measures), and at rules is required, but only an interpolation It is supposed that “for some melon m, evi- the semantic level, where a semantics in between the conclusions of a parallel rules dence that m is red intemally has strength terms of a possibility distribution over a set set). These expert systems, as expected by 0.5, and m is green externally with strength of classical interpretations has been proved Elkan, do “knowledge-intensive tasks such of evidence 0.8.” It is not clear what Elkan to be sound and complete with respect to as diagnosis, scheduling, or design,” and means by “strength of evidence” in the light the syntax. Due to the fact that a possibility include Cadiag-2,” Taiger,*O RUM,2’ of the above comments. We shall assume distribution encodes a preferential ordering Milord?2 OPAL.*’ All these systems were they are indeed degrees of uncertainty, over a set of possible interpretations, possi- or are used in applications in one of the rather than degrees of red and degrees of bilistic logic has been shown to capture an above-mentioned fields. These systems use green. But then the only way to anchor this important class of nonmonotonic reasoning some form of fuzzy set or possibility-the- discussion in the fuzzy logic debate is to consequence relations and has capabili- ory-based inference mechanisms that is interpret these degrees in possibility theory. ties for handling partial inconsistency in much more sophisticated than the three Elkan’s watermelon sentence can be under- knowledge bases5 Moreover, possibilistic formulas proposed by Zadeh in 1965 stood as N(redinside(m))2 0.5 and N(green- assumption-based truth maintenance sys- (Equations 1-3) -and to which fuzzy set outside(m))t 0.8, expressing that the avail- temd6 based on possibilistic logic have and possibility theory methods cannot be able information makes us certain to the been defined for dealing with uncertain reduced. There are many other important degree of 0.5 that m is red inside, and to the justifications and ranking environments in works on fuzzy set and possibility theory- degree 0.8 that it is green outside. A direct a label; they have been successfully ap- based inference systems in temporal, quali- application of Equation I2 leads to plied to a data-fusion appli~ation.’~ tative, and abductive reasoning, that, for the sake of brevity, we do not mention here.

AUGUST 1994 17 Fuzzy logic is not as simple as Elkan seems References 14. D. Dubois and H. Prade, “Combination of to believe. In this respect, the absence of Fuzzy Information in the Framework of 1. D. Dubois and H. Prade, “An Introduction Possibility Theory,” in Data Fusion in Ro- any mention in Elkan’s discussion of to Possibilistic and Fuzzy Logics,” Non- botics and Machine Intelligence, Academic Zadeh’s possibility theory and approximate standard Logics forAutomated Reasoning, Press, New York, 1992, pp. 481-505. reasoning approach4,18is quite revealing. Academic Press, New York, 1988, pp. 287- 326. 15. D. Dubois and H. Prade, “Possibilistic In the literature, the expression “fuzzy Logic, PreferentialModels, Nonmonotonic logic” usually refers either to multiple- 2. D. Dubois, H. Prade, and J. Lang, “Fuzzy and Related Issues,” Proc. Int’l Joint Con$ valued logic (as in the first part of Elkan’s Sets in Approximate Reasoning,” Fuzzy Art$cial Intelligence (IJCAI ’91), Morgan paper) or to fuzzy controllers. However, Sets and Systems, Vol. 40, No. 1, March Kaufmann, San Francisco, Calif., 1991, pp. 1991. pp. 143-244. 419-424. the two domains have very little in common, due to the fact that control engineers 3. D. Dubois and H. Prade, Possibility Theory: 16. D. Dubois, J. Lang, and H. Prade, “A Possi- usually do not know about logic, and logi- An Approach to Computerized Processing bilistic Assumption-Based Truth Mainte- of Uncertainty. Plenum Press, New York, nance System with Uncertain Justifications, cians do not know about control. In that 1988. and its Application to Belief Revision,” sense, the first part of Elkan’s article has L.A. Zadeh, “Fuzzy Sets as a Basis for a Lecture Notes in Art$cial Intelligence, Vol. very little relevance to his discussion on 4. 515, Springer-Verlag, Berlin, 1990, pp. 87- Theory of Possibility,” Fuzzy Sets and Sys- 106. fuzzy control. tems, Vol. 1, No. 1, Jan. 1978, pp. 3-28. If the success of fuzzy logic is paradoxi- 17. F.F. Monai and T. Chehire, “Possibilistic 5. D. Dubois and H. Prade, “Epistemic En- cal, it is certainly not because of Elkan’s Assumption-Based Truth Maintenance trenchment and Possibilistic Logic,” Art$- System: Validation in a Data Fusion Appli- collapsing property. More importantly, cia1 Intelligence, Vol. 50, No. 2, July 1991, cation,” Proc. Eighth Con$ Uncertainty in Zadeh’s view of fuzzy logic seems to go far pp. 223-239. AI, 1992, Morgan Kaufmann, San Fran- beyond multiple-valued logic, and is as 6. L.A. Zadeh, “Fuzzy Sets,” Information and cisco, Calif., 1992, pp. 83-91. much a framework for handling incomplete Control, Vol. 8, No. 4, June 1965, pp. 338- information as a methodology for captur- 353. 18. L.A. Zadeh, “ATheory of Approximate Reasoning,” Machine Intelligence, Vol. 9, ing graduality in propositions. The concept 7. R.R. Yager, “Connectives and Quantifiers in John Wiley & Sons, New York, 1979, pp. of fuzzy truth values refers as much to the Fuzzy Sets,” Fuzzy Sets and Systems,Vol. 149-194. idea of a partially unknown truth value as 40, NO. 1, Mar. 1991, pp. 143-244. 19. K.P. Adlassnig and G.Kolarz, “CADIAC-2: to intermediate truth values. This is why 8. D. Dubois, H. Prade, and R.R. Yager, eds., Computer-Assisted Medical Diagnosis we have emphasized the crucial distinction Readings in Fuzzy Sets for Intelligent Sys- Using Fuzzy Subsets,”Approximate Rea- between the truth-functional handling of tems, Morgan Kaufmann, San Francisco, soning in Decision Analysis, North-Hol- gradual properties and the possibilistic Calif., 1993. land, Amsterdam, 1982, pp. 219-247. treatment of uncertainty (which is not fully 9. R.C.T. Lee, “Fuzzy Logic and the Resolu- 20. H. Farreny, H. Prade, and E. Wyss, “Ap- compositional). tion Principle,” J. ACM, Vol. 19, No. 1, Jan. proximate Reasoning in a Rule-Based Ex- It is certainly true that the huge quantity 1972, pp. 109-119. pert System Using Possibility Theory: A Case Study,” Proc. Information Processing of fuzzy set literature -whose quality is 10. D. Dubois and H. Prade, “Gradual Inference ’86,North-Holland, Amsterdam, 1986, pp. unavoidably inconsistent -does not con- Rules in Approximate Reasoning,” Infor- 407-413. tribute much toward helping newcomers mation Sciences, Vol. 61, No. 1-2, Apr., have a synthetic, well-informed, and bal- 1992, pp. 103-122. 21. P.P. Bonissone, S.S. Cans, andK.S. Decker, “RUM: A Layered Architecture for Reason- anced view of the domain. Fuzzy controllers 11. T. Takagi and M. Sugeno, “Fuzzy Identifi- ing with Uncertainty,” Proc. Int’l Joint have encountered great success by provid- cation of Systems and its Applications to Con$ Art$cial Intelligence (IJCAI), Mor- ing an efficient way of implementing an Modeling and Control,”IEEE Trans. Sys- gan Kaufmann, San Francisco, Calif., 1987, tems, Man and Cybernetics, Vol. 15, No. 2, pp. 891-898. interpolative mechanism, not only in small, 1985, pp. 116-132. but also in very large and complex prob- 22. L. Godo et al., “Milord: The Architecture lems. However, this should not obscure 12. D. Dubois and H. Prade, “Possibility The- and the Management of Linguistically Ex- ory as a Basis for Preference Propagation in other existing applications, and the great pressed Uncertainty,” Int ’1 J. Intelligent Automated Reasoning,” Proc. First IEEE Systems, Vol. 4, No. 4, Winter 1989, pp. potential of fuzzy set and possibility theory Int’l Con$ Fuzzy Systems (FUZZ-IEEE’92), 471-501. for AI applications in general. IEEE Press, Piscataway, N.J., 1992, pp. 821-832. 23. E. Bensana, E. Bel, and D. Dubois, “OPAL A Multi-Knowledge-Based System for 13. B. De Finetti, “La Logique de la Probabil- Industrial Job-Shop Scheduling,”Int’l J. itC,” Actes du Congr2s Int’l. de Philosophie Product Research, Vol. 26, No. 5, 1988, pp. Scientijque, Paris, 1935, Hermann et Cie 795-819. Editions, 1936, pp. IVI-IVY.

18 IEEE EXPERT

.-- I Didier Dubois is a research scientist at IRIT, Uni- Henri Prade is a researcher at IRIT, University of Philippe Smets is coordinator of the Institut de versity of Toulouse, and is director of research at Toulouse, and director of research at the French Recherches Interdisciplinaires et de Developpe- the French National Center for Scientific National Center for Scientific Research. His re- ments en Intelligence Artificielle at the Univer- Research. His research interests focus on the mod- search interests include fuzzy sets and possibility sit6 Libre de Bruxelles, where he is also a profes- eling of uncertainty in various branches of infor- theory, approximate reasoning nonclassical logics, sor of medical statistics and director of the mation technology, especially AI, decision analy- AI and database systems, and operations research. Laboratory for Medical Statistics. His research sis, and operations research. He is the coauthor of He is the coauthor (with Didier Duhois) of a book interests are in approximate reasoning, Bayesian numerous technical papers, and coauthor with on fuzzy sets (Academic Press, 1980) and a book theory, fuzzy set theory, and belief functions. He Henri Prade of a book on fuzzy sets (Academic on possibility theory (Plenum Press, 1988), and he received his MD from the UniversitC Libre de Press, 1980) and a book on possibility theory has coedited three other books and published more Bruxelles in 1963, his master's degree in experi- (Plenum Press, 1988). He is a member of the Ua than 200 technical papers. He is also a member of mental statistics from North Carolina State Uni- SombC research group, which wrote Reasoning the Lta SombC research group, authors of Reason- versity, and his Ph.D. in medical statistics from with Incomplete Information, and Belief Revision ing with Incomplete Information, and Belief Revi- the UniversitC Libre de Bruxelles in 1978. and Updating (John Wiley & Sons, 1990 and sion and Updating (Wiley, 1990 and 1994) Prade Philippe Smets is on the editorial board of the 1994). Dubois is on the editorial board of Fuzzy is an associate editor of IEEE Transactions on International Journal for Approximate Reason- Sets and Systems, the Joumal of Intelligent Manu- Fuzzy Systems and is on the editorial board of ing and Fuzzy Sets and Systems. Philippe Smets facturing, the Intemational Joumal ofApproni- several journals, including Fuzzy Sets and can be reached at IRIDIA, Univ. Libre de Brux- mute Reasoning, and the Intemational Joumal of Systems, the Intemational Joumal ofApproximate elles, CP 194/6, 1050 Bruxelles, Belgium; Inter- General Systems, among others. He is also an Reasoning, the Informution Sciences Series on net: [email protected] associate editor of IEEE Transactions on Fuzzy Intelligent Systems, and the Intemational Joumal Systems. He received a doctorate degree in engi- of Intelligent Systems. He received a doctorate neering in 1977 from ENSAE, Toulouse, and the degree in engineering in 1977 from ENSAE, Doctorat d'Etat in 1983 from Grenoble University. Toulouse, and the Doctorat d'Etat in 1982 from Didier Dubois can be reached at IRIT, Univ. Paul Paul Sabatier University, Toulouse. Henri Prade Sabatier, 3 1062 Toulouse Cedex, France; Internet: can he reached at IRIT, Univ. Paul Sabatier, 3 1062 [email protected] Toulouse Cedex, France; Internet: [email protected]

IEEE/IAFE Conference on Computational Intelligence for Financial Engineering (CIFEr) April 9-11,1995, New York City, Crowne Plaza Manhattan Sponsors The IEEE Neural Networks Council The Intemational Association of Financial Engineers The IEEE Computer Society The IEEE/IAFE CIFEr Conference is the first major collaboration between the professional engineering and financial communities, and will be the leading forum for new technologies and applications in the intersection of computational intelligence and financial engineering. Intelligent computational systems have become indispensable in virtually all financial applications, from portfolio selection to proprietary trading to risk management. Topics in which papers, panel sessions, and tutorial proposals are invited include, but are not limited to, the following: Financial Engineering Applications Computer & Engineering Applications & Models Asset Allocation Risk Management Neural Networks Stochastic Processes Trading Systems Complex Derivatives Machine Intelligence Dynamic Optimization Corporate Financing Currency Models Probabilistic Reasoning Knowledge & Data Engineering Forecasting Technical Analysis Fuzzy Systems Time Series Analysis Hedging Strategies Portfolio Management Parallel Computing Harmonic Analysis Options and Futures Standards Discussions Pattem Analysis Signal Processing Risk Arbitrage Genetic Algorithms Non-Linear Dynamics Keynote Speaker Conference Committee General Co-chairs: Robert C. Merton Tutorials Chairs Tomaso Poggio, Whitaker Professor Baker Professor Of Business Administration Douglas Stone. Frank Russell R&D, Tacoma WA MIT Artificial Intelligence Laboratoly and Harvard Business School Joe R. Brown, MCC, Austin, TX Brain Sciences Department International Liaison David Schwartz, Mitsubishi Bank. NY John Marshall, Professor of Finance Toshio Fukuda,Dept. of Mechanical Eng.. Finance Chair St. John's University Nagoya University, Japan Christine Alan. CPA. Motorola, AZ Program Committee Co-chairs: Organizational Chair Exhibits Chair Andrew W. La, Professor of Finance Scott H. Mathews,MPVC Financial Eng.. Steve piche, MCC, ~~~m,TX MIT Sloan School of Management Bothell, WA Robert Marks, Professor of Electrical Eng. Plenary Chair University.- of Washington,- Seattle Meeting Management Douglas Stone, Frank Russell R&D. Tacoma WA International Chair POT MO= 2003 Main Street. Suite 090, Publications Chair Apostolos N. Refenes Informationcont.ct: Imine. CA 92714 Donald Wunsch, Dept. of Electrical Engineering. London Business School (714)752-8205 FEX(714) 752-7444 Texas Tech University Emak 74710.2266@com~userve.com

I Fuzzy Logic &I Interface Between Logic and Human Reasoning I Christian Freka, University of Hamburg, Germany 1 Charles Elkan addresses two distinct areas between the structure of these propositions One of Lotfi Zadeh’s main motivations of fuzzy logic: formal expressiveness and and operations, on one hand, and the knowl- for introducing the notions of fuzzy sets and practical usefulness. He describes as a edge structure they are supposed to repre- fuzzy logic was his observation that real- paradox that although the theory of fuzzy sent, on the other. When we represent for- world knowledge generally has a different logic is not generally accepted, it is suc- mal domains (for example, card games or structure and requires different formaliza- cessfully used in many real-world applica- mathematical theorems), establishing this tion than existing formal systems. Contrary tions. He also calls paradoxical the fact that correspondence may not cause major prob- to established practice, a one-to-one corre- these applications are predominantly found lems. However, when we represent knowl- spondence between natural-language in the control domain. edge about a real domain, the correspon- propositions and predicate calculus propo- I will not discuss here the alleged equiva- dence between our formalism and the sitions can be shown to be inadequate. In lence between fuzzy and two-valued logic; represented structure becomes a major issue. particular, the instantaneous switch from by choosing criteria established for the more A representation system consists of truth to falsity can easily distinguish propo- restricted two-valued formalism, Elkan does sitions in classical logic from those in nat- a represented world, and the relations not have a suitable framework for a mean- ural language. In addition, numerous as- and operations in it; ingful comparison. To point out prerequi- sumptions of the formally correct treatment a representing world, and the relations sites for the practical usefulness of knowl- of the propositions cannot be established in and operations in it; and edge representation formalisms, I will focus the corresponding source knowledge. the correspondence between the two on the role of fuzzy logic in linking two worlds.’ formally incommensurable worlds: the nat- The fuzzy logic interface. Zadeh recog- ural world of human perception and experi- When representing knowledge about the nized the power of a formal approach to ence that leads to subjective cognitive con- real world, it is inherently impossible to knowledge processing as well as the ad- cepts, and the formal world of classical logic prove something about the represented vantages of using soft knowledge in human that yields universal truth conditions. real-world knowledge; this part of the rep- reasoning. He thus took a first step in incre- Given the premise that there is no one- resentation system is outside the formal- mentally relaxing constraints imposed on to-one mapping between human concep- ism. We only can prove something within existing formalisms to accommodate im- tual structures and the framework of classi- the representing formalism. Thus, the rep- portant properties of natural inference. This cal logic, it is not important for the analysis resented real world and its representation step was to generalize the classical notion of a formal representation structure if two are formally incommensurable. of a set to the notion of a fuzzy set that al- logically equivalent expressions are evalu- In expert systems, the knowledge engi- lowed gradual membership. The choice of ated identically; what we have to ensure is neer establishes the correspondence numerical degrees of membership was that derivations accepted in human reason- between the real and formal worlds, but he largely made for formal reasons: it ing can also be derived in our formalism. cannot prove its correctness; he depends on provided a transparent way of formally his perception and intuition to determine treating the new notion. Using the familiar Classical logic and human knowledge. In the equivalence between the two. Usually, a language of mathematics, the theory can AI, propositions and various kinds of logic knowledge engineer relies upon assump- easily be implemented in computer sys- formalisms serve to represent and derive tions to determine the validity of operations tems, while at the same time offering a knowledge about formal or real domains. on a representation. These assumptions better approximation to the associated Traditionally, most effort has been put into stem from his knowledge about formal human concepts. the development of logically correct and logic, rather than from knowledge about Because human notions and concepts consistent operations within the fomzal rep- specific properties of human reasoning. I form the basis for reasoning in expert sys- resentation; however, little attention has Nevertheless - as Elkan’s article shows - tems, the success of these systems depends been paid to the correspondence problem this approach appears to be widely accepted upon the correspondence relation between for the treatment of human knowledge. human concepts and their formalization.

