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  October 20, 19981 Fuzzy Logic - A Mordern