20 IEEE EXPERT I

Studying the formal properties of the repre- knowledge, on the one hand, and systematic we should not be surprised if some barriers sentation is insufficient. and formal methods for dealing with it, on must be removed before fuzzy logic will be Zadeh realized that it was much more the other. This contribution might have a widely applied to more delicate areas of important to have a good model of the se- much more significant impact on human fuzzy reasoning. mantics of human concepts and perform thought and the role of classical logic in For judging the quality of a representa- reasonable operations than to have a bad systems analysis than the fuzzy set notion tion formalism, I have proposed taking a model and perform verifiably correct opera- will have on the success of expert systems. representation-theoretical viewpoint: The tions. He never insisted that his initial pro- As the transition from crisp sets to fuzzy correspondence between the represented posal for a fuzzy logic should be viewed as sets is a rather moderate step toward domain and the formalism is at least as the final solution for representing human accounting for the nature of human concepts, important as the representation’s formal knowledge about the world; rather, he of- we should not expect it to solve all our prob- properties taken by themselves. This view- fered a model based on established notions lems. In particular, fuzzy sets and fuzzy logic point permits a high-level characterization that could easily be grasped by engineers do not answer the fact that human concepts of the overall representation problem. I and researchers alike as a step toward for- develop and are modified in an open world, have also argued that real-world knowl- malizing human reasoning. Because of this, while formal concepts are fixed in closed edge and formal knowledge are formally Zadeh’s basic notion of a fuzzy set stimu- worlds, for the most part. Therefore, it is not incommensurable. As long as the laws of lated enormous research activity in soft surprising that successful applications of human reasoning are not well understood, a knowledge processing. fuzzy logic are so far found mainly in well- good model of human reasoning should be Zadeh’s work also helped establish a defined closed domains like control prob- expected to preserve some paradoxes; ex- radically different view of the status of lems which, to a large extent, share the prop- perimentation with the model may deepen expert knowledge. No longer is it viewed erties of synthetic, formal problems. The the understanding and help resolve them. as a collection of absolute truths piped into way gradual membership is represented in an inference engine to derive all sorts of fuzzy sets quite naturally suits such applica- Acknowledgments unexpected results; rather, it is now consid- tion domains. I acknowledge stimulating discussions on this topic at the Tasso workshop 1993 in Bonn; at the ered as a system of more or less soft con- The further we move from representing panel discussion on Fuzzy Logic and AI at UCAI straints that are applied to specific situa- human knowledge about clearly delineated ’93 in Chambery, France; at the GI-Workshop tions to make reasonable decisions. problems to representing concepts relating “Fuzzy-Systeme ’93” in Braunschweig, Germany; Soft knowledge is processed differently to open domains, the more we will have to and valuable comments by Gerhard Dirlich. than logic clauses -the reasoning power is overcome certain rigidities of the classical typically due to processing breadth rather formal approaches. Referenter 1, S.E. Palmer, “Fundamental Aspects of Cog- than depth. The ability to use shallow pro- nitive Representation,” Cognition and Care- cessing to merge knowledge from different gorization, Lawrence Erlbaum, Hillsdale, sources produced useful decisions. (Elkan N.J., 1978, pp. 259-303. uses the terms “deep” and “shallow” in two Classical logic has proved extremely useful different senses: to distinguish general for solving formal problems specified in Christian Freksa is a professor in the Computer Science Department and the Cognitive Science knowledge from specific knowledge, and to two-valued terms. Fuzzy logic is proving Graduate Program at the University of Hamburg, distinguish extensive and restricted knowl- particularly useful for quasi-formal prob- Germany. His research interests include qualita- edge propagation. I use the terms here in the lems involving gradual transitions between tive spatial and temporal reasoning, and repre- second sense, which is the usual sense.) various system states. For adequately for- senting incomplete, uncertain, and fuzzy knowl- Elkan appears to attribute the fact that fuzzy malizing less rigid domains, like the open edge about the physical world. He studied computer science and AI at the Technical Uni- systems employ only a few rules to the do- world of human fuzzy concepts, we must versity of Munich, the University of San Fran- main’s simplicity. However, this fact can relax the constraints on the formalisms cisco, and the University of Califomia, Berkeley, also be attributed to the important capabil- even more. Specifically, numerical gradua- where he received his Ph.D. in 1981. His work ity of summarizing complex knowledge tion of membership used in classical fuzzy has appeared in several journals and anthologies, including Approximate Reasoning in Decision into a dense and transparent description. logic is hardly justified for the representa- Analysis, (North-Holland, 1982),Arti$cial Intel- tion of cognitive concepts; instead, less ligence Methodolog): Systems, Applications, Success and limitations.The fuzzy set constraining ordering relations like partial (North-Holland, 1985), and in the Artificial In- paradigm introduced a new concept of soft orderings may be appropriate. telligence Journal (1 992) Christian Freksa can knowledge that helped characterize an im- Considering the fact that it took 25 years be reached at the Fachbereich Informatik, Uni- versitat Hamburg, Vogt-Kolln-Str. 30,22527 portant aspect of knowledge about complex to put fuzzy logic into wide use in the well- Hamburg, Germany: fax 49-40-54715-385; In- environments. It also provides a language to understood engineering domain of control, temet: [email protected] bridge the gap between soft and shallow

AUGUST 1994 21 I Known Concerns About Fuzzy Logic Oscar N. Garcia, George Washington University

I thank Charles Elkan for bringing into the Where is the catch? First, the fourth line of his Theorem 1 - is what deductive tau- open questions about fuzzy inferences that of Definition 1 in Elkan’s paper indicates tologies (those involving implication, and seem to bother him and others. I hope the that each side of a “logically equivalent” particularly those known as the inferential result of this discussion will be a clearer formula has the same evaluation. This is implication tautology’) should be used in understanding of many-valued logics in not a fair imposition, and Elkan need not fuzzy logic if limited by Definition 1. This general, and fuzzy logic in particular. My choose such a formula to make his point. question is worthy of investigation, and has comments address three topics: questions Just consider requiring the valuation of two led to multiple alternatives to Zadeh’s orig- about Theorem 1, the “watermelon” exam- “logically equivalent” formulas: inal definition of implication; however, it is ple, and the issue of fuzzy logic in control. beyond my concern here. My acquaintance t(A AND 4)H t(i(A OR 4)) Much of the confusion surrounding The- (5) with expert systems applications indicates orem 1 stems from its rather unclear state- which, of course, only occurs in the biva- that, in practice, value sets are categorized ment. I interpret the theorem to say the lent case following Definition 1. Equiva- as designated (truth-like), antidesignated following: lences are tautologies, and while the argu- (false-like), and neutral (those for which ments of t on each side of equivalence 5 are insufficient knowledge exists for the model “Letfl(A,B) = (A AND 4)and “logically equivalent” in classical logic, to be useful). A typical example for the real f2(A,B) = B OR (4AND 4).Using Def- they are not so in fuzzy logic where the law interval [0,1] would be antidesignated A = inition 1, ifone were to require the follow- of the excluded middle does not hold. [0,0.4), neutral N = [0.4,0.6], and desig- ing four equivalences - Thus, it is not surprising that the attempt to nated D = (0.6,1]. (The complement of a (1) fl(A,B) HP(A,B)AND evaluate these formulas using the classical designated value is antidesignated and (2) fl(4,B) -f2(4,B) AND bivalent logic interpretation of “logical vice-versa, while neutral values are the (3) fl(A, 43) Hf2(A, iB)AND equivalence” would not yield sound complements of other neutral values.) The (4) fl(4,lB) Wf2(4, 4) results. It can be easily shown that the ma- object of expert systems is to mimic, as nipulation offl orfr in classical logic leads closely as possible, the reasoning of expert -then such a logic system would also re- to a disjunction of a variable and its com- humans in terms of the best causality rela- quire that t(A) = t(B) or that t(A) = 1 - t@)” plement. We should not take a tautology tions known to them, and to incorporate I can prove this supposition or “theo- that supports a rule base in one logic, use it them in a knowledge-based model, often rem” following the valuation 1 of Defini- in another logic that does not support that represented as a rule base. tion 1 for values of A and B in the interval tautology, and expect it to work - and Because fuzzy logic is known as a “nor- [0,1]. Such valuation yields validity for the then go on to claim that a “collapse” of one mal” logic (all of its truth-assuming tau- first of the four equivalences above except logic to another has been proved. The re- tologies are included in classical logic, for 0 < t(A) < r(B) < 1 - t(A) when tf~‘“1)- quirement of “logical equivalence” in Defi- though the converse is not true, as can be t(f2) has the value t(B) - t(A) if t(B)< 1/2 nition 1 is therefore suspect. Elkan raises shown in the case of equivalence 5)the and the value 1 - t(A) - t(B)if t(B)> 1/2. the question of why it is that intuitionistic puzzlement shown by Elkan is not novel. The area where the equivalence is not satis- logic is capable of rejecting the law of the Indeed, the tautology involving the law of fied is an isosceles triangle in the square excluded middle while fuzzy logic is not. the excluded middle from classical bivalent [ 1,O] x [ 1,0] not including the isosceles While this is not directly relevant to the logic does not hold in fuzzy logic, nor in sides. Similarly, for the other three equiva- claimed “collapse,” it is clear that intuition- many other normal many-valued logics. lences, the non-overlapping triangles istic logic is not used to the extent that For those systems in which operations in- where the equivalences are not satisfied fuzzy logic is used in controller design. volving the designated and anti-designated would cover the whole unit square - ex- Another issue that might be troubling defined sets coincide with those of classi- cept for the isosceles sides, which consti- Elkan - implicit in his choice of the func- cal logic, Shaw has enumerated the possi- tute the two diagonals of the square. Thus, tions called f 1 and f2 in my interpretation ble homomorphisms of any ordered desig- either t(A) = t(B)in one diagonal of the nated system into one of 12 groups defined square, or t(A) = 1 - t(B) in the other. by the conjunction table of their designated (D),antidesignated (A), and neutral (N)

22 IEEE EXPERT

. ... I

subsets.* (A deeper and more thorough al- it was incomplete. The problem I find here been possible with short inference chains gebraic approach to the theory of many- is not directly related to the logic, but raises more interesting questions yet: valued logics, including fuzzy, intuitionis- rather to the use and interpretation of the Under what circumstances are long chains tic, and probabilistic inferencing is given model. Elkan has not expressed what he indispensable? How could long chains of by B~lc.~)For example, Shaw shows that calls “implicit” background knowledge in inferences be avoided? However - make the table for element conjunctions from terms of rules. While he uses the equiva- no mistake - even a set of one-layer rules these fuzzy logic subsets is lence operator to define watermelon(x),a requires some form of inference, and rule knowledge engineer trying to identify wa- sets will increase their sequential complex- 6AND termelons might have given two rules (with ity when hysterisis is taken into account. AAAA different logical meaning) from (inside- Elkan repeats conventional wisdom NANN red(x) + watermelon(x))AND (outside- when stating, “The basic problem is that DAND green(x) -+ warermelon(x))to indicate that the ways in which items of uncertain This table is also valid for Lukasiewicz’s the two predicates contribute separately to knowledge are combined must be carefully n-valued logic, which, as one would ex- the implication of watermelon(x). controlled to avoid incorrect inferences. pect, shares many analogies with fuzzy Alternatively, the engineer might have Fixed, domain independent operators . . . logic. The Lukasiewicz’s logics are first given the one rule with the conjoined an- do not work” to which I add: regardless of defined in terms of negation and implica- tecedents. In the former case, in many ex- the logic system. We should not expect to tion, and other operations are defined in pert systems shells the connotation that the find an exact function f such that t(A*B) = terms of these two. The table of conjunc- conditions insidered(x) and outsidegreen(x) f(t(A),t(B)) for a logical operator * unless tions above is also valid for Kleene’s give “separate” results to argue the conse- we know either the functional relations of (strong) three-valued logic based on nega- quent from two different viewpoints would occurrence between A and or, equiva- tion, conjunction, and disjunction opera- yield a heuristic function of the two valua- lently, know that they are independent (and tors. Kleene’s logic has no true-assuming tions - somewhere between the values of if that were the case, an exact analytical tautologies (formulas that always assume insidered(m) and of outsidegreen(m).If the model could be built!) It is then not surpris- the highest truth value if more than one knowledge engineer had selected the ing that knowledge engineering and incre- designated value is available). If only the model (insidered(x)AND outsidegreen(x)) mental learning methods are used in con- operators of Definition 1 are used, and dif- 4watermeZon(x) for this example, it junction with parameter determination to ferent independent operators are defined as would connote the necessity to satisfy “si- compensate for this lack of generic knowl- part of fuzzy logic, then Elkan’s point that multaneously” both related conditions, and edge, not the weakness of a logic system. only DeMorgan-like tautologies are possi- it can be argued that the conservative an- So, what is new? The dogma of generality ble in fuzzy logic is well taken, but of no swer would be the “weakest link” answer versus efficiency strikes again, and knowl- great consequence as long as viable deduc- (the minimum of the two valuations). edge engineering and machine leaming are tive laws are available. (A discussion of not exempted. what those deductive laws could be and Fuzzy logic in control. It seems reason- Elkan’s ability to generate interest in their relation to implication is treated able that the longer a chain of implications both the topic of nonclassical logics for AI nicely by Trillas, who characterizes a with uncertain predicates is - whatever and the need for more general understand- generic “modus ponens generating func- the definition of the approximate deductive ing of many-valued logics and basic re- tion.”’) As these references point out, there law - the more uncertain the result at the search on how it is applied, are important is not only practical but theoretical credi- end of the chain will be (as in computing contributions that should be acknowledged. bility to the inferences proposed for fuzzy the range of values in worst-case designs). It is a good thing that the relatively smooth logic well beyond the limitation to DeMor- So it seems that it would be a good thing, in imprecisions of natural-language semantics gan equivalences suggested by Elkan. general, to have short inference chains and - when contrasted with crisp symbolic Elkan’s acceptance of the so-called col- a small number of rules whenever possible. approaches -are available without exces- lapse as an established fact in his conclu- Furthermore, the fuzzifying and defuzzify- sive complexity when simpler, closed- sions could be considered disingenuous by ing that takes place at times reminds me of form, and linear designs are not forthcom- the finality with which he considers the the reshaping done in the analog transmis- ing. This occurs frequently around those hypotheses of Theorem 1 to be “apparently sions of digital pulses to avoid signal dete- transitional regions of system operation reasonable conditions.” rioration through consecutive repeaters to where decision changes interface, and distort information. points to the value of vagueness in process- The watermelon problem. Elkan’s ver- The fact that so many applications have ing natural language - usually considered sion of this example is a revision of the in the negative - as a useful, approximate, 1993 AAA1 conference publication, where the watermelon model was in error because

AUGUST 1994 23 real-world engineering design tool, a fact iot popularly noticed by researchers in iatural-language processing. We can use fuzzy reasoning, as we do in everyday dis- zourse, when more exact approaches are too complex, time-consuming, costly, or uejust not available.

Acknowledgment Thanks to Massoud Moussavi for some inter- :sting and clarifying discussions on this topic.

References 1, E. Trillas and L. Valverde, “On Mode and Implication in Approximate Reasoning,” Approximate Reasoning in Expert Systems, Elsevier, New York, 1985. 2. K.A. Shaw, “A Classification of Designated Logic Systems,” master’s thesis, George Washington Univ., Washington, D.C., 1988. 3. L. Bolc and P. Borowick, Many-Vulued Logics: Theoretical Foundations, Springer- Verlag, New York, 1992.

Oscar N. Garcia is director of the Interactive Systems Program at the National Science Foun- dation, and a professor in the Department of Electrical Engineering and Computer Science at the George Washington University. His research interests are in artificial intelligence, with emphasis on speech recognition, knowledge acquisition and representation, multivalued logics to represent uncertainty, and human-computer interaction. He has also worked in computer architectures and parallel processing, testing of digital circuits, and arithmetic codes. He coau- thored Knowledge-Based Systems: Fundamen- tals and Tools, published by IEEE Computer Society Press. He is a fellow of the IEEE and the AAAS, and past president of the IEEE Computer Society. He also served several terms as director of the American Federation of Information Pro- cessing Societies. He was awarded the Special Group Award for his work in the Computer Sci- ence and Engineering Model Curricula, the Richard E. Merwin Award for Distinguished Service to the IEEE Computer Society, the 1991 Professional Leadership Award from the IEEE US Activities Board, and the 1994 IEEE Richard M. Emberson award. He earned his BS and MS degrees in electrical engineering from North Carolina State University, and his PhD from the University of Maryland. Oscar Garcia can be reached on the Internet at: [email protected]

IEEE EXPERT I

Elkan Goes Wrong -Again GeorgeJ. Klir and Bo Yuan, State University of New York, Binghamton

Elkan’s article has three basic parts: a ors of Definition 1 are logically equivalent to the fuzzy logic involved. However, the mathematical part consisting of one defini- f and only if their truth values are equal for theorem is still stated incorrectly or, altema- tion and one theorem; a discussion of the dl possible assignments of truth values in tively, its proof is incorrect. The proof de- roles played by fuzzy logic in expert sys- 0,1J to logic variables involved. pends on eight logical equivalencies, only tems and control systems (based upon the The principal result (and the only mathe- me of which is included in the statement. mathematical part); and his appraisal of the natical result) of Elkan’s papers, which pur- The last paragraph of the proof is thus math- roles of fuzzy logic and its likely signifi- ~ortto demonstrate “technical limitations of ematically incorrect. It would be correct if cance in the future. Here we discuss the Fuzzy logic,” is Theorem 1. What is this re- logical equivalencies representing the seven major fallacies we found in the first two; sult? The answer depends on which version implications listed in the paragraph were due to space limitations, we will not ad- 3f the paper you use. In the original version: included as conditions in the statement of dress the third, though we disagree with the theorem. Without these seven logical For any two assertions A and B, either almost all of the author’s opinions. equivalencies as conditions, the theorem r(B)=t(A)or t(B)=l - t(A). As is well known, Elkan’s article is a must be reformulated as follows: revised version of his original paper, pub- The theorem is supposed to apply to the Given the formal system of Definition I, lished last year. These two versions are not system of fuzzy logic introduced by Defini- for any two assertions A and B, if fully compatible, especially in the mathe- tion 1. However, as explained above, the 7(A A 43)and B v (4A 4)are logi- matical part. Here, we point out discrepan- definition is based on the logical equiva- cally equivalent, then the truth values cies between the two versions and address Lence of two-valued logic and hence it is not t(A)and t(B) are constrained by the in- both alternatives. a definition of a fuzzy logic system. The equalities t(A)+t(B)>I or r(B)lr(A). In Definition I, Elkan introduces a par- proof of the theorem is based on the fact ticular system of fuzzy logic by choosing that the sentences -(A A 43) and B v In this case, the last paragraph of Elkan’s the standard fuzzy operators for conjunc- (4A 43)(and seven other pairs of sen- proof is incorrect and must be excluded. tion, disjunction, and negation, and by re- tences obtained by exchanging and comple- Assume that the statement of Theorem 1 quiring that “t(A)=t(B)if A and B are logi- menting A and B) are logically equivalent in and its proof are made compatible in one of cally equivalent,” where t(A)and t(B)are, classical two-valued logic. However, these the two ways we suggest. What then is the respectively, the degrees of truth of arbi- sentences are not equivalent in a fuzzy logic meaning of the resulting theorems - one trary propositions A and B. Clearly, t(A) that employs the logic operators of Defini- with the single logical equivalence as a and t(B)are values in [0,1]. In the original tion 1. Hence the theorem has no relevance condition, and one with the eight logical paper, the term “logically equivalent” is to this fuzzy logic. Let us tum now to the equivalencies as conditions? These theo- defined as “equivalent according to the revised version of the theorem: rems basically show that the truth values of rules of classical two-valued propositional propositions within the system of fuzzy Given the formal system of Definition 1, calculus.” This is, of course, nonsense, logic introduced by Definition 1 become if -,( A A 43)and B v (4A 4)are since one logic system (in our case, a par- appropriately constrained when additional logically equivalent, then for any two ticular system of fuzzy logic) cannot be extraneous conditions are imposed. With assertions A and B, either t(B)=t(A)or defined in terms of logical equivalence of the eight conditions, the constraint is obvi- t(B)=l -t(A). another system (the more restrictive classi- ously more severe than with only one of cal two-valued logic). The fundamental difference between the them. If, for example, we required our sys- In the revised version, the meaning of the original and revised version of the theorem tem to satisfy A v 4 = 1 then the truth term in Definition 1 is not explicated. It is reflects the difference in the two versions values would become constrained to the set only remarked that “depending on how the of Definition. 1. In the revised version, the { 0,1], and the system would collapse to the phrase ‘logically equivalent’ is understood, logical equivalence of -,(A A 43)and B v classical two-valued logic. Definition 1 yields different formal sys- (4A 4)is employed as a condition in All this is well known, and Elkan’s theo- tems.” Since the role of Definition 1 is to stating the theorem rather than a fact in rem (when properly fixed) does not offer characterize a system of fuzzy logic, logical proving it. If the notion of logical equiva- anything new. It is absurd, however, to con- equivalence in this definition must be ex- lence in the revised Definition 1 is under- strain a system by extraneous requirements pressed in terms of all possible truth values stood as applying to all truth values in and then claim that the original system has of fuzzy propositions, that is, in terms of all [0,1], in spite of its confusing characteriza- “technical limitations.” This is what Elkan real numbers in [O,l]. Specifically, two ex- tion by the author (as discussed above), then attempts to do in his papers. The fact that pressions in fuzzy logic based on the opera- the revised version of Theorem 1 is relevant every system of fuzzy logic must violate,

AUGUST 1994 25 under the assumption of truth functionality, relevant crisp sets. While these two areas References D. Dubois and H. Prade, “An Introduction to some properties of Boolean algebra (and, have distinct application domains, they can 1. Possihilistic and Fuzzy Logics,” Non-Stan- hence, the classical two-valued logic) is a be combined, resulting in statistics with dard Logics for Automated Reasoning, Acad- simple consequence of the decision to for- imprecise probabilities6 or in fuzzified evi- emic Press, New York, 1988, pp. 287-315. mulate logics that can deal with propositions dence theory; for example. 2. G.J. Klir and T.A. Folger, Fuzzy Sets, Uncer- that are not required to be either true or false, In his discussion of fuzzy controllers, tainty, and Information, Prentice Hall, Engle- wood Cliffs, N.J., 1988. but may be true or false to various degrees.’ Elkan’s lack of understanding is again re- 3. N. Rescher, Many-valued Logic, McGraw- Elkan’s remarks about the connection vealed. He fails to understand that most of Hill, New York, 1969. between fuzzy logic and intuitionistic logic the simple fuzzy controllers on the market 4. A.A. Zinov’ev, Philosophical Problems of also contain some errors. For example, it is (we may call them the first generation of Many-valued Logic, D. Reidel, Dordrecht, not sufficient to characterize fuzzy logic by fuzzy controllers) are not explicitly based on Holland, 1963. the rejection of the law of excluded middle. fuzzy logic, but rather on the approximation 5. 2. Wang and G.J. Klir, Fuzzy Measure The- ory, Plenum Press, New York, 1992. The system of fuzzy logic determines which of relevant control functions by fuzzy num- 6. P. Walley, Statistical Reasoning with Impre- properties of Boolean algebra are rejected. bers that represent chosen linguistic states of cise Probabilities, Chapman and Hall, New The system introduced by Definition 1, for the variables involved. This is similar to York, 1991. example, rejects not only the law of excluded classical control, which is also not explicitly 7. J. Yen, “Generalizing the Dempster-Shafer middle, but the law of contradiction as well? based on classical two-valued logic. It is Theory to Fuzzy Sets,” IEEE Trans. Systems, Man, and Cybernetics, Vol. 20, No. 3, May- This differs from intuitionistic logic, which well established that fuzzy controllers of this June 1990, pp. 559-570. rejects the law of excluded middle and the kind are universal appro xi mat or^.^.^ 8. J.J. Buckley and Y. Hayashi, “Fuzzy implication -, 44 A, but does not reject While most existing fuzzy controllers Input-Output Controllers are Universal Ap- the law of contradiction and the opposite are rule based, research on combining rule- proximators,” Fuzzy Sets and Systems, Vol. 58, No. 3, Sept. 1993, pp. 273-278. implication A + 4.3,4 Other systems and model-based approaches in designing 9. B. Kosko, Neural Networks and Fuuy Systems: of fuzzy logic do not reject any of the men- fuzzy controllers is ongoing. Models em- A Dynamical Approach to Machine Intelli- tioned laws; instead, they reject distributivity ployed in these controllers are expressed, gence, Prentice Hall, Englewocd,N.J., 1991. and idempotence.*Furthermore, de Mor- in general, in terms of relations among rel- gan’s laws are valid only in some systems of evant fuzzy variables. Hence, the use of George J. Klir is distinguished professor of sys- fuzzy logic. Another error is to consider the fuzzy set theory (not necessarily fuzzy tems science in the Watson School of Engineering logical equivalence in Definition 1 as intu- logic in the narrow sense) involves both and Applied Science at the State University of New York, Binghamton. He is author or coauthor itionistic equivalence. A distinctive feature of parts of the controller -the rule-based of 13 hooks, including Fuzzy Sets, Uncertainty, intuitionistic logic is the operator of negation part as well as the model-based part. and Information (Prentice Hall, 1988) and Fuzzy upon which it is based. For any proposition Measure Theory (Plenum Press, 1992), and is the editor of seven books. He has also written more A, where t(A) E [0,1], the intuitionistic nega- than 300 research papers, and has been the editor tion, 4,is defined by Elkan’s papers do not contribute to knowl- of the Int’l Journalof General Systems since 1974, edge. The mathematical part is fallacious; and editor of the Int’l Book Series on Systems Sci- 1 when t(A)= 0 and, while some critical errors in the origi- ence and Engineering since 1985. He is past presi- 0 otherwise dent of the International Federation for Systems nal version are corrected in the revised Research, and the North American Fuzzy Informa- This negation is not involutive, nor is it con- version, new errors are introduced and tion Processing Society. He is currently president tinuous - it acts as a defuzzifier. Clearly, some statements become less specific. of the International Fuzzy Systems Association. He there is no compatibility between intuition- Even if we fix all the mathematical errors was recently awarded an honorary doctoral degree from the University of Economics in Prague, and istic logic and the fuzzy logic introduced in to help Elkan obtain his intended result, we the Gold Medal of Bernard Bolzano in mathemati- Definition 1. find only that the result is trivial and well cal sciences from the Czech Academy of Sciences. known: If one takes an axiomatic system Klir can be reached at the Department of Systems Science and Industrial Engineering, SUNY, Bing- Fuzzy logic applications. Elkan’s discus- and adds to it additional requirements, the hamton. N.Y. 13902-6000. sion of fuzzy logic in expert systems reveals system becomes more constrained. Given a his confusion between degrees of truth in free choice of requirements, one can con- BoYuan is a PhD candidate in systems science at fuzzy logic and degrees of evidence strain the system as he or she wishes. This the Watson School of Engineering and Applied expressed in terms of some fuzzy measures is precisely what Elkan attempts, in an am- Science at the State University of New York, Bing- hamton. His research interests are in fuzzy set (probabilities, belief measures, and so on).5 ateurish way. He tries to find requirements theory, fuzzy logic, approximate reasoning, fuzzy While the former are a matter of compatibil- that would constrain a given system of control, neural networks, genetic algorithms, and ities of given objects with relevant fuzzy fuzzy logic so severely that only two truth their applications in industrial engineering. He predicates, the latter result from information values are allowed. He then argues that this received his BS and MS degrees from Shanghai Teachers University in 1985, and 1988, respec- deficiency regarding the classification of a shows technical limitations of fuzzy logic. tively. Yuan can be reached on the Internet at given (incompletely characterized) object in This sort of argumentation is absurd. ba05074@hingvmb

26 IEEE EXPERT I

A Misconcention of Theorv and Applicatio6 EH. Mamdani, Queen May& Wesgield College, London

The argument in Charles Elkan’s article has form of a calculus. However, after reading features of the domain knowledge intro- three steps. First, he provides a theorem Boole’s “Laws of Thought” it is difficult to duced in an application also contribute to that “proves” that fuzzy logic is deficient discern whether Boole is concerned with a its success. There is a common misconcep- because it collapses to a two-valued logic. descriptive explanation of how people ac- tion that models are created and then ap- He then shows what makes the current ap- tually think, or with a prescriptive model of plied, and that success then legitimizes a plications of fuzzy logic successful, how they ought to think. AI research work- model. This view is superficial, because an although this success may seem paradoxi- ers have seldom addressed this key distinc- application’s requirements seldom match cal. His final step shows how such a suc- tion properly. the underlying axioms of the model cess cannot be guaranteed as applications Within AI there are three distinct areas exactly. The fixes that are added (defuzzifi- scale up in the future -thus resolving the of research: the descriptive, the prescrip- cation in fuzzy logic control) are instru- paradox. I expect other commentaries will tive, and what I call the applicative con- mental in the industrial success -but deal with the misconceptions regarding the cerns. In the first area, researchers deal often sit uncomfortably in the original the- theorem. I focus attention here on the re- with descriptive theories about cognitive ory. This is true of all applications inspired mainder of Elkan’s argument. processes. These theories are very hard to by prescriptive models. The source of Elkan’s paradox is the link prove experimentally (or more specifically The links between these three groups between the first two steps formed by his to disprove experimentally - if one ap- (descriptive, prescriptive and applicative) statement that “One way to defend a calcu- plies the Popperian view) because the level must be properly understood if one is to lus is to show that it succeeds in interesting of control in experimental studies on avoid the methodological trap Elkan has applications.” But first a couple of very human cognition is far below that in the fallen into. In AI, the work of each group different but relevant pointers. natural sciences. The second group of re- inspires the direction of the others -but Fuzzy logic control is successful because searchers are concemed with prescriptive that is all they do. The results of one group it replaces the classical PID controller. models: different reasoning systems and a can never be used to legitimize the approach When tuned, the parameters of a PID con- variety of logics. Here, the issue is one of of another. Weak though these links are, troller affect the shape of the entire control correctness of these models, variously de- they still play a significant role in scientific surface. Because fuzzy logic control is a fined. Again, it is not possible to use nat- advances. My point is not to belittle the rule-based controller, the shape of the con- ural-science methods to devise controlled interplay between the three areas, but to trol surface can be individually manipu- experiments that demonstrate the correct- point out that a misunderstanding of their lated for the different regions of the state ness of these models; correctness can only relationship is clearly the source of Elkan’s space, thus limiting possible effects to be dealt with by means of philosophical perception of the paradox. neighboring regions only. Furthermore, the arguments (more on this later). What then is the relationship between use of fuzzy mathematics provides interpo- I belong to the third group of AI fuzzy logic control applications and fuzzy lation between the adjoining regions, re- researchers, whose main concern is to logic itself? Precisely the same as that be- sulting in an overall smooth control surface build industrially successful artifacts. Such tween Boole’s laws of thought (a descrip- - an important requirement in the control artifacts are successful in their own right, tive theory?), Boolean logic (a prescriptive of continuous systems. This also suggests and do not owe their success to the under- model?), and logic circuits (an application) that fuzzy sets are an efficient way of rep- lying theory or a mathematical model. It is -namely, an effective tool presented itself resenting continuous variables in rule- sad how many AI workers have lost the that met many, though surely not all, of the based systems. ability to distinguish between applications application needs. However, the widespread Secondly, I have always felt that fuzzy and well-designed controlled experiments success of logic circuits cannot be used to logic has similarities with Boole’s logic. set up to disprove a particular theory. Ap- legitimize Boole’s logic any more than the That logic, originating over 150 years ago, plications address the scientific needs of a industrial success of fuzzy logic control was the first system of reasoning in the specific domain, and cannot replace experi- legitimizes the philosophical correctness of ments conducted to test a theory. Many fuzzy logic. Therefore, the question of a paradox - a central idea in Elkan’s paper

AUGUST 1994 27

I edited by Frances Brazier and Dag Johansen

The 11 papers that follow the introduction discuss experience with actual systems and applications -does not arise. Similarly, his argument that fuzzy logic control is not worthy of The5e papers report on in- on the philosophical deficiencies of fuzzy industrial consideration because of its lack terprocess communication, logic focuses on a theorem without fully of complex form and structural sophistica- the Chorus dpprodch to d discussing the assumptions and axioms it is tion, as Elkan effectively does in the final inicrokernel-based IJNIX based upon; this does nothing to argue part of his paper, is to subscribe to an anti- system, and Plan 9 from inventions culture. Accentuating form with- AT&T, the 05F framework against the adoption of fuzzy logic control. for applications and security The terms “logic” in logic circuits and out attention to the content is like praising issues, the evolution of an “fuzzy logic” in fuzzy logic control are beauty and ignoring the brain. To use the authentication service, the purely incidental, and a matter of historical colloquial term, the scientific mythology Isis system, and tools for evolution. within AI has created a “bimbo science.” monitoring and controlling The AI approach puts a much higher The scenario worth keeping in mind is value on prescriptive mathematical models that since its inception, fuzzy logic has had 192 pages 1993 than they actually deserve. These models its detractors and antagonists not least be- ISBN 0-8186-4292-0 cannot be legitimized by controlled experi- cause the tag “fuzzy” is seen as debasing to Catalog # 4292-03 ments or by application, nor can they be the somber image of science. So incensed $40.00 Members $32.00 justified by some underlying descriptive are some that they will clutch at any straw theory (in spite of Boole). Prescriptive to rid us of fuzzy sets research, even IEEE COMPUTER models can only be argued over at a philo- through a paper based on mistaken inter- SOCIETY PRESS sophical level - an ability few AI pretations and modish posturing. This sce- researchers possess. Philosophical disputa- nario leaves me saddened, for reasons ex- Call 1-800-CS-BOOKS tions about prescriptive models within in- plained above. formed groups such as Uncertainty in AI, It is the word “paradox” I find most baf- have, nevertheless, helped to enlighten fling in Elkan’s article. Science at its best is many difficult points. In the end, however, often counterintuitive; but paradoxical? Our such disputations can never completely accepted understanding of the scientific settle the matter. method is based on natural science and de- Because AI researchers are mostly scriptive theories. But applying descriptive trained in mathematical skills, another fre- theories to computer science - which is quently applied but false way of legitimiz- dominated by prescriptive theories -can- ing prescriptive models is on the grounds not, in my opinion, work. New prescriptive of mathematical symmetries or some in- theories often alienate many researchers, but trinsic sophistication of potential function. they also inspire others to build novel appli- On rare occasions when models are cations. It may be that some of these appli- abstracted from applications, the concern is cations are a runaway success. Rather than no longer what led to the success of the talking of “paradoxes,” what is required at application. Rather, the academic game of this point is a rigorous attempt to discover looking for the symmetries and the sophis- the secret of that success. Because this in- tication of the form or the soundness of the vestigation is descriptive in nature, the tradi- calculus begins. tional scientific method is likely to yield dividends. In the case of fuzzy control, this process is now underway.

E.H. “Abe” Mamdani is a professor in the elec- Having rightly or wrongly detected a para- tronic engineering department of Queen Mary & dox, one then has to resolve it; in doing so, Westfield College, University of London. His Elkan commits further errors. He has a lot research in fuzzy logic control began in the early to say about the small number of rules, the 1970’s, and he has acted as a consultant to sev- shallowness of fuzzy rule bases, and so on eral European industrial companies on applications of knowledge-based systems. His current - implying that some beauty of the form research interests include reasoning under uncer- often plays a significant role in assessing tainty, agent-based systems, and soft computing. the worth of a model (and the intellectual He is a fellow of the IEEE and the Royal Acad- enterprise of a researcher) rather than the emy of Engineers. Abe Mamdani can be reached through the Intemet at [email protected] content or industrial usefulness. To argue

IEEE EXPERT

I Fuzzy~~ Logic A Misplaced Appeal Francis.Jefy Pelletiel; University of Alberta

I have long found puzzling the acceptance system. Supporters of fuzzy logic are with- “autodescriptivity” in the logic: a way of and apparent success of fuzzy logic. We out doubt tempted to respond to this by mirroring the semantics within the syntax. philosophically oriented logicians have focusing on the assumption that logical This autodescriptivity is regarded by some pretty much sneered at fuzzy logic ever equivalence in classical (or intuitionistic) authors as necessary for the adequacy of since it was introduced with that name.* logic is a warrant for formulas having the any many-valued logic,* for without it, the Yet what can I say when I own an excellent same truth value in fuzzy logic. I do not apparent many-valuedness is only illusory fuzzy logic camera? I am grateful to Elkan wish to enter this debate; instead, I will because we cannot say anything in a many- for his explanation of this point of tension. take this opportunity to point to some other valued way. There are a number of ways of Basically, Elkan explains that the notion features of a logical nature that have been accomplishing this, depending on what of “fuzzy logic” as it is used in control sys- used to criticize fuzzy logic and its claims sorts of operators are available within the tems has nothing to do with the term as it is of usefulness in various tasks. language. The direct way is to have so- used in logic. That is, it has nothing to do Presentations of fuzzy logics have gen- called parametric operators in the language: with fuzzy logic as a formal system with erally been semantic in nature, while the For each k, where 0 5 k 5 1, there is a unary rules of formation, evaluation, and infer- syntax - axioms and rules of inference - sentence operator Jk.The truth of such sen- ence. Fuzzy controllers are so-called be- has generally been ignored. The basic se- tences is evaluated thus: cause of a certain analogy with fuzzy logic, mantic notion is that propositions can take = 1, if t(@)= k but in fact they do not embody, implement, any real value in [O.. .l]intuitively corre- t(Jk[@]) = 0, otherwise. or instantiate fuzzy logic. sponding to “degrees of truth” of the For Elkan, the relationship between proposition. Many advocates of fuzzy That is, a Jk operator says that the formula it fuzzy controllers and fuzzy logic is rather logic, especially those who want to replace operates on takes exactly the value k. Al- like that between on-off light switches and classical logic as the medium of represen- though there are other approaches, I will predicate logic: Yes, there is a certain anal- tation for ordinary reasoning and the de- adopt this direct approach - that the lan- ogy between on-off and true-false, but it’s scription of natural-language phenomena, guage being used to “express the semantics” only an analogy, a way of looking at light would like to “use” the semantics of fuzzy contains the parametric operators directly. switches. There is nothing in the light logic. That is, they are not interested (There are many other ways to get their switch corresponding to the connectives of merely in asserting theorems, nor in the effect. Some writers allow constants - sentence logic nor to predicates, names, uninterpreted formulas of fuzzy logic, but symbols that denote the truth values - and quantifiers of predicate logic. To iden- rather would like to be able to claim that a others have “threshold operators,” and so tify the two, or to say that light switches proposition is true to a certain degree, that on. With suitable such other operators, we implement or instantiate predicate logic, it can be compared to another proposition can indirectly define the parametric opera- would be to ignore most of predicate logic which is true to some different degree, and tors. Since there are innumerable truth val- and mistakenly fixate on just one insignifi- that certain conclusions can be drawn from ues in the real range [O.. . I], the methods of cant aspect. According to Elkan, we should this comparison. autodescriptivity mentioned here cannot not be surprised that critiques of fuzzy For example, it might be that “Sally is really be applied. Instead, we must con- logic have no impact on fuzzy control the- wealthy” is true to degree 0.7 while “Mike sider the fuzzy logic generated by the TU- ory; the areas of fuzzy logic that get criti- is wealthy” is true to degree 0.4. Now, we tionals in the [O.. .1] interval. Attempts by cized are simply not employed in the con- might wish to draw certain conclusions fuzzy logicians to incorporate ever more trol arena (whether practical or theoretical). from this information, such as that Sally is inclusive - some would say obscure - Elkan’s theorem shows one of the diffi- wealthier than Mike, or wealthier to a cer- operators indicates to me a lack of appreci- culties surrounding fuzzy logic as a formal tain degree than Mike. To do this, we need ation of what a logic is. For, if it can be some way to “use the semantics.” Techni- shown that there is no algorithmic, deter- ~~ ~~ cally speaking, we wish to have a kind of ministic procedure to determine the truth It was studied by Lukasiewicz and A * J value of an arbitrary expression, then it is Tarski in 1930 under the name “infinitely many- valued logic$,”’ and received intensive qtudy by very unclear that there can be any use of Feveral mathematical logicians in the 1950s and the formalism.) early 1960s *

AUGUST 1994 29

I Observation 1: Fuzzy propositional logic Observation 2: There is no normal form Observation 3: Full fuzzy predicate logii is not argument-complete. The first short- for fuzzy monadic predicate logic in is not recursively axiomatizable. The rea coming of fuzzy logics concerns proposi- which quantifiers have widest scope. The underlying reason that fuzzy logic fails to tional logic (and hence any fuzzy logic, second shortcoming of fuzzy logic is found be of any logical interest does not have to because they all contain propositional logic in the attempt to add quantifiers, even a do with the elementary fragments of propo as a part). There is no theory of argumenta- simple monadic predicate logic. Fuzzy sitional fuzzy logic and monadic fuzzy tion for fuzzy propositional logic such that logic dictates that a universally quantified predicate logic, even though it is cute to whenever all premises of the argument are formula, such as VxFx, takes the least note that even these elementary parts of designated, then so is the conclusion. (Intu- value of all the substitution instances for x fuzzy logic are not usable in the desired itively, some of the truth values are consid- in the formula Fx,or the greatest lower form. Instead, it is that full predicate logic ered “good’ or designated, while the others bound if there is no such least value. An is not really a logic. are undesignated. Exactly which ones are existentially quantified formula takes the This result was proved by Scarpellini3 foi designated might vary from application to greatest value of all the substitution infinite-valued Lukasiewicz logics, and the application. The point is that an argumenta- instances, or the least upper bound if there proof carries over to all the well-known tion theory is designed to take us from is no such greatest value. modifications (such as adding parametric “good” premises to “good” conclusions, A sentence like Jk[VxFx]says that the operators or various arithmetic operators) o and never mislead us by deriving a “bad” greatest lower bound of the Fx’s values is this logic, which includes any of the fuzzy conclusion from “good” premises. The exactly k. There is no formula that has any predicate logics ever described in the litera- present observation says that this cannot be quantifier outside the scope of Jk that has ture. The thrust of the proof is that the set of done, ever; and this holds for any decision the same truth value. For example, b”ml,[Fx] unprovable formulas of ordinary two-value’ on what is designated, so long as at least says that every individual instance of Fx predicate logic can be mapped one-to-one one number is designated and at least one has a greatest lower bound of exactly k, into the set of valid (designated) formulas o is undesignated.) which is clearly wrong. 3xJk[Fx]says that fuzzy logic, for any closed or open range of This result does not depend on there there is some particular individual that is F values (k...l) that we designate. But the set being (or lacking) any particular syntactic to exactly degree k, which is also wrong of unprovable formulas of ordinary predi- machinery around (other than the paramet- because there might not exist an object that cate logic is not recursively axiomatizable, ric operators); rather, there simply can be has the greatest lower bound value. But the and therefore neither is the set of valid for- no such theory of argumentation. The proof lack of a normal form makes it unlikely that mulas of fuzzy logic. Hence, they cannot of this is via the fact that fuzzy logics are there can be any method to detect theorem- even be adequately characterized or talked not semantically compact. That is, it is not hood in fuzzy monadic predicate logic. Cer- about coherently, except by example. Fur- true for fuzzy logic that a set of formulas is tainly resolution will not work. thermore, fuzzy control theorists do not satisfiable just in case every finite subset of So, not only can we not tell when a con- merely wish to appeal to examples of valid it is satisfiable. For example the infinite set clusion is validly derived in fuzzy logic formulas of fuzzy Iogic, but to be able to (Observation l), but we cannot even tell characterize them in some way or other. r={lJk[piio5k<~1 when a formula (even of monadic predicate is not satisfiable, since the sentence letter p logic) is a theorem. Surely together these must take on one of the values 0 5 k 5 1, two observations should give fuzzy control whereas the membership condition in r theorists pause; they show that fuzzy logic Lest my message be thought entirely critica says it doesn’t. Yet any finite subset of r is as an abstract theory reduces to stating in- of fuzzy control theory, let me point out tha satisfiable. Similar sets can be described tuitive principles without any way to gen- I believe that everything its proponents wis using quantified sentences, such as eralize or use them. And, since fuzzy con- to do can be adequately carried out. (My trol theory is surely committed to using camera works!) However, their appeal to { J,[Fa113Jl/z[Fazl, > I-’= J,,dF%I,... something - it follows that what it is com- fuzzy logic is misplaced. Every fuzzy logic 7Jo[VxFx]] mitted to using is not fuzzy logic, just as application has an analogue in finitely Having noted this fact, it is an easy step to Elkan said. many-valued logic, and each one of these is the conclusion that fuzzy logic, even fuzzy logically well-behaved. There are correct propositional logic, is not argument-sound, theories of argumentation for them, there are since all proofs are finite. Thus, there can resolution-like theories of theorem-detec- be no adequate scheme for making infer- tion for them, and they are axiomatizable. ences in general within fuzzy logic. The only apparent advantage to fuzzy logic is that it seems to be a grand generalization of all those finitely many-valued log-

30 IEEE EXPERT

I I

ics -after all, we never know in advance times. Such a logic does not have any of the Jeff Pelletier is a professor of philosophy and which particular finite value might be shortcomings that fuzzy logic does, and computing science at the University of Alberta. He has written more than fifty articles on philoso- needed for a specific application. However, would seem to be the sort of thing that could phy of language and logic, computational linguis- it is an illusion to think that fuzzy logic is the form a logically adequate background theory tics and automated theorem proving, and ancient correct generalization. It cannot be used and for fuzzy control systems. Greek philosophy. His is author of Mass Terms: it has no reasonable logical foundation. But a Some Philosophical Issues (Reidel, 1979), Par- “variable precision” finitely many-valued References menides, Plato, and the Semantics ofNot-Being (U. Chicago Press, 1990), and The Generic Book logic can do the sort of things desired. In 1. J. Lukasiewicz and A. Tarski, “Untersuchen (U. Chicago Press, 1994), and is currently writing such logics we have a superstructure of (say) uber den Aussagenkalkul,” 1930; translated a book about automated theorem proving in non- three values. Having determined that some as “Investigations into the Sentential Calcu- classical logics. He is also interested in cognitive sentence takes one of these top-level values, lus” in Tarski’s Logic, Semantics, Meta- science verification of AI constructs. Pelletier was the Luce Professor of Cognitive Science at the we can then expand it to determine which of mathematics, Oxford Univ. Press, 1956, pp. 38-59. University of Rochester and is a past president of those three values it has at a lower level. For the Society for Exact Philosophy. He received his example, we grade essays as “good,” “so- 2. N. Rescher, Many-valued Logics, McGraw BS in mathematics and education, and his MA in so,” and “bad,” but then given that an essay Hill, New York, 1969. philosophy from the University of Nebraska in has been categorized as “so-so” we can look 1965 and 1966, and his PhD in philosophy from UCLA in 1971. He also received MS degrees in more closely at whether it is a good, so-so, or 3. B. Scarpellini, “Die Nicht-Axiomatieier- linguistics (1978) and computing science (1982) bad example of being so-so. And this process barkeit des Unendlichwertigen Pradikaten- from the University of Alberta. Jeff Pelletier can can continue for some finite number of kalkuls von Lukasiewicz,” J. Symbolic Logic be contacted at the Department. of Philosophy, Vol. 27, No. 2, June 1962, pp. 138-153. University of Alberta, Edmonton, Alberta, Canada T6G 2E5; Intemet: [email protected]

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.- I On the Purportedly Paradoxical Nature of Fuzzy Logic Enrique H. Ruspini, SRI International

Elkan’s original paper purportedly showed literature and issues, but mostly out of ig- For example, if (C,U,1) are negation, dis- that fuzzy logic was paradoxical in nature norance or plain confusion - are appropri- junction, and conjunction operators, respec- due to its reliance on formal bases that pre- ateIy addressed by other respondents. tively, that is - clude truth values other than 0 and 1. Elkan Starting from an axiomatic characteriza- t(-P) = C(t(p))3 has now modified some of his claims and tion of fuzzy logic proposed by Gaines, and t(P v 4) t(4))‘ arguments slightly, although he still de- assuming that logical equivalence in fuzzy = W@)? A t(4)) pends on that result as his major source of logic means equivalence in the sense of t07 9) = IMP), insight into the technology. classical logic (thus implying that all clas- that satisfy the laws of excluded middle We are now told, for example, that sical logic theorems are also fuzzy logic and contradiction, then the corresponding fuzzy logic is paradoxical because it is theorems), Elkan shows that Gaines’ ax- logics can be neither idempotent nor dis- successful in many applications while its ioms imply that the only possible truth val- tributive. If Elkan had probed further, he foundations remain under attack. Perplex- ues are 0 and 1: fuzzy logic collapses into could have proved that all continuous ing as this situation might be from a socio- conventional logic. truth-functional multivalued logics “col- logical viewpoint, it is hardly a logical Anybody acquainted with fuzzy logic, lapse’’ as well. self-contradiction, and describing it as a however, would not have much difficulty The definition of equivalence that Elkan paradox is totally inappropriate. questioning Elkan’s notion of logical equiv- describes as “apparently reasonable” is, Nor is a paradox implied by the claim alence; it is well known that many theorems therefore, patently unreasonable. The sup- that most theoretical fuzzy logic papers of propositional logic are not valid in fuzzy posedly shocking result is just a well- seem to deal with representation and rea- logic. Assuming otherwise immediately known fact of little relevance to the prac- soning methods, while most fuzzy logic leads to the result that Elkan finds so para- tice of fuzzy logic. Simply stated, Elkan applications have resulted in embedded doxical. Applying, for example, the axioms has found that fuzzy logic and the classical controllers. The embedded controllers have of fuzzy logic to the law of the excluded propositional calculus are different logical been developed, of course, upon founda- middle av la, which is not a theorem of systems. tions provided by the representation and fuzzy logic, leads to the equation Not much is gained either by looking inferential methods of fuzzy logic. Elkan is into seemingly more congenial quarters for max(t(a), 1-t(a))= 1, not only unaware of this fact, but his over- alternative definitions. Elkan turns, for all analysis of the technology is colored by which only has the solutions t(a)= 0 and example, to intuitionistic propositional the strange notion that the depth and qual- t(a)= 1. calculus (IPC) as another place to borrow ity of deductive procedures in a controller Many theorems of classical propositional notions of equivalence, feeling that his are inferior to those in “sophisticated” rea- logic may also be used to derive this result. result is strengthened by the fact that the soning systems. Elkan’s unnecessarily lengthy proof - law of the excluded middle - a previous For reasons of space, I will not discuss based on the conventional propositional source of trouble - fails for both IPC and here Elkan’s statements about application logic equivalence of the formulas 7(u A ,b) fuzzy logic. IPC is, however, based on a of fuzzy logic to control and other intelli- and b v (lu A ,b) - actually assumes the negation operator with different semantics gent reasoning systems, but will confine validity of the law of the excluded middle than that of fuzzy logic (one is involutive my comments to the formal result (Theo- (to see this, simply expand the latter and while the other is not). Once again, one rem 1) that remains the major basis of his note the conjunct b v lb). does not need a proof as extensive as claims about the purported paradoxical Elkan’s “shocking” discovery has been Elkan’s. The Godel translation l(la A nature of fuzzy logic. Other assertions long known, and is discussed in elementary la) of the law of the excluded middle is about the methodology -arising in some textbooks on fuzzy and multivalued logics.’ a theorem in IPC but not in fuzzy logic. cases from superficial analyses of relevant Assuming that it is leads once again to the same incorrect conclusion: Fuzzy logic collapses. Elkan’s theorem is, therefore,

32 IEEE EXPERT just as true for IPC equivalence as it was Those who have read Elkan’s original Elkan’s arguments, arising from a meaning- for classical equivalence, but it is also just paper wondered at the time why he had to less result and a superficial and confused as meaningless as before: All that has been seek definitions in other logics rather than evaluation of the state of the art in fuzzy proven is that fuzzy logic is neither classi- proceeding along the lines I have sketched logic, do not provide any substantial insights cal nor intuitionistic logic. here. In the present article, Elkan at last into the methodology, its advantages, or its This explanation, however, still does not considers a definition based on the seman- shortcomings. Given the weakness of his answer a basic question: What is the mean- tics of the negation, disjunction, and con- arguments, one can only be astonished at his ing of the word equivalent in Gaines’ junction operators, but not on that of the conclusion that proponents of fuzzy logic Axiom 4: implication connective (see his last para- are guilty of fallacious non-sequitur think- graph in the section on paradox). He con- ing (post hoc, ergo propter hoc). Those who [(a)= t(b)if a and b are logically cludes, however, that this leads to an ex- propound the technology found their claims equivalent? tremely weak system where the only on solid theoretical results and on thousands In classical logic, logical equivalence be- equivalences are the De Morgan axioms. of examples of its successful application. All tween two formulas a and p may be de- This statement, unlike previous claims, that Elkan produces, on the other hand, is an fined either as the validity of the formula is not only irrelevant but false and mislead- irrelevant theorem and a rather shallow and a ts p, or as the equality of the truth values ing. Simple application of fuzzy logic oper- mistaken discussion of a minor segment of of the formulas a and p for all possible ators for disjunction, conjunction, and the literature. assignments of truth values to their con- negation immediately shows that the fol- stituent propositional symbols. A quick lowing laws of propositional logic also References inspection of the truth table of the H con- hold in fuzzy logic: commutativity of dis- 1. G. Klir and N. Folger, Fuzz? Sets, (incrrtuintj nective shows that these definitions are junction and conjunction; associativity of Information, Prentice Hall, Englewood equivalent. disjunction and conjunction; distributivity Cliffs, N.J., 1988, pp. 52-59. While this is very reasonable, something of disjunction (conjunction) with respect to seems to be amiss here. How can we con- conjunction (disjunction): idempotence of 2. N. Rescher, Many-Vulued Logic, McCraw- sider Axiom 4 before we even define logi- disjunction and conjunction; identity with Hill, New York, 1969, pp. 138ff. cal equivalence? If equivalence means that respect to T and 1;absorption with re- 3. R. Lopez de Mantaras, Approximate Rea- the truth value of a is always equal to that spect to disjunction and conjunction; ab- soning Models, Ellis Horwd, Chichester, of p, why do we need an axiom to state that sorption by T and I;involution; and, England, 1990. this should be the case? surely enough, the De Morgan laws. In multivalued logics, equivalence in the All these properties give fuzzy logic Enrique H. Ruspini is a senior computer scientist with the Artificial Intelligence Center of SRI sense of the validity of a tf p is not the considerable strength as a reasoning for- International. He received his doctoral degree same as equivalence in the sense of equality malism, but their consideration alone - in from the University of California, Los Angeles, of the truth values of ~1 and p. For example, the absence of definitions for the implica- and has extensive research experience in the these notions yield the same relation in the tion connective + and for the deductive areas of approximate and commonsense reason- Lukasiewicz L3 logic, but not in the 3- rules of fuzzy logic (such as the general- ing, the calculus of evidence, knowledge-based valued logic of Bochvar (where, if and systems, knowledge acquisition, inductive rea- a p ized modus ponens) - cannot be the bases soning, and the representation and manipulation have the third value 1/2, then a ts p also of any substantive argument, either pro or of uncertainty. He was an early contributor to the has the third value 1/2). In these logics it is con, regarding the adequacy and correct- development of fuzzy-set theory and its applica- possible to consider several characteriza- ness of fuzzy logic as a deductive method- tions, and pioneered its introduction to automatic tions of the notion of logical equivalence, ology. Curiously, Elkan does not seem to classification and pattern recognition in 1969. His recent research has focused on the applica- each having different formal properties.* feel that there is any need to discuss these tion of fuzzy-logic techniques to the develop- In multivalued logics in general, and matters, interpreting the independence of ment of intelligent control and signal processing fuzzy logic in particular, equivalence is his theorem from any notion of implication systems. He is a Fulbright Fellow and a SRI usually defined in terms of the semantics of as a sign of its universality and strength Institute Fellow, and is one of the founding the connective. Several such definitions members of the North American Fuzzy Informa- + rather than as yet another indicator of its tion Processing Society and a recipient of that have been proposed, notably by Zadeh, and lack of relevance. society’s King Sun Fu award. Dr. Ruspini was by Trillas and Valverde.’ Seeking a wide the General Chairman of the Second IEEE Inter- characterization of fuzzy logics, Gaines national Conference on Fuzzy Systems (FUZZ- chose not to specify a particular semantics IEEE ‘93) and of the 1993 IEEE International Conference on Neural Networks. He can be for the implication operator, instead requir- reached at the Artificial Intelligence Center, SRI ing only the use of a reasonable notion of International, 333 Ravenswood Ave., Menlo equivalence compatible with equality of Park, Calif. 94025; Internet: [email protected] truth values.

AUGUST 1994 33 Semantic Uncertainty of the FuzzXied Laws of Logic Burhan Tiirken, University of Toronto

The confusion surrounding Charles Elkan’s pressions (normal forms) for every meta- (2) If we assume (n, U :) is a De Mor article is generated by a lack of clear under- linguistic expression. The first is to assign logic such that boundary, monotonicity, standing of the four levels of knowledge the symbols fl, U, and to the basic meta- associativity, and commutativity conditi representation: linguistic, metalinguistic, linguistic connectives AND, OR, and NOT, together with the involutive complemen propositional, and computational. When we respectively. Next, we form the canonical tion are imposed, then we have FDNF ai attempt to convert knowledge expressed in expressions of the basic metalinguistic ex- FCNF for the second-level fuzzy logics: natural language into computable knowl- pressions asAnB,AUB, andA‘. Then we FDNF(2)(A AND B) =(A flB) U (A edge, at least three significant transforma- derive all other propositional expressions n tions occur between these four levels. with an application of AnB,A UB, or AC, FCNF@)(A AND B) = subject to the particular interpretations. (AUB) n (AUBC)n (ACUB)n Linguistic expressions. Linguistic expres- In the second approach, we first give an (AUB) n (AUBC) n (ACUB) sions are natural language expressions, interpretation to a metalinguistic expres- such as sion and define its meaning with a truth (3) If we assume (n,U,c) is a De Mol table. We then determine its normal forms logic such that boundary, monotonicity, “inventory is low and demand is high,” from the truth table by the application of associativity, commutativity, and idemp “inventory is low or demand is high,” or the “canonical form” generation algorithm. tency conditions together with the invol “inventory is not low,” In this approach, two distinct but equiva- tive complementation are imposed, ther where “inventory” and “demand” are lent canonical forms are generated: the have FDNF and FCNF which are equiv nouns, and “low” and “high” are adjectives. disjunctive normal form (DNF) and the lent to the fuzzified extensions of the cl In the terminology of fuzzy set theory, I the conjunctive normal form (CNF). For exam- sical normal forms:’ nouns are linguistic variables, the adjec- ple, DNF and CNF for “A AND B’ are FDNF(’) (A AND B) = A nB = tives are linguistic values, and “and,” “or,” DNF(A AND B) = A nB= DNF(A AND B) and “not” are linguistic connectives that generate interval-valued fuzzy CNF(A AND B) = FCNF(’) (A AND B) = A metalinguistic expression is a map- (AUB) n (AUBC) n (AWB) (AUB) n (AUBC) n (ACUB)= ping from natural language to a symbolic CNF(A AND B) language. For example, the metalinguistic Fuzzy normal forms. It has been shown that forms of the linguistic expressions above fuzzy normal forms can be generated from In particular, it has been shown’ that are: “XI is A AND is B,” “Xi is A OR the fuzzy truth table directly.2 Depending Xz Xz FDNF(’) (A AND B) is B,” and “XI is NOT A,” where XI and X, on the set of axioms we impose, we get at FCNF(3)(A AND B) are the metalinguistic representations of least three different classes of.fuzzy logics the linguistic variables, A and B are the with their corresponding normal forms: In a similar manner, we can obtain FDNF metalinguistic representations of the lin- (1) If we assume (n,U;) is a De Morgan and FCNF for the three classes of fuzzy guistic values, and AND, OR, and NOT are logic such that only boundary and monoto- logics and for all other metalinguistic the metalinguistic representations of the nicity conditions together with the involu- expressions. linguistic connectives. In short form, these tive complementation are imposed, then we metalinguistic expressions are represented have the following FDNF and FCNF ex- Computational expressions. At this level, as “A AND B,” “A OR B,” and “NOT A.” pressions for the first-level fuzzy logics: symbolic elements of sets are assigned numeric values, and conjunction, disjunc- FDNF(’) (A AND B) = (BnA)U (AnB) Propositional expressions. In the classical tion, and complement operators are chosen. two-valued logic, there are at least two ap- FCNF(’) (A AND B) = In Aristotle’s logic, the assignments are proaches that generate propositional ex- (BUA) n (BWA) n (BUR) n pA: Xi + [ 0,1},and ,ug: Xz+ { 0,1} (AUB) n (ACUB)n (AUBC)

34 IEEE EXPERT I

Acknowledgments In Zadeh’s fuzzy set theory and its logic, subinterval of [0,1J that is bound by its This work was supported in part by the Manu- the assignments are lower bound PFDNF@)(A OR NOTA) E [0,11, facturing and Research Corporation of Ontario and its Upper bound pFcNF(Z)(A OR NOTA) = 1. (MRCO), and the Natural Science and Engineer- pA :XI + [0,1], and pB : X, + [0,1] ing Council of Canada. Furthermore, for Zadeh’s logic, the compu- Conclusions tational expression of the metalinguistic I have demonstrated that there are three References 1. I.B. Tiirksen, “Interval-Valued Fuzzy Sets expression “A AND B’ at the third level are basic transformations between four levels Based on Normal Forms,” Fuzzy Sets and of knowledge representation. Each metalin- Systems, Vol. 20, No. 2, Sept. 1986, pp. pFDNF(A AND E) (a,b, = aAb guistic expression is transformed to at least 191-210. FFCNFIA AND B) (a,b, = two propositional expressions known as the 2. I.B. Tiirksen, “Fuzzy Normal Forms,” to (avb)A (avN(b))A (N(a)vb) fuzzy disjunctive and conjunctive norms appear in Fuzzy Sets and Systems, 1994. forms: FDNF and FCNF, respectively. where a E A, b E B are elements of fuzzy A consequence of this FDNF(.), FCNF(.) Burhan Tiirksen is a principal investigator in sets A,B - that is, a = F~(x,)and b = pB(x2) bounds is that classical expressions such as the Expert Systems Laboratory of the Manufac- turing and Research Corporation of Ontario at - and A = min, v = max, and N(.) is the “excluded middle,” “contradiction,” and the University of Toronto. His research interest5 standard complement. “equivalence,” and any combination of two include fuzzy sets and logics, approximate rea- or more vague evidences, must be reinter- soning, knowledge representation and inference, Interpretations preted. The type-I fuzzy representation of and manufacturing and process industries, em- We can now reinvestigate and reinterpret linguistic expressions provides only a my- phasizing management decision support systems and intelligent systems control. Tiirksen edited these expressions. the law of excluded middle for both the opic interpretation of the book series Fuzzy Logicfor Decision and idempotent and nonidempotent operators These interpretations need to be restated: Control, and is on the editorial board of Fuzzy as examples of our classification discussed The fuzzified versions of the laws of classi- Sets and Systems, Approximate Reasoning, Deci- above. For the idempotent class, an exam- cal logic hold to the degree specified by a sion Support Systems, Information Sciences, ple is the min-max-standard complement type-11, second-order, semantic uncertainty Expert Systems and Applications, Turkish Oper- ations Research Transactions, Fuzzy Logic Re- triple. For the excluded middle expression, computed by the membership of the mem- ports and Letters, and the Encyclopedia of Com- FDNF(3)and FCNF(3)are computed to be bership grades, that is, p(~A(x))= p2A(x). puter Science and Technology. Thus, we cannot state, for example, that the Tursken received his PhD in systems manage- (U A N(a)) V (U A U) V (N(a)A N(a))5 law of excluded middle is satisfied or not. ment and operations research in 1969, and his a v N(a) MS and BS in industrial engineering in 1962 and We can, however, state that the excluded 1960, respectively, all from the University of which results in middle expression is satisfied to a degree Pittsburgh. He is a senior member of the IEEE contained in the interval specified by: and a member of Alpha Pi Mu, IIE, CSIE, 0.5 5 a vN(a)5 1.0 for a E [0,1] CORS, IFSA, NAFIPS, APEO, APET, TORS, This is a type-I1 semantic uncertainty, that [IIFDNF(A OR NOTA) (a3Na)), and ACM. pFCNF[A OR NOTA) (a,N(a))l Readers can contact him at the MRCO-Intelli- is, p(pA(x))= a v N(a).However, it reduces gent Fuzzy Systems Laboratory, Department of to singletons as opposed to intervals. It is Reinterpretations for contradiction, Industrial Engineering, University of Toronto, clear that in Zadeh’s fuzzy logic we ought equivalence, and so on can be stated in a Toronto, Ontario M5S 1A4, Canada; Internet: not to state that the excluded middle expres- similar manner. In fact, this is the source of turksen @ fuzzy.ie.utoronto.ca sion holds or does not hold. We ought to controversy surrounding Elkan’s paper. instead state that it is satisfied to the degree The essence of fuzzy set theory is that all specified by a v N(a) for a E [O,l]. vague statements should at least be inter- For the nonidempotent class, consider preted first with type-I semantic uncertainty the bold intersection-union-standard com- at the primary, elemental level. But when plement De Morgan logic where TB(a,b) = two or more vague concepts are combined max(0, a+b-1), SB(a,b)= min(1, a+b), and with a linguistic connective, then we are N(a) = 1-a. For this case we obtain confronted with a type-11, second-order, semantic uncertainty. This generates an in- o.o pFDNF‘2’[A OR NOTA) terval where the location of a specific de- pFCNF‘2’(A OR NOT A) = .o gree of membership is nonspecific in that and have an interval of graded values interval. where the excluded middle expression is satisfied to a continuum of degrees in a

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~~ The Promising Future of Fuzzy Logic Nader Vadiee and Mohammad.Jamshidi,University of NmMexico

As Charles Elkan points out in his article, Fuzzy logic-based models are actually ef- developed for Japan’s Stock Exchange the foundation of fuzzy logic is the notion forts in building our perceptual models and Market in Tokyo. The Nikkei average has of partial truth and degrees of truth in any maps of reality, and not the reality itself. been reportedly gone consistently higher proposition stating facts about real-world using fuzzy logic.’ However, real applica- objects, whether these objects are entities, Fuzzy logical equivalence. Like any other tions of fuzzy expert systems have, for the events, relations, algorithms, systems, or notion in fuzzy logic, the notion of logical most part, been kept out of the public eye machines. Facts and propositions are un- equivalence is based on degrees of truth. The because much of the work is proprietary. certain, ambiguous, and incomplete - and fourth postulate in Elkan’s Definition 1 is not As far as the domain independence of more importantly, they are goal-oriented, necessary, and can be replaced by classical fuzzy operators is concerned, it is well intentional, and subjective to the observer’s implication relations. For example, known that max-min operations are default perceptual capabilities, mental constructs, operations, and there are many different t(A = B) = t( (A+@ AND (B+A)}= and meaning systems. In this philosophical definitions suggested by the fuzzy logic min(r(A+B), t(B+A)}. view, the universe is seen as holistic, dy- research community for “and,” “or,” and namic, and chaotic. This expression can be used as the defini- implication operation. Of course, aggre- Fuzzy logic is basically a theory of tion for degree of equivalence in fuzzy gation operators are important and context human perception and cognition. It is con- logic. For the special case where A and B dependent, but they can be a part of the cerned with the marvelous paradigm and have truth values of 1 or 0, the degree of knowledge to be learned and gathered from methodology discovered by evolution and logical equivalence is equal to 1 for the the expert. realized in our brains to cope with com- case of t(A) = t(B).Consider the following Consider Elkan’s watermelon example plexity, holism, dynamism, and chaos in special case. Using the classical implica- about the context dependency of the “and’ the world around us. tion relation that was generalized by Zadeh aggregators: If being red inside and green The goal of a fuzzy expert system is to for fuzzy logic,’,’ we have outside are believed to be mutually rein- take in subjective, partially true facts that forcing pieces of evidence toward being a t(A+B) = max(min[ t(A), t(B)],I-t(A)] are randomly distributed over a sample watermelon, then the logical proposition @+A) = max{min[ t(B),r(A)], I-@)) space, and build a knowledge-based expert could read: system that will apply certain reasoning for t(A) = 0, t(A = B) = 1 - t(B).For t(B) If X is red inside and X is green outside, and aggregation strategies to make useful being a number between 0 and 1, the de- then X is a watermelon is very true. decisions. These decisions are again ap- gree of equivalence will be in a range from proximate, and have partial degrees of truth 0 to 1. In this example, Elkan is using the fourth and likelihood; the decisions and derived As seen in the above equations, for the postulate to reach an intuitively incorrect facts are reliable to the best of our available case where t(A) = t(B),t(A = B) is always conclusion. Based on the definition of the knowledge. greater than 0.5 in fuzzy logic, which fuzzy logical “and” operation, t(red inside The important fact about these systems means a strong logical equivalence. Two and green outside) is simply the degree to is that decisions made by them can be itera- propositions could be logically equivalent which an object is “red inside” and “green tively and adaptively improved, and as in a fuzzy sense without [(A) = t(B). outside” and does not have anything to do more such randodfuzzy facts accumulate, We agree with Elkan’s point that the last with being a watermelon. The degree of the results will converge to real precise postulate of Definition 1 is the most con- being a watermelon depends on the other facts. In this view of reality, no proposition troversial piece. He has in fact provided his circumstantial information as well as the is always 100% true for 100% of the ob- own answer for preserving the continuum degree of being red inside and green out- servers and experts. Absolute certainty, of degrees of truth. side. This “other piece of information” is absolute truth, and absolute objectivity are the degree of logical equivalence that must impossible because they require infinite Fuzzy expert systems. The types of uncer- be provided by the expert. pieces of information, infinite number of tainty captured by fuzzy logic are vague- Fuzzy expert systems have been used in samples, and infinitely many observers. ness, incompleteness, and ignorance. An many applications. For example, Parkinson example of this is the fuzzy expert systems & Duerre have used both expert systems and fuzzy expert systems to choose the most suitable new “technology” for oil

36 IEEE EXPERT

I I

recovery. In the case of classical expert tem. Like any other notion, causality is not advantage of fuzzy logic in control systems. systems, the sharpness of the boundaries of a matter of black or white, or yes or no; To achieve quick design periods, simple crisp variables involved in this application instead, the cause-and-effect relation itself rules have been used thus far to put the led to wrong conclusions based on the $- is a matter of degree. As Elkan correctly designer in the ball park, and although ap- then rules. The fuzzy expert system took observes, the advent of the fuzzy chip, proximate and crude, through tuning and care of all the limiting (worst-case) prob- which came on the market in 1987, is a adaptation the rules are fine tuned for better lems and made natural conclusions. Al- major force behind the spread of industrial performance of the overall system. It is true though these worst-case problems are not applications of fuzzy logic control. that most current applications of fuzzy logic the most common for this application (in In reference to the use of words such as could use other rule-based formalisms, but which the recovery technologies are some- “image stabilization” for fuzzy logic cam- these come with costs in terms of memory, what outdated), their occurrence will be- corder image stabilizing systems, or “grade efficiency, development times, and longer

come the rule rather than the exception in logic “ for fuzzy logic, Elkan brings out the compilation of vague linguistic types of future years. As it is now, major oil reserves common difficulties that English-speaking knowledge. For example, consider the fol- in the US cannot be recovered by the old Western communities have with this new lowing type of proposition: techn~logies.~ technology, and with the innocent word Most experts believe if X is A, then Y is “fuzzy.” It is not surprising in light of this B is very true and fairly likely. Fuzzy control. Most of the current applica- bias that manufacturers chose alternative tions of fuzzy logic are fuzzy expert control words in their advertisements in the US, There are techniques that can handle this systems. Fuzzy controllers are expert con- and to a lesser extent, in other English- type of vague logical proposition that have trol systems that smoothly interpolate be- speaking countries. elements of both probability and possibil- tween otherwise crisp (or predicate logic- As far as the standard architectures of ity.’,*Elkan brings up the brittleness of based) rules. Rules fire to continuous fuzzy control are concemed, a small num- rule-based systems caused by a missing degrees and the multiple resultant actions ber of rules are an advantage for fuzzy con- piece of information. This is not the case are combined into an interpolated result. trol systems. This is evidently a result of for fuzzy rule-based expert systems. As The basis of fuzzy control is provided by interpolative reasoning and the ability to mentioned earlier, this is due to the inter- processing uncertain information and sav- aggregate the overlapping pieces of fuzzy polative capabilities of fuzzy logic’s con- ing energy through the use of commonsense information. Elkan brings up the point indi- tinuous aggregation of the rules and elastic rules and natural-language statements. cated by Sugeno and his colleagues -that semantics assigned to the symbols, as de- The use of sensor data in practical con- the knowledge recorded in a fuzzy confined by the membership functions. trol systems involves several tasks that are troller typically reflects immediate rela- Fuzzy control, as we mentioned earlier, usually done by a person, such as an astro- tions between the inputs and outputs of the constitutes a major application area of naut adjusting the position of a satellite or system to be controlled, as opposed to a fuzzy logic. With most control systems, putting it in the proper orbit, or a driver deep causal model of the system? Although based on some real data from certain sen- adjusting a car’s air conditioning unit, and this point of view is accurate, it is also true sors, some decision must be made through so on. All such tasks must be performed that this is the exact manner in which a decision process. Fuzzy controllers are based on an evaluation of the data accord- human experts summarized their expertise nonlinear controllers that provide rather ing to a set of rules that the person has - by capturing the causal links between reasonable robustness and adaptiveness learned from experience or has been trained the inputs and outputs of the systems and with the changing environment - be it in. Often, if not most of the time, these rules putting them in the form of a set of linguis- unmodelled dynamics in the system, exter- are not crisp (based on binary logic), that is, tic rules. The expert might have deep nal disturbance, or simply a lack of precise they involve common sense and human knowledge of the system’s causal relation- knowledge about the plant that is being judgment in the decision making process. ships, but it is hard to access that type of controlled. Such problems can be addressed by a set of knowledge in the form of linguistic proto- The subjectivity in fuzzy modeling is a fuzzy variables and rules that, if calculated cols. For example, the knowledge of an blessing rather than a curse. The subjectiv- and executed properly, can make expert operator with 20 years of experience at an ity in the definition of the terms is compen- decisions. electric power substation cannot be tapped sated for by the subjectivity of the condi- Fuzzy logic has given a new definition in a few simple linguistic rules to offer a tional rules used by an expert. Because the to the causality in dynamic systems. Fuzzy deep knowledge about the transience and set of variables and their meanings, as rep- relational equations‘,* are indications of the stability of a power system. resented by corresponding membership notion of degree of causality between input Short development times have been a big functions, are compatible and consistent and output variables in a dynamical sys- with the set of conditional rules used, the overall outcome tums out to be objective, meaningful, and reliable. Fuzzy mathemat-

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_- ical tools and the calculus of fuzzy if-then principle-based approaches in modeling ison of Crisp and Fuzzy Logic Methods for rules opened the way for the automation the real world belong to the childhood Screening Enhanced Oil Discovery Tech- years of science. The scientific thought that nologies,” Fuzzy Logic and Control: Hard- and use of a huge body of human expertise ware and Software Applications, M. that has gone untapped for years in indus- began with Aristotelian logic and was fol- Jamshidi, N. Vadiee, and T. Ross, eds., try. Fuzzy logic has provided a mechanism lowed by Laplacian determinism has Prentice Hall, Englewood Cliffs, N.J., 1993. to share, communicate, and transfer a reached its limitations -particularly when Vader Vadiee is a teaching and research assis- wealth of human technical expertise into it comes to understanding human systems. ant at the CAD Laboratory of Intelligent and computers. This has reversed the trend of In the last hundred years, we have witnessed Robotics Systems, Department of Electrical and machine tyranny: We are now forcing com- the development of quantum mechanics, Zomputer Engineering, University of New Mex- puters to think like people, rather than the and with it, probabilistic notions of micro- ico, Albuquerque. He received his BS and MS kgrees from Shiraz University (formerly other way around. This is the beginning of cosm, relativistic mechanics for macro- Pahlavi University), Shiraz, Iran, in 1975 and a new era in the applications of AI, neural cosm, and more recently, fuzzy logic and 1978, respectively. He will receive his PhD from networks, fuzzy logic, genetic algorithms, chaos theories. The emergence of these the University of New Mexico in September, and probabilistic reasoning within a bigger theories have a philosophical implication 1994. Vadiee has more than 20 years of teaching and research experience at engineering schools, picture called computing.6 The funda- that points toward a probabilistic and pos- soft where his work has focused on robot control, mental issues of AI can only be solved with sibilistic picture of reality. neural networks, and fuzzy logic. His doctoral an orchestrated application of fuzzy logic, Fuzzy logic -with the help of probabil- dissertation is on cognitive systems in integrated neural networks, genetic algorithms, and ity theory -will provide yet another pow- neuro-fuzzy architecture for nonlinear control probabilistic reasoning. erful tool in an engineer’s or scientist’s and system identification. He is co-editor of Fuzzy Logic and Control: Sofhvare and Hard- toolbox for coping with complexity and wure Applications, and is a member of the IEEE nonlinearity in real-world systems. It will Control and Computer Societies. Vadiee can be Elkan is distinguished from most critics of also furnish answers that are never 100% reached at the CAD Laboratory for Intelligent fuzzy logic because he seems to have sin- accurate and certain, but are acceptable and Robotic Systems, Dept. of Electrical and Computer Engineering, Univ. of New Mexico, cerely studied the subject from both a theo- within the constant constraints of real time, Albuquerque, N.M., 87 13 1 ; Internet: retical and applied point of view. It seems energy, memory, and resources. Manimar @carina.unm.edu to us, however, that Elkan’s primary con- Mohammad “MO”Jamshidi is a professor of tact with fuzzy logic has been through open References electrical and computer engineering at the Uni- 1. N. Vadiee, “Fuzzy Rule-Based Expert Sys- literature rather than industrial applications versity of New Mexico, Albuquerque where he tems I,” in Fuzzy Logic and Control: Hard- and the tremendous activity across the in- is director of the Computer-Aided Design Labo- ware and Software Applications, M. Jam- ratory for Intelligent and Robotic Systems and dustrial world. shidi, N. Vadiee, and Ross, eds., Vol. 2, T. holds the AT&T Professorship of Manufacturing Some of the shortcomings that Elkan Prentice Hall, Englewood Cliffs, N.J., 1993. Engineering. He is also a consultant with the Los attributes to applied fuzzy logic are due to 2. N. Vadiee and M. Jamshidi, “ATutorial on Alamos and Oak Ridge National Laboratories, the gap that exists between theory and ap- Fuzzy Rule-Base Expert System Models I: and is the founder of TSI Enterprises. He has Mathematical Foundations,” J. Intelligent plication, despite the revolution in the in- more than 300 publications, including 32 books. and Fuzzy Systems, Vol. 1, No. 2, 1993, pp. dustrial use of fuzzy logic. We believe, His latest book is Soff Computing: Fuzzy Logic, 171-188. Neural Networks, and Distributed AI (Prentice however, that it is too soon to scrutinize 3. M. Jamshidi, T.J. Ross, and N. Vadiee, Hall, 1994). He is editor of several publications, this gap. For example, at the University of Fuzzy Logic With Industrial Applications, including the International Journal of Comput- New Mexico’s CAD Laboratory for Intelli- National Short Course Notes, Univ. of New ers and Electrical Engineering, and was found- Mexico, Albuquerque, N.M., 1991-94. gent and Robotic Systems, fuzzy logic ing editor of IEEE Control Systems Magazine, 4. M. Jamshidi, N. Vadiee, and T. Ross, eds., technology is being put on a chip to be em- and cofounding editor of International Journal Fuzzy Logic and Control: Hardware and of Environmentally Conscious Manufacturing. bedded in a new generation of controllers Software Applications, Vol. 2, Series on He earned his BS cum laude from Oregon State with large industrial and technology trans- Environmental and Intelligent Manufactur- University, Corvallis, in 1967, and his MS and fer implications.’ we are trying to intro- ing Systems, Prentice Hall, Englewood PhD from the University of Illinois at Urbana- duce the next generation of fuzzy expert Cliffs, N.J., 1993. Champaign in 1969 and 1971, respectively. He 5. M. Sugeno et al., “Fuzzy Algorithmic Con- is a member of several honor societies, a fellow systems capable of handling truth quali- trol of a Model Car by Oral Instructions,” of the IEEE, and recipient of the IEEE Centen- fiers, quantifiers, rule interaction, chaining, Fuzzy Sets and Systems, Vol. 32, No. 2, nial Medal and IEEE Control Systems Society and hierarchical rule structures. 1989, pp. 135-156. Distinguished Member Award. He is the hon- By starting to think in terms of a holistic, 6. F. Aminzadeh and M. Jamshidi, eds., Soff orary chaired professor at Nanjing Aeronautical relativistic, probabilistic, and possibilistic Computing: Fuzzy Logic, Neural Networks, Institutes, Nanjing, China, and Xia’n Institute of and Distributed Artificial Intelligence, Technology, Xia’n, China. Jamshidi can be knowledge structures, we believe scientific Prentice Hall, Englewood Cliffs, N.J., 1993. reached at the CAD Laboratory for Intelligent thinking is entering its new major stage of 7. G.W. Parkinson and K. Duerre, “ACompar- and Robotic Systems, Dept. of Electrical and maturity. Crisp, binary, deterministic, first- Computer Engineering, Univ. of New Mexico, Albuquerque, N.M., 87131.

______~ 38 IEEE EXPERT I

Toward a Framework for Fuzzy Dynamic Systems Pei-Zhuang Wang, Sie-Keng Tan, and Shaohua Tan, National Uniuenity ofsingapore

Fuzzy logic, according to Lotfi Zadeh, can (5) Distributive laws: a unary operation ‘, and the nullary opera- be broadly considered as the union of (a v b) A c = (a A c) v (bA c) tions o and i on S, that satisfies the axioms fuzzified crisp logics. Its primary aim is to (a A b) v c = (a v c) A (bv c) 1 to 7 stated in Definition I. provide a formal, computationally oriented (6) Involution law: Thus, fuzzy logic theory can be seen as a system of concepts and techniques for (aC)C=a theory based on the structure of a soft alge- dealing with modes of reasoning that are (7) De Morgan’s laws: bra. It is clear that every Boolean algebra is approximate rather than exact. Charles (a v b)’ = a‘ A b‘ a soft algebra, but not vice versa. Hence, Elkan’s claims are derived mainly from (a A b)‘ = ac v b‘ soft algebra is a more general system than entangled interpretations of fuzzy logic (8) Laws of Excluded Middle: Boolean algebra. Consequently, proposi- stemming from his mathematical approach avaC=i tions that are valid in classical logic may to the formal system and intuitionistic ap- UAUC=O not be valid in fuzzy logic. For example, if proach to the practical system. Here we we view each assertion A as a set in a uni- examine the mathematical structures of Here, the nullary operations o and i are verse U and identify the truth value t(A)of classical and fuzzy logic, and then point commonly known as the least element and the assertion A by its characteristic function out that Elkan’s view of the standard ver- the greatest element of the Boolean algebra. pA: U + [ 0,1 }, then the two compound sion of fuzzy logic is not valid. We then Due to the pointwise definition of the statements (A A B‘)’ and B v (A‘ A B‘) are attempt to envisage fuzzy logic, in its prac- operations used in the theory of two-valued logically equivalent according to the rules tical aspect, as a dynamic system that will logic, we can consider classical logic as a of classical two-valued propositional calcu- enhance control and expert systems. theory that is based upon the particular lus; however, in the context of fuzzy logic,

Boolean algebra ({0,1], v, A, 1,0, 1) these two statements with truth value in Mathematical aspect of fuzzy logic. It is where 1 and 0 represent respectively the [0,1] are not equivalent. (For example, take well known that the fundamental rules of true and false of a certain statement or an t(A) = 0.3 and t(B)= 0.6, then t((A A E)’)= classical logic are governed by the structure assertion, and the operations v, A, and 0.7, whereas t(B v (AcA BC))= 0.6. of a Boolean algebra, defined as follows: are defined according to the usual rules of In his article, Elkan views a standard Definition 1. A Boolean algebra (B. v. the logical connectives or; and, and not, version of fuzzy logic as a system that sat- A, c, 0, i) is a system consisting of a non- respectively. Zadeh’s innovation of fuzzy isfies the four postulates given in the fol- empty set B together with two binary oper- logic, on the other hand, is an attempt to lowing definition: ations v and A, a unary operation ‘, and two generalize the classical two-valued logic. Definition 3. Let A and B be arbitrary nullary operations o and i on B, that satis- Instead of the two values true and false assertions. Then fies the following axioms for any elements represented by the set { 0, 1 }, he considered t(A A B) = min { t(A),f(B)} (1) a, b, c E B: the interval [0,1] to be the range of the truth r(A v B) = max t(A),t(B)] (2) value of any assertion, and replaced the (1) Commutative laws: t(iA) = 1 - t(A) (3) binary operations v and A on { 0,1] by the avb=bva t(A)= t(B)if A and B are logically binary operations max and min on [0,1]. u~b=b~a equivalent, (4) The unary operation is also replaced by c (2) Associative laws: where c(a)is defined to be 1-a for any where “logically equivalent” means equiv- (a v b)v c = a v (b v c) a E [0,1]. Under these operations, the sys- alent according to the rules of classical (a ~b)A c = a A (bA c) tem ([0,1], max, min, c, 0, 1) satisfies all two-valued propositional calculus. (3) Absorption laws: the axioms of a Boolean algebra except the Certainly, under these postulates, one can (a v b) A b= b laws of excluded middle. Such a system is prove that for any assertions A and B, either (a Ab) v b = b known as a soft algebra defined as follows: t(A)= t(B)or t(A)= 1 - t(B).However, the (4) Idempotent laws: Definition 2.Asofalgebra (S, v,A, ‘, 0, i) main issue here is that postulate 4 is gener- ava=a is a system consisting of a nonempty set S UAU=U together with two binary operations v and A,

AUGUST 1994 39

- ally not valid in the realm of fuzzy logic, as needs. Other semantically dependent 2. E.H. Ruspini, “Approximate Reasoning: shown in the above example. Thus, as a formulations of fuzzy operations and infer- Past, Present, Future,” Tech. Note 492, SRI rigorous mathematical system, postulate 4 ence relations have also been propo~ed.~?~ Int’l, 1990, pp. 1-22. should not be included as a postulate in the Elkan’s paper does bring up some valid 3. S.K. Tan, P.Z. Wang, and E.S. Lee, “Fuzzy formal system of fuzzy logic. Although points in the discussion of the status quo of Set Operations Based on the Theory of Falling Shadows,” J. Mathematical Analy- Elkan has stated that in fuzzy logic applica- fuzzy control. Indeed, the present fuzzy sis andApplicarions, Vol. 174, NO. 1, 1993, tions it is unclear whether or not postulate 4 controllers are mostly structurally shallow, pp. 242-255. is assumed, and that in theoretical work it is and in most cases, the controllers simply S.K. Tan, P.Z. Wang, and X.Z. Zhang, often used explicitly, he still imposes this deal with no more than a simple static 4. “Fuzzy Inference Relations Based on the postulate in the formal system of fuzzy fuzzy mapping of the sensory and actuation Theory of Falling Shadows,” Fuzzy Sets logic and then claims that the standard ver- signals. However, this is not the whole pic- and Systems, Vol. 53, 1993, pp. 179-188. sion of fuzzy logic collapses mathemati- ture of fuzzy control. In fact, the success of Pei-Zhuang Wang is professor and head of the cally to two-valued logic. This type of con- Aptronix’s simulation of the two-stage Fuzzy Information Processing and Fuzzy Com- demnation, in our view, is an impediment to inverted pendulum using a fuzzy controller puting National Laboratory, Beijing Normal the growth of our knowledge. is a fuzzy logic application that is not struc- University, Beijing, China. He is also a turally shallow. When fuzzy logic is used researcher at the Institute of Systems Science, National University of Singapore. He has served Practical aspect of fuzzy logic. At the as a way of quantization, it can serve as our in a number of national and international soci- present stage, fuzzy mathematics is viewed quantity basis for modeling dynamic sys- eties including as the vice president of the Inter- in two ways. First, it is a theory that con- tems in the real world. This leads to the national Fuzzy Systems Association (1991 -93), forms with the precise and rigorous princi- notion of fuzzy dynamic systems. Obvi- and chair of the Chinese Chapter of the Intema- ples of mathematics to deal with fuzzy ously, fuzzy dynamic systems are more tional Fuzzy Systems Association. He is honorary chair of Aptronix, San Jose, Calif. He is on objects. In other words, it is a strictly math- complex, as they describe dynamic evolu- the editorial board of several international and ematical theory to study the objects in a tion of certain fuzzy quantities, not simple local technical journals on fuzzy logic, and has fuzzy environment. Second, fuzzy mathe- points or numbers. Undoubtedly, in the been given awards by various organizations matics is a metamathematical theory that light of such a theory, many important is- including China’s National Personal Ministry and Ministry of Education. He has published involves fuzzy proof techniques and fuzzy sues such as stability, controllability, and five books and more than 100 technical papers in theorems with their applications. This latter observability can properly be addressed, international journals and conferences. His re- status has yet to be fully developed. We and it may also serve to bring the seem- search interests are in fuzzy mathematics theory, have already seen many successful applica- ingly diverging model-based or rule-based probability, and AI systems. tions of fuzzy logic that the use of conven- methodologies into a unifying framework. Sie-Keng Tan is a senior lecturer in the Depart- tional mathematics could not achieve. We ment of Mathematics, National University of envisage that as fuzzy mathematics devel- Singapore. He earned his BS and PhD from ops further, applications will be even more Nanyang University, Singapore, and his MPhil from Queen Mary College, University of Lon- convincing and prominent. An appropriate theory for fuzzy systems don, England. His main research interest is in the Elkan and other researching technolo- has not yet been developed in fuzzy con- mathematical structure of the theory of tmth- gists perhaps view fuzzy logic as fuzzy trol. The main task is to establish a frame- value flow inference of fuzzy logic. S.K. Tan is a mathematics in its second status, and this work in which fuzzy controllers of deeper member of the International Fuzzy Systems Association (IFSA). might explain why Elkan has been unable structures can be described properly and to find a real-world expert system that uses handled with ease. Elkan has predicted a Shaohua Tan is a senior lecturer in the Depart- fuzzy logic as its primary formalism for tough time ahead for fuzzy logic in general, ment of Electrical Engineering, National Univer- sity of Singapore. He has been working in the reasoning under uncertainty. and for fuzzy control in particular. We, too, areas of systems and control, digital signal pro- Fuzzy logic, in the practical aspect, deals predict a tough time ahead in working out a cessing, speech processing, and artificial neural mainly with fuzzy quantization, its mean- meaningful and acceptable framework for nets. His current research interests are in intelli- ing and means. In this respect, Elkan re- fuzzy-based dynamic system theory. How- gent control and pattern recognition using neural net and fuzzy inference techniques. He received gards those operators in fuzzy logic as ever, we remain optimistic. We believe that his BE degree in mechanical engineering from fixed and domain-independent. In fact, fuzzy such a framework will emerge. the Beijing Institute of Technology, China, in quantization is introduced precisely for the 1982, and ME and PhD degrees in electrical purpose of generating domain-specific Referentes engineering from the Katholieke Universiteit quantities. The numerous forms of fuzzy Leuven, Belgium, in 1984, and 1987, respec- 1. 1.R.Goodman and H.T. Nguyen, Uncer- tively. Shaohua Tan can be reached at the De- operations suggested in the literature were tainty Models for Knowledge-Based Sys- partment of Electrical Engineering, National created to cater to the domain-specific tems, Elsevier, New York, 1985. University of Singapore, 10 Kent Ridge Cres- cent, Singapore 05 11; Internet: eletansh@leo- nis.nus.sg

40 IEEE EXPERT Misretwesentations and Challenges A Responlse to Elkan Ronald R. Yagel; Iona College

The comments made by Charles Elkan can and hence to appreciate the pervasive nature of its be classified into three categories. The first ability to model continuity and graduality in t(A v (i4 v YB)) are those that are technically incorrect and all concepts. In using fuzzy logic, we are t((4V B) A (B V iB)) should not have been allowed to pass an not confined to only using the idea of fuzzi- = min[t(4 v B), t(B v 41. unbiased review process. The second are ness (graduality) in the definition of the those that are truly challenging, and in point Thus we have the requirement predicates (rednesdgreenness), but we can of fact help show the great representative also apply the concept of fuzziness to the t(4v B) = min[t(4 v B), t(B v -41. power of fuzzy subsets. The third category operators used to connect the predicates. In are those pertaining to the practical applica- For this condition to hold for every B re- addressing this important issue, we must tion of fuzzy control. Some of these latter quires that call upon fuzzy logic’s ability to provide comments are quite reasonable -though connectives lying between the logical and t(B v 1B) = I not as damning as Elkan tries to make them. and logical or. Consider the definition of I shall address these issues in turn. for all B. However, this condition is the law watermelon suggested by Elkan, The first category - those that are com- of the excluded middle and is exactly what waterme/on(x)= redinside(x) i pletely incorrect - is dominated by Theo- fuzzy logic was constructed not to support. greenoutsid@). rem 1. Rather than wasting considerable In fact, I have suggested a measure of space on addressing this “theorem” when I fuzziness based upon the lack of satisfac- Elkan correctly shows that if we interpret am certain that other respondents will ef- tion of this condition.2 Furthermore, the i as a pure logical and, defined as the fectively show its complete absurdity, I condition t(B v 4)= 1 implies that min(A) we end up with a result that doesn’t shall make only a few comments. The key provide the appropriate property of rein- max[t(B), 1 - t(B)]= 1. issue here is of course the last premise, forcement. Similarly, using a pure logical Hence, either or, defined as the max(v) also leads to un- t(A) = t(B)if A and B are logically satisfactory results. The key point is that in equivalent. t(B)=lort(B)=O(l-@)=I) fuzzy logic we are not restricted to these In most texts on logic,’ the definition of Thus, Elkan has essentially assumed that two extremes as we are in binary logic. logical equivalence is specified the other the logic is a binary logic. Recently my colleagues and I introduced way. Usually, one says that A and B are As to the second category, Elkan raises a new class of fuzzy connectives, called logically equivalent if propositions A and B the issue of defining the concept of water- uninorms? that provide the exact type of attain the same truth value for all models of melon in terms of the constituent concepts aggregation postulated as being required by the constituent atoms. However, in this of redness on the inside and greenness on Elkan. Consider the situation that for some case, rather than defining the undefined the outside. His basic contention is that the object m we have t(redinside(m))= a, and concept of logical equivalence in terms of a definition of watermelon in terms of these t(greenoutside(m))= b. Our problem is to well-defined idea of attaining the same constituents should exhibit a characteristic provide an aggregation operator Ito imple- truth value, Elkan tries to define the idea of of reinforcement. Essentially, he correctly ment the connection between these values. attaining the same truth value from the requires that multiple confirmations to the Formally, letting d = t(waterme/on(m))we undefined concept of logical equivalence. constituents’ criteria should reinforce each require some aggregation R such that Once having made this error, the author other, while disconfirmations of the con- d = R(a, b). then compounds it by imposing a require- stituents’ criteria should also reinforce each ment that is completely antithetical to the other in the other way. The issue raised here The question is, what form should R take to idea of fuzzy logic: is an interesting and challenging question. capture the type of reinforcement desired However, rather than showing the limita- by Elkan? As I will show, uninorms pro- t(i(A A iB))= t(B V (4AiB)) tions of fuzzy logic, this problem illustrates vide the appropriate aggregation. These First we note that the power of fuzzy logic to model sophisti- uninorms, which generalize the idea oft- cated aggregation requirements. norms (and operators) and t-conorms (or t(i(A A 4))= t(4V B), It is fundamental to a comprehensive operators) and lie between these extremes, understanding of the agenda of fuzzy logic

AUGUST 1994 41

I do exactly what Elkan requires. both are below the neutral value. of the implication operative. A uninorm is a mapping3 Finally, consider the case where one is below and one above, a I6and b 2 6. In R: [0, 11 X [0, 13 4 [0, 11 this situation we see To me, Elkan’s reference to the 1980 com- having the properties: ment by Mamdani and Sembi5is most dis- U I R(a, b) turbing. Rather than seeing these remarks as (1) Commutativity: R(a, b) = R(b, a) b 2 R(b, a) = R(a, b) I believe Mamdani meant them- as a state-

(2) Monotonicity: R(a, b) 2 R(c, if a ~ c and hence we get d) ment of the power of the symbiotic relation- andb>d a I R(a, b)I b ship between the paradigms of AI (in this (3) Associativity: R(R(a,b), c) = and thus there is no reinforcement. case rule-based systems) and the knowl- R(a, c)) R(b, With the aid of these uninorm fuzzy ag- edge-representation capability of fuzzy (4) There exists an identity element 6 E gregation operators, we can capture the logic - Elkan has chosen to interpret this [0, 11 such that for all a, R(a, 6) =a. type of aggregation Elkan desires. as a sign of the weakness of fuzzy logic. If 6 = 1, this reduces to the t-norms that Finally, Elkan’s comments on the use of However, if we discard the obvious misrep- are essentially pure and aggregations that fuzzy logic in heuristic control - while in resentations, Elkan’s paper can serve as a include the min operator, while if 6 = 0, it some points are quite valid - manifest a challenge to fuzzy researchers to continue reduces to the t-conorms which are essen- type of “fuzzy bashing” that is all too com- improving the valuable tool of fuzzy logic. tially pure or aggregations and include the mon in the AI community. For example, max operator. Thus, the logical and and or Honda’s choice of the term “grade logic” References are extremes of this class. has much less to do with their concern for 1. E. Mendelson, Introduction to Mathemati- cal Logic, Van Nostrand Reinhold, New Of particular interest is a property of any scientific resistance to fuzzy logic York, 1964. these uninorms called the upward reinforce- methodology than to the simple marketing 2. R.R. Yager, “On the Measure of Fuzziness ment characteristic. For the uninorm, the expedient that “fuzzy” is not the type of and Negation Part I: Membership in the upward reinforcement characteristic is cap- word that sells cars. Unit Interval,” Int. J. General Systems, Vol. tured in the following: In a recent book on fuzzy modeling and 5, 1979, pp. 221-229. 3. R.R. Yager and A. Rybalov, “Uninorm Ag- control: we look carefully at the process of R(a, b) 2 a if b > 6 gregation Operators,” Tech. Report #MI]- building fuzzy logic controllers. The rea- 1407, Machine Intelligence Institute, Iona R(a, b)5 a if b < 6 sons we found for the success of these con- College, New Rochelle, N.Y., 1994. We select a value 6 to be our neutral trollers are not in complete agreement with 4. R.R. Yager and D.P. Filev, Essentials of Fuzzy Modeling and Control, John Wiley & point; values above 6 are considered as those Elkan suggests. Sons, New York, 1994. confirming and those below 6 as discon- First of all, the fact that most fuzzy con- 5. E.H. Mamdani and B.S. Sembi, “Process firming. (Actually, 6 can be a range; how- trollers are built with a small number of rules Control Using Fuzzy Logic,” Fuzzy Sers, P.P. ever, for the present purpose we’ll consider should be seen as one of the powers of this Wang and S.K. Chang, eds., Plenum Press, 6 as a point.) Now, assume that both a and technology. An essential feature of the fuzzy New York, 1980, pp. 249-266. b are above 6. In this case we have approach is the ability to generalize -in a Ronald R. Yager is director of the Machine Intel- way, to reduce the necessity for detail. ligence Institute and professor of information a 5 R(a, b) Elkan fails to mention a feature I think is systems at Iona College, and is a research fellow b I R(b, a) = R(a, b) essential to the success of the fuzzy model- of the Knowledge Engineering Institute, Guangzhou University, China. He is on the scien- and thus ing approach: the partitioning of the input tific committee of the Fuzzy Logic Systems Insti- variable space into regions that allow a R(a, b) 2 max[a, b]. tute, Iizuka, Japan, and is copresident of the Inter- simplification of the modeling process. national Conference on Information Processing Hence there is a reinforcement in the posi- Closely related to this is the idea of partial and Management of Uncertainty, Paris. He is editor-in-chief of the Intemational Joumal oflntelli- tive direction when both criteria are “con- matching, which lets us smoothly combine gent Systems. He also serves on the editorial board firmed.” solutions from different regions as we get of a number of otherjournals, including the IEEE Now assume that both a and b are below near the boundary. Transactions on Fuzzy Systems, the Journal of 6. In this case we have that Elkan correctly observes that most fuzzy Approximate Reasoning, and Fuzzy Sets and Sys- controllers are shallow (requiring no chain- tems. He has published more than 350 articles and a 2 R(a, b) has edited 13 books. He is co-author of a new ing between the rules) and usually directly b 2 R(b, a)= R(a, b) book, Essentials of Fuzzy Modeling and Control connect the input to the output. I think it is (John Wiley & Sons, 1994). He received his un- and thus here that these systems might have trouble dergraduate degree from the City College of New in the future. However, the reason for these York, and his PhD from the Polytechnic Institute R(a, b) I min(a, b). potential problems is not found in the para- of Brooklyn. RonaldYager can be reached at the Machine Intelligence Institute, Iona College, New Hence there is negative reinforcement if digm of fuzzy modeling, but in the choice Rochelle, NY 10801; Internet: nyl @iona.bitnet

42 IEEE EXPERT Why the Success of Fuzzy Logic is not Paradoxical Lo@ A. Zadeh, Univelsity of California, Berkeley

Elkan’s paper consists of two almost unre- with (4v B) A (Bv 4),which is in tum fuzzy logic in the narrow sense plays a lated parts. In the first section, Elkan arrives equivalent to very minor role in fuzzy control, just as at the conclusion that an apparently reason- zlassical logic plays a very minor role in B v (4A 43). able version of fuzzy logic collapses mathe- classical control theory. matically to two-valued logic. In the second Consequently, we can assert the logical In his article, Elkan fails to differentiate section, he questions the value of fuzzy logic equivalence between fuzzy logic in the narrow sense and in control applications and concludes that fuzzy logic. In the first part, he interprets l(A A 4)EB v (4A YB), (1) fuzzy logic does not provide an effective tool fuzzy logic in its narrow sense. But in the for dealing with the problem of uncertainty which is the example used in Elkan’s proof. second part, he interprets fuzzy logic in its in knowledge-based systems. As I see it, the What we see, then, is that Elkan’s exam- wide sense, since most applications of fuzzy first conclusion is based on faulty reasoning, ple uses a disguised form of the law of the logic - especially in the realm of control - while the second reflects a misconception of excluded middle. As should be expected, do not involve fuzzy logic in the narrow what fuzzy logic is and a misunderstanding Equation 1 is not a logical equivalence in sense. However, narrow fuzzy logic plays an of the role it plays in control and knowledge- multivalued logic because the law of the essential role in the management of uncer- based systems applications. excluded middle does not hold, in general, tainty in expert systems.] In what follows, It is easy to show why Elkan’s mathe- in multivalued logic. In sum, what Elkan fuzzy logic will be used in its wide sense. matical analysis is faulty. What he really shows in a roundabout way is that the law What are the reasons for the rapid shows is that fuzzy logic is not consistent of the excluded middle does not hold in growth in the number, variety, and visibil- with the law of the excluded middle. This, multivalued logic. There is no justification ity of fuzzy logic applications? The reasons of course, applies in general to multivalued whatsoever for jumping from this obvious are not those given in Elkan’s article. What logical systems. fact to the conclusion that fuzzy logic col- fuzzy logic offers, above all, is a methodol- The law of the excluded middle asserts lapses to two-valued logic. ogy for representing and analyzing depen- that the truth value of any logical expres- Turning to his analysis of fuzzy logic dencies that are approximate rather than sion of the form B v 4 is T (true). The applications, Elkan’s conclusion reflects a exact. In this methodology, the key con- law of contradiction asserts that the truth misunderstanding of what fuzzy logic is, cepts are: value of any logical expression of the form and a faulty analysis of the reasons for its a linguistic variable, whose values are B A 4 is F (false). success. First, it must be clarified that the words rather than numbers; Immediate consequences of these laws term “fuzzy logic” is used in two different a canonical form, which expresses the in two-valued propositional calculus are as senses. In its narrow sense, fuzzy logic is a meaning of a proposition as an elastic follows: logical system that is an extension of multi- constraint on a variable; valued logic and serves as a foundation for If p is logically equivalent to q then p is a fuzzy if-then rule and rule qualifica- approximate reasoning. What is important also logically equivalent to q A (Bv 4). tion, in particular probability qualifica- to note is that even in its narrow sense, the tion and possibility qualification; If p is logically equivalent to q then p is agenda of fuzzy logic is quite different from interpolative reasoning; and also logically equivalent to q v (BA 43). that of traditional multivalued systems. a fuzzy graph. In its wider sense - the sense in which Ifp is logically equivalent to q then it is predominantly used today - fuzzy Through the use of techniques based on p A (B v 4)is logically equivalent to logic is a much broader theory that is these concepts, fuzzy logic makes it possi- V (BA iB). fuzzily synonymous with “fuzzy set the- ble to exploit the tolerance for imprecision Now let us consider Elkan’s Theorem 1. ory,” that is, the theory of classes with un- and uncertainty. In so doing, fuzzy logic Starting with the valid equivalence sharp boundaries. In this perspective, fuzzy has proved to be successful where tradi- logic in the narrow sense is one of the tional approaches have failed or yielded y(A A 43) VB =A many branches of fuzzy logic, among inferior results. which is an expression of De Morgan’s which are fuzzy arithmetic, fuzzy probabil- Most fuzzy logic applications involve law, we can replace the right-hand member ity theory, possibility theory, fuzzy rela- the use of what might be called the calcu- tions, and so on. It should be noted that lus offuzzy rule^.^.^ The use of fuzzy rules

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I in conjunction with interpolative reasoning experienced user with a fuzzy rule-based What this implies is that the problem of greatly reduces the number of rules that are system. In the case of knowledge-based inference from probability-qualified propo- needed to describe imprecise dependen- systems, what has to be replaced is an ex- sitions may not have a satisfactory solution cies, and makes it much easier for humans pert rather than an operator. This is an in- within the framework of classical probabil- to articulate them. Consider, for example, herently more complex problem, no matter ity theory. the rules that people use (consciously or what approach is used. In this connection, it should be noted subconsciously) in parking a car, filling a Basically, what differentiates control ap- that Elkan gives the impression that there tub with hot water, crossing a traffic inter- plications from knowledge-based systems are many expert systems that do not em- section, or riding a bicycle. How would applications is that in control the main prob- ploy fuzzy logic and that provide effective Elkan describe the rules that govern human lem that has to be addressed is that of impre- ways of dealing with uncertainty and im- behavior in these and similar instances? cision. By contrast, in the case of knowl- precision. This is not the case. As a test, What is actually used in most control edge-based systems, one has to come to which of the systems that he as in mind applications is a subset of the calculus of grips with both imprecision and uncertainty. could provide an answer to the following fuzzy rules, which can be called the calcu- In applying fuzzy logic to control sys- question: lus offuzzy In this calculus, a tems, it is generally sufficient to employ graph^.^.^ If X is small then it is very likely that Z is functionf: Vis approximated to by a categorical rules - rules that involve no U + large. fuzzy graphfY, which is a disjunction of quantifiers, probabilities, or possibilities. Cartesian products of the form In the realm of control, the calculi of fuzzy If X is large then it is not likely that Z is rules and fuzzy graphs provide the neces- large. f* =&A, X Bi sary tools for exploiting the tolerance for What is the probability that 2 is large if where A; and , i=l,..., n, are values of imprecision and lead to systems that are Bi Xis medium? linguistic variables, and C,represents the simpler, more robust and have higher ma- disjunction (union) of Cartesian products chine IQ than systems designed by conven- What this example points to is that the A; X Bj . For example, a fuzzy graph of a tional methods. Recently published books6-’ conventional approaches to the manage- function may be expressed in a coarse way as provide easily understandable accounts of ment of uncertainty in expert systems fail the methodology of fuzzy logic control and in four important respects: = small X small + medium X large + f* explain why the applications of fuzzy con- large X small (1) They do not provide the means for trol are growing rapidly in visibility, vari- dealing with the fuzziness of which is equivalent to the set of rules ety, and number. It is very likely that it will antecedents and consequents. not be long before familiarity with fuzzy If X is small then Y is small. (2) They assume that probabilities can be control will be an essential qualification for If X is medium then Y is large. estimated as crisp numbers. control engineers and system designers. If X is large then Y is small. (3) They do not offer a mechanism for In the case of knowledge-based systems, inference from rules in which the The use of fuzzy graphs results in data two sources of difficulty are that the rules qualifying probabilities are fuzzy. compression, which is one of the key - are frequently probability-qualified, and (4) The rules for composition of probabil- though perhaps not widely recognized - that the qualifying probabilities are not ities depend on unsupported assump- advantages of using fuzzy rules. Elkan’s compositional. More specifically, assume tions about conditional independence. analysis makes no reference to this point, that we have two rules of the form and fails to identify the use of the fuzzy Fuzzy logic addresses some - but not all Ifp, then q (PI) graph concept as one of the principal tools - of these problems.’ More specifically, Ifp2thenq2 (P2) in the application of fuzzy logic to control. fuzzy logic allows the antecedents and/or Today, fuzzy logic applications in con- wherepl, ql,p2, and q2 are propositions, consequents and/or qualifying probabilities trol and consumer products are far more and PI and P2are qualifying probabilities. to be fuzzy. Furthermore, fuzzy logic makes visible than fuzzy logic applications in Assume that we wish to compute the quali- it possible to estimate probabilities as fuzzy knowledge-based systems. Does this mean, fying probability, P, in the combined rule rather than crisp numbers. There remain, as Elkan surmises, that fuzzy logic is lim- however, two problems. First, the composi- If @, andpZ)then (ql and q2). ited in its applicability to simple systems? (PI tion of qualifying probabilities can lead to Not at all. What it means is that fuzzy logic The problem is that P cannot be computed fuzzy probabilities that are insufficiently can be applied easily and effectively to the as a function of PI and P, without making specific or, equivalently, insufficiently in- conception and design of “high machine some assumptions about conditional inde- formative. Second, inference in fuzzy logic IQ” control systems and consumer prod- pendence or, equivalently, invoking the reduces, in general, to the solution of a non- ucts - applications that in most cases in- maximum entropy principle. Such assump- linear program. Standard techniques for the volve replacing a trained operator or an tions tend to be ad hoc and hard to justify. solution of such programs may be computa-

1 44 IEEE EXPERT

I tionally expensive. What we do not have as Cadiag-2, the well-known large-scale med- spect, the skeptics will find it hard to un- yet are approximate, inexpensive techniques ical diagnostic system.IO-”Another well- derstand why they failed to realize that for inference from fuzzy-probabili ty-quali- known and commercially available system fuzzy logic is a phase in a natural evolution fied fuzzy if-then rules. However, we do is FRIL,I4 which is Prolog-based and has a of science - an evolution brought about have an effective method of inference from highly sophisticated system for the man- by the need to find an accommodation with possibility-qualified rules within a branch agement of uncertainty. Still another exam- the pervasive imprecision of the real world. of fuzzy logic known as possibilistic logic? ple is the Yamaichi Securities Fund, and There are many statements in Elkan’s there are many more (see Table 1 on page References L.A. Zadeh, “The Role of Fuzzy Logic in articles that relate to ill-posed questions or 46). 1.5-1 7 1. the Management of Uncertainty in Expert reflect a misunderstanding of what fuzzy Elkan also seems to suggest that expert Systems,” Fuzzy Sets and Systems, Vol. 11, logic is, as well as an inadequate familiar- systems that combine grades of member- 1983, pp. 199-227. ity with its literature. I will comment here ship using operators other than max and 2. L.A. Zadeh, “Outline of a New Approach to on just a few of these statements. min are not valid examples of the use of the Analysis of Complex Systems and Deci- In the section on fuzzy logic in expert fuzzy logic. This position is hard to under- sion Processes,” IEEE Trans. Systems, Man, and Cybernetics, Vol. 3, 1973, pp. 28-44. systems, Elkan states, “there is still debate stand since the use of t-norms, t-conorms, 3. L.A. Zadeh, “On the Analysis of Large to and other connectives is now a standard as what types of uncertainty are captured Scale Systems,” Systems Approaches and by fuzzy logic.” Obviously if the bound- part of fuzzy logic.18 Environment Problems, H. Gottinger (ed.), aries of what constitutes fuzzy logic are not The issue of the management of uncer- Vandenhoeck and Ruprecht, 1974, pp. 23- defined, this is not a well-posed issue. In tainty in expert systems presents many 27. this context, what is important to realize is complex and difficult problems. There is 4. L.A. Zadeh, “The Calculus of Fuzzy If- that any theory X can be fuzzified by gen- no system at present that is free of serious Then Rules,” AI Expert, Vol. 7, No. 3, 1992, pp. 22-27. eralizing the concept of a crisp set in X to a shortcomings, and it would be unrealistic 5. L.A. Zadeh, “Fuzzy Logic, Neural fuzzy set, leading to a theory that can be to expect that such systems will be devel- Networks and Soft Computing,” Comm. called fuzzy X. For example, classical oped in the foreseeable future. But Elkan’s ACM, No. 37, March 1994. probability theory can be generalized to statement that “experience shows that 6. D. Driankov, H. Hellendoom, and M. Rein- fuzzy-probability theory; topology to fuzzy fuzzy logic is rarely suitable in practice for frank, An Introduction to Fuzzy Control, topology; neural network theory to fuzzy reasoning about uncertainty” reflects inex- Springer-Verlag. Berlin, 1993. neural network theory; control to fuzzy perience in the use of fuzzy logic. I advise 7. M. Jamshidi, N. Vadiee, and T. Ross, eds., Fuzzy Logic and Control, Prentice Hall, control; arithmetic to fuzzy arithmetic; Elkan to study with care the extensive liter- Englewood Cliffs, NJ, 1993. modal logic to fuzzy modal logic; resolu- ature on the management of uncertainty in 8. A. Kandel and G. Langholz, eds., Fuzzy tion to fuzzy resolution; temporal logic to expert systems based on the use of fuzzy Control Systems, CRC Press, Boca Raton, temporal fuzzy logic; Mycin to fuzzy logic. A good starting point would be the 1994. Mycin; chaos to fuzzy chaos, and so on. treatises by Dubois-Prade on possibility 9. D. Dubois and H. Prade, “Epistemic En- Many such generalizations have already theory and approximate reasoning, and the trenchment and Possibilistic Logic,” Art@ been described in the literature and many books on fuzzy expert system^.'^.'^ There cia/ Intelligence, Vol. 50, 1991, pp. 223- 239. more will be made in the future. What is is little doubt that, in coming years, the 10. K.P. Adlassnig, “A Fuzzy Logical Model of gained from fuzzification is greater gener- growth in familiarity with fuzzy logic will Computer-Assisted Medical Diagnosis,” ality and better approximation to reality. lead to its wide acceptance as a key compo- Methods of Informution in Medicine, Vol. Given that any theory can be fuzzified, nent of information systems and knowl- 19, 1980, pp.141-148. the question of what types of uncertainty are edge engineering methodologies. 11, K.P. Adlassnig, “Fuzzy-Set Theory in Med- captured by fuzzy logic loses much of its ical Diagnosis,” IEEE Trans. Systems, Mun, und Cybernetics, Vol. 16, 1986, pp. 260- meaning. For example, when probability 265. theory is fuzzified, it becomes a part of 12. K.P. Adlassnig and G. Kolarz, “Representa- fuzzy logic. In this broad perspective, then, I compliment Elkan on writing a provoca- tion and Semiautomatic Acquisition of fuzzy probabilistic uncertainties fall within tive article that is likely to contribute to Medical Knowledge in Cadiag- 1 and Ca- the scope of fuzzy logic. The same applies further discussion of the strengths and limi- diag-2,” Computers and Biomedical Re- to any type of uncertainty that I can think of. tations of fuzzy logic. Fuzzy logic has been search, Vol. 19, 1986, pp. 63-79. In the same section, Elkan reports that and still is somewhat controversial. With 13. K.P. Adlassnig, “Uniform Representation of Vagueness and Imprecision in Patient’s his search of the literature revealed no pub- the passage of time, however, the contro- Medical Findings Using Fuzzy Sets,” in lished reports of an expert system that uses versies will abate and fuzzy logic is likely Proc. Cybernetics und Systems ‘88,Kluwer fuzzy logic as its primary formalism. This to become a standard tool for the concep- Academic, Dordecht, 1988. pp. 685-692. is somewhat surprising, since there are, in tion and design of intelligent systems. In- 14. J.F. Baldwin, “Fril: A Fuzzy Relational fact, many such examples. Among them is deed, it would not be surprising if, in retro- Inference Language,” Fuzzy Sets Systems,

AUGUST 1994 45 Table 1. Fuzzy logic tools and products. (Source: Sammy Wong and Nelson Wong, Computer Science Dept., Chinese University of Hong Kong.)

COMPANY PRODUCT DESCRIPTION

American NLX 230 fuzzy microcontroller Has 8 digital inputs, 8 digital outputs, 16 fuzzifiers; holds Neuralogix 64 rules. Evaluates 30M rules/sec. ADS230 fuzzy microcontroller PC-compatible system uses NLX 230 with analog and development system digital I/O. NLX 110 fuzzy pattern correlator Correlates eight I-Mbit patterns; expandable to 256 n-bit Vol. 14, 1984, pp. 155-174. patterns 15. K.S. Leung, W.S. Wong, and W. Lam, “Ap- NLX 11 2 fuzzy data correlator Performs pattern matching on serial data streams plications of A Novel Fuzzy Expert System Shell,” Expert Systems: The Int’l J. Knowl- Aptronix Fide (Fuzzy Interference Runs under MS Windows on 386/486 PCs. Supports edge Engineering, Vol. 6, 1989, pp. 2-10. Development Environment) development, fuzzy simulation, debug tracing, and 3D 16. N. Vadiee and M. Jamshidi, “ATutorial on display of control surfaces. Real-time code generation for microcontrollers. Software implementation of fuzzy Fuzzy Rule-Based Expert System Models I: logic in C. Complete tutorial information and phone Mathematical Foundations,” J. Intelligent support. and Fuzzy Systems, Vol. 1, NO.2, 1993, pp. 171-188. Bvte Craft FUZZY-C Preprocessor translates fuzzy source code into C source code. 17. S.D. Wee, R.E. Larew, and F.C. Hadipriono, Fuzzy Systems Manifold editor Runs under MS Windows 3.1 on 386/486 PCs. Edits “Angular Fuzzy Logic for An Expert Sys- Engineering rules in a matrix display. Lets users view fuzzy sets tem for Pavement Maintenance and Reha- graphically. bilitation Strategy in Ohio,” Proc. Fifrh Manifold graphics editor Runs under MS Windows 3.1 on 386/486 PCs. Color Int’l. Fuzzy Systems Association World graphics display of rules and fuzzy sets. Lets users view Congress ‘93,Vol. 2, 1993, pp. 2 1 1-214. designs in 3-D map and slice formats. 18. H.J. Zimmerman, Fuzzy-Set Theory and its Hitachi American H8/3OO and H8/500 Microcontrollers Applications, 2nd ed., Kluwer Academic Hyperlogic Cubicalc Software for developing fuzzy-logic applications. Runs Publishers, Dordrecht, 1990. under MS-Windows with 286 or higher processor. 19. C. Negoita, Expert Systems and Fuzzy Sys- Simulates fuzzy and nonfuzzy systems. tems, Benjamin Cummings, 1985. Cubicalc-RTC A superset of Cubicalc. Provides runtime compiler support and libraries for linking. Compatible with Lotfi A. Zadeh is professor emeritus of electri- Microsoft C and Borland C. cal engineering and computer sciences at the Cubicalc runtime source code Generates C source code for use in compiling to a University of California, Berkeley, where he is specific processor. director of the Berkeley Initiative in Soft Com- Cubicard Includes Cubicalc-RTC and PC-based hardware for puting. Until 1965, his work centered on system analog and digital 110. theory and decision analysis. Since then, his Inform Software Fuzzytech Explorer Edition Introductory fuzzy-logic system. Software runs under research interests have shifted to the theory of MS Windows. Accepts two inputs, one output, five fuzzy fuzzy sets and its applications to artificial intelli- membership sets per variable, and 125 rules. Includes gence, linguistics, logic, decision analysis, ex- tutorial. pert systems, and neural networks. His current Fuzzytech MCS-96 Edition Full fuzzy development system for MCS-96 research focuses on fuzzy logic and soft comput- microcontrollers. Generates optimized assembly code. ing. He is an alumnus of the University of Fuzzytech Online Edition Lets users debug and modify fuzzy-logic systems while Teheran, MIT, and Columbia University, and they are running. Generates C source code. was awarded honorary doctorates by the Paul- Integrated Systems RT/Fuzzy Module Simulation and code generation of fuzzy logic for real- Sabatier University, France, and the State Uni- time systems. versity of New York, Binghamton, in recognition Metus Systems Group Metus Fuzzy-logic development and simulation system, Runs of his development of the theory of fuzzy sets. under MS DOS. Provides high-level modeling and Iow- He was also awarded honorary doctorates by the level development for embedded applications. University of Dortmund, Germany (l993), and Modico Fuzzle 1.8 PC-based fuzzy-logic shell. Generates source code for C the University of Granada, Spain (1994). Zadeh and Fortran. has held visiting appointments at the Institute for Motorola Fuzzy-logic kernel foi Fuzzy processing kernels for 68HC05 and 68HCll Advanced Study in Princeton, N.J., MIT, IBM microcontrollers microcontrollers. Includes fuzzy knowledge-base Research Laboratory, SRI International, and generator to create code for kernel. Stanford University. He is a fellow of the IEEE, Fuzzy-logic educational kit Interactive training tool provides good introduction for AAAS, ACM, and AAAI, and is a member of the understanding and using fuzzy logic. Runs under MS National Academy of Engineering and the Russ- Windows. Includes demonstration version of Fide (from ian Academy of Natural Sciences. He has been Aptronix). the recipient of numerous honors, including the Togai lnfralogic TlLShell t fuzzy C Complete fuzzy development system generates C code IEEE Education Medal (1973), the IEEE centen- development system and includes debug, fuzzy-simulation, and graphical- nial medal (1984), the Honda Foundation’s analysis tools. Tutorial included. Honda Prize (1989), the Certificate of Commen- Microcontroller evaluation Fuzzy development systems for Hitachi H8/300, H8/500, dation for AI Special Contributions Award from packages and HMCS400; Intel 8051; and Mitsubishi 37450. the International Foundation for Artificial Intel- Microcontroller production Unlimited production license ligence (1992), the IEEE Hamming Medal licenses (1992), the Grigore Moisid Prize (1993), and the Rufus Oldenburger Medal from the American FC 110 Digital fuzzy-logic processor (IC) Society of Mechanical Engineers (1993). Lotfi FC 110 development system Hardware and software development system for FC 110. Zadeh can be reached at the Computer Science Versions support IBM PC/AT bus, Sbus, and VMEbus. Division, Dept. of EECS, University of Califor- nia, Berkeley, Berkeley, CA 94720; Internet: [email protected]

46 Elkan’s Reply The Paradoxical Controversy over Fuzzy Logic The responses to my article provide an that I have is whether the distinction is re- knowledge becomes implicit background exceptionally wide range of perspectives ally well defined. On the one hand, there knowledge that must be used tacitly in tun- on the current state of research on fuzzy may be multiple types of imprecision and ing the allowed interactions between the logic and its applications. Overall, I find vagueness. Is the domain-independent im- items of explicit shallow knowledge. To that with most commentators I agree more precision involved in “around 1.80m” the quote Garcia, “The dogma of generality than I disagree. I shall try here to steer a same as the human-specific imprecision versus efficiency strikes again, and knowl- middle course between simply repeating involved in “tall”? On the other hand, it edge engineering and machine learning are points of agreement and narrowly counter- may be possible to model some types of not exempted.” ing points of disagreement. imprecision probabilistically. For example, the degree of truth of the assertion “1.80m Fuzzy logic in expert systems. Only three The foundations of fuzzy logic. Some is tall” might be modeled as the probability of the responses give references in an at- commentators take a more extreme posi- that an individual with height 1.80m would tempt to dispute the claim that there are tion than I do concerning the coherence of be labeled as tall given incomplete knowl- very few deployed expert systems that ac- fuzzy logic. I do not agree with Attikiouzel edge, that is, given no other information on tually use fuzzy logic as their principal that “if one wishes to write a program or the individual. formalism for reasoning about uncertainty. build a machine that will perform inference Overall, I am wary of the enterprise of Moreover, most of the references given in the same way as human beings, then one even making an attempt to classify the actually support this claim. must build the basic equations of probabil- types of uncertainty. A complete and con- Before I discuss these references one by ity theory into it, or face the inevitable out- sistent analysis of all the many varieties of one, it is worth emphasizing that I use the come that it will not perform as required’ uncertainty involved in human thinking term “expert system” to designate a reason- Neither humans nor machines always re- and revealed in human language is a philo- ing system that applies a large base of ex- quire formal rigor to act successfully in the sophical goal that we should not expect to plicit knowledge to perform a task requiring world, nor is success always guaranteed by achieve soon. Moreover, this aspiration is a complex inference, such as diagnosis, rigor. Successful controllers and expert variant of the quest for formal rigor criti- scheduling, or design. A fuzzy controller is a systems can use heuristic, shallow knowl- cized above as neither necessary nor suffi- knowledge-based system of a different na- edge and therefore they can use arbitrary cient for engineering success. As Freksa ture. If a fuzzy controller is called an expert reasoning formalisms such as certainty points out, it is always the case that “the system, this blurs some important distinc- factors or fuzzy logic. I also do not agree represented real world and its representations. As Zadeh writes, “what differentiates that “Proponents of fuzzy logic appear to tion are formally incommensurable.” applications to control from applications to be unaware of Cox’s work and that of Therefore, however ideal the logics that [general] knowledge-based systems is that Jaynes and Tribus”; for evidence see the one has at hand, knowledge engineering is in control the main problem which has to be debate in a recent issue of IEEE Transac- always a tentative activity that can never addressed is that of imprecision. By con- tions on Fuzzy Systems.‘ succeed completely. trast, in the case of knowledge-based sys- However, I am uncomfortable with the More varieties of uncertainty may well tems, one has to come to grips with both dogmatism evinced by many of the advo- exist in the case of shallow knowledge than imprecision and uncertainty.” cates of fuzzy logic or some of its many in the case of deep knowledge, because shal- As I discussed in my paper, another im- variants. For example, Dubois, Prade, and low knowledge is intrinsically domain- portant difference is that most controllers do Smets say that I fail “to understand the specific and of restricted generality. As not have to remember and reason about the important distinction between ... properties Garcia points out, the reasoning in my history of the portion of the outside world whose satisfaction is a matter of degree” watermelon example relies on important that they deal with. Most fuzzy controllers and uncertainty “induced by incomplete background knowledge that is not expressed have no internal state, while expert systems states of knowledge.” Later they write that in terms of rules. But it is not a fair reply to retain considerable state information. the AI community has forgotten this dis- the example to call for this implicit back- Dubois, Prade, and Smets give five refer- tinction. It appears to me that the AI com- ground knowledge to be made explicit. The ences, the latest of which is five years old. munity has not forgotten this very binary deep knowledge that underlies a given frag- The Cadiag work of Adlassnig and his col- distinction, but rather has implicitly ment of shallow knowledge may often be leagues is indeed impressive.2 However, it rejected the claim that it is a uniquely im- impossible or too expensive to make is especially difficult to deploy medical portant distinction. A particular concem explicit. It is precisely then that the deep expert systems in the real world, in compar-

AUGUST 1994 47

I ison, say, to applications in manufacturing. edge base of fuzzy logic sentences, and com- as asking for a schema of logical equiva- Both the cited paper and more recent papers pilation into single-level rules “to simplify lences, in which A and B may be replaced on Cadiag-23,4state only that Cadiag-2 sys- and speed computation” is mentioned by by any assertions, including assertions of tems are undergoing clinical trials. several commentators, Berenji in particular. the form lC. Similarly, the paper on Taige6 does not As Garcia and other commentators point claim that the system has been deployed, out, the theorem can also be proved by con- The success of fuzzy control. Perhaps the and I could not find any further papers on sidering much simpler equivalences such as most important contention of my paper is this system. The cited paper on RUM6 A ~4=l(A v4)orA AA=BA lB. that the success of fuzzy controllers has states it is a “development environment,” The reason the proof given uses a more little to do with the theory of fuzzy logic or and the only published application built complicated equivalence is that, as just men- fuzzy sets. Several commentators confirm using it is described as a “prototype.”’Fi- tioned, it is more natural in some intuitive this. For example, Klir and Yuan say that nally, OPAL’ is described in the cited paper sense. Intuitively speaking, in A A 4 = “most of the simple fuzzy controllers on as “under development,” and Milord9 is B A 43the two sides are irrelevant to each the market ... are not explicitly based on said to be a “shell.” More recent versions other, andA A 4 = l(A v 4)is obviously fuzzy logic.” Dubois, Prade, and Smets of Milord use finite multiple-valued logics similar to the law of excluded middle. write that “Takagi and Sugeno have pro- rather than fuzzy logic.’o The phrase “obviously similar” in the posed an interpolation mechanism ... this Vadiee and Jamshidi say that “The statement above is vague. One interpretation kind of ‘inference’ (which is widely used in Nikkei average has reportedly gone consis- of the theorem is that if we reject the law of fuzzy control) has nothing to do with un- tently higher using fuzzy logic.” This state- excluded middle, then we must also reject certainty handling,” and Pelletier writes ment is difficult to understand, let alone to many other equivalences that are not obvi- that “those areas of fuzzy logic that get believe; the only citation is to the authors’ ously similar to this law, but that are never- criticized are simply not employed in the own unpublished course notes. The other theless interchangeable with the law using control arena.” application they mention is a system for only the first three postulates of Definition It is a general property of systems that choosing oil recovery methods. According 1. When Yager gives a derivation of the law use only shallow knowledge that numerical to the journal paper on this system it uses of excluded middle from t(-(A A 4))= uncertainty values can be tuned, if neces- the Clips shell, which is not founded on t(B v (4A +)), this is an alternative state- sary, to overcome arbitrariness in the opera- fuzzy logic.’ ment of the theorem, not a demonstration tors used for combining uncertainty values. Zadeh gives three examples of expert that the theorem is absurd. Alternatively, within reason, the operators systems using fuzzy logic as their primary Overall, I am saddened by the hostility can be adjusted to match given numerical formalism for reasoning about uncertainty: visible in the comments by Yager and by values. As Chandrasekaran reminds us con- Cadiag-2 again, FRIL,I2 and a system for Klir and Yuan. I will refrain from respond- cerning Mycin, a system based on shallow securities trading with no citation. Recent ing line by line to their remarks on the dif- medical knowledge: “The fine structure of papers indicate that FRIL is a “program- ferent versions of my theorem and its proof. uncertainty didn’t really matter.” Several ming language”” and that the trading sys- It is quite usual in the history of mathemat- commentators support my specific tem has only been “te~ted.”’~Zadeh also ics for a theorem that attracts interest to be contention that this property is one reason cites papers on systems for acupuncture restated and reinterpreted over time, and for for the success of heuristic controllers using diagnosis and pavement maintenance from similarities with previous results to be no- fuzzy logic. For example, Wang, Tan, and the 1993 International Fuzzy Systems As- ticed later. For a similar but friendly exege- Tan write that “...numerous forms of fuzzy sociation World Congress, but I do not sis of the development of the statement and operations ... were created to cater to the have access to these papers. proof of a far deeper and more important domain-specific needs.” theorem the reader can consult Proofs and I do not agree with Ruspini that the term The theorem. Except for Klir and Yuan, no Refutations by Imre Lakatos.I5 “paradox” should only be used to mean commentators dispute the mathematical The theorem is technically correct as “logical self-contradiction,” so I believe that validity of the theorem given in my paper, stated and proved both here and in my it is fair to call the lack of connection in but several commentators disagree with the AAA1 ’93 paper. Klir and Yuan say that fuzzy systems between theory and practice assumptions made in its statement. Dubois, either the statement or the proof of the the- an apparent paradox. All paradoxes have the Prade, and Smets say it relies “at best on a orem is incorrect, because the “proof de- property that once resolved, they no longer logical equivalence the rationale of which pends on eight logical equivalencies, only appear paradoxical. To paraphrase a state- is far from natural in the scope of fuzzy one of which is included in the statement.” ment by Tiirksen, there are no paradoxes, logic.” In my opinion, the opposite is true. This claim is based on a misreading of the only limited or partial understanding. The The equivalence between l(A A 4)and statement of the theorem, where the condi- paradox that fuzzy controllers have had real B v (4A 4)is a natural one to use (per- tion “if l(A A 43) and B v (4A 43) are industrial success, while fuzzy logic itself is haps inadvertently) in compiling a knowl- logically equivalent” must be understood still under attack mathematically, is resolved

4a IEEE EXPERT I

by understanding the distinction between a “Fuzzy sets provide for a general yet com- an Expert System,” Expert Systems with scientific experiment designed to confirm or pact characterization of system state that Applications, Vol. 6, No. 4,Oct.-Dec. 1993, pp. 44148. requires fewer rules.” However, interpola- disconfirm a theory and an engineering ap- 5. H. Farreny, H. Prade, and E. Wyss, “Approxi- plication of the theory. Fuzzy controllers are tion is a purely local operation, where the mate Reasoning in a Rule-Based Expert Sys- applications, not experiments that could :onclusions of a few rules describing re- tem Using Possibility Theory: A Case Study,” validate theoretical claims about fuzzy sponses to nearby input parameter configu- Proc. 10th World Computer Congress (IFIP), North-Holland, Amsterdam, 1986, pp. logic. On this point I agree with Mamdani: rations are blended. It is therefore difficult 407413. “There is a common misconception that to see how interpolation could reduce the 6. P.P. Bonissone, S.S. Cans, andK.S. Decker, models are created and then applied and the amount of knowledge needed to capture a “RUM: A Layered Architecture for Reason- success then legitimizes a model.” complex, multidimensional inputloutput ing with Uncertainty,” Proc. 10th Int’l Joint Conf Artificial Intelligence (IJCAI), Morgan Overall, the response by Mamdani is mapping by more than one order of magni- Kaufmann, San Francisco, Calif., 1987, pp. particularly trenchant and thought-provok- tude compared to other approaches. 891-898. ing. Where we disagree, I think the cause is Klir and Yuan write that “. . . fuzzy con- 7. P.P. Bonissone and S. Dutta, “MARS: A a misunderstanding. I do not argue that trollers of this kind [that do interpolation] Mergers and Acquisitions Reasoning Sys- tem,” Computer Science in Economics and fuzzy control “is not worthy of industrial are universal approximators.” This fact is Management, Vol. 3, No. 3, 1990, pp. consideration because of its lack of com- true, but less significant than it may appear 239-268. plex form and structural sophistication.” at first sight. Given suitable smoothness 8. E. Bensana, G. Bel, and D. Dubois, “OPAL: Rather, I argue that this simplicity is vital constraints, many mathematical formalisms A Multi-Knowledge-Based System for Indus- trial Job-Shop Scheduling,” Int’l J. Produc- to the industrial success of the current gen- can be used as universal approximators of tion Research, Vol. 26, No. 5, May 1988, pp. eration of fuzzy controllers, but that fuzzy multidimensional inputloutput mappings. 795-819. controllers for more complex applications For example, any continuous function can 9. L. Godo et al., “Milord: The Architecture and will run into the same problems of com- be approximated to any desired degree of the Management of Linguistically Expressed Uncertainty,” Int’l J. oflntelligent Systems, plexity that other knowledge-based sys- accuracy by a polynomial of sufficiently Vol.4, No. 4, 1989, pp.471-501. tems do today. It is the case that the “philo- high order. Neural networks with hidden IO. C. Sierra and L. Godo, “Modularity, Uncer- sophical deficiencies of fuzzy logic” do layers are also universal approximators.” tainty, and Reflection in Milord 11,” Proc. 1992 IEEE Int ‘1 Coni Systems, Man, and Cybemet- something “to argue against the adoption of The important question is how complex an ics, IEEE Computer Society Press, Los Alami- fuzzy logic control”: These deficiencies are approximation must be allowed to be to tos, Calif., Vol. I, 1992, pp. 255-260. what makes scaling-up difficult. achieve a given level of precision. As rec 11. W.J. Parkinson et al., “Using an Expert Sys- Many research teams are actively working ognized by Kosko and Isaka,lXthe number tem to Explore Enhanced Oil Recovery Methods,” Computers and Electrical Engi- on scaling-up fuzzy controllers. A common of rules required by a fuzzy controller - neering, Vol. 20, No. 2, March 1994, pp. feature of the research prototypes developed which is the number of patches used to ap- 181-197. by these teams is the use of ideas for organiz- proximate its control surface - grows ex- 12. J.F. Baldwin and S.Q. Zhou, “A Fuzzy Rela- ing large intelligent systems first proposed ponentially with the dimensionality of the tional Inference Language,” Fuzzy Sets and Systems, Vol. 14, No. 2, Nov. 1984, pp. by mainstream AI researchers. For example, controller and the level of precision 155-174. the SRI autonomous robot mentioned by demanded. From a formal point of view, 13. J.F. Baldwin and T.P. Martin, “Fast Opera- Berenji uses “several deliberation levels to fuzzy controllers thus do not enjoy a clear tions on Fuzzy Sets in the Abstract FRIL determine the relevance level of each control advantage over other formalisms for ap- Machine,” Proc. IEEE Int’l Conf FuZiy Sysrems, IEEE Press, Piscataway, N.J., 1992, rule ...; to identify current goals and their proximating smooth functions. Of course pp. 803-810. state of achievement; to activate control rules they are still pragmatically very useful. 14. H. Tanaka, “Fuzzy Modeling in Operations according to the current context; and to Research,” Japanese J. Fuzzy Theory and blend their control recommendations.” The References Systems, Vol. 4, NO. 1, 1992, pp. 75-83. 15. Imre Lakatos, Proofs and Refutations: The main novelty here compared to classical 1. J.C. Bezdek, ed., IEEE Trans. Fuzzy Systems, special issue on fuzziness versus probability, Logic of Mathematical Discovery, robot architectures is the idea of interpolat- Vol. 2 No. 1, Feb. 1994. Cambridge Univ. Press, New York, 1976. ing smoothly between different suggested 2. K.P. Adlassnig and G. Kolarz, “Cadiag-2: 16. R.A. Brooks, “Intelligence Without Repre- actions -but this idea is also found in other Computer-Assisted Medical Diagnosis Using sentation,” Artificial Intelligence, Vol. 47, No. 1-3, Jan. 1991, pp. 139-159. AI work, such as that of Brooks.16 Fuzzy Subsets,” in Approximate Reasoning in Decision Analysis, North-Holland, Ams- 17. H. White, “Connectionist Nonparametric The ability to interpolate between the terdam, 1982, pp. 219-247. Regression: Multilayer Feedforward Net- conclusions of several rules is an important 3. K.P. Adlassnig et al., “Approach to a Hospi- works Can Learn Arbitrary Mappings,” advantage of fuzzy control methodologies. tal-Based Application of a Medical Expert Neural Nemorks, Vol. 3, No. 5, 1990, pp. 535-549. As Yager writes, “the fact that most fuzzy System,” Medical Informatics, Vol. 11, No. 3, July-Sept. 1986, pp. 205-223. 18. B. Kosko and S. kaka, “Fuzzy Logic,” Scien- controllers are built with a small number of 4. K.P. Adlassnig, H. Leitich, and G. Kolarz, tificAmerican, Vol. 269, No. 1, July 1993, rules should be seen as one of the powers “On the Applicability of Diagnostic Criteria pp. 76-83. of this technology,” and as Berenji writes, for the Diagnosis of Rheumatoid Arthritis in

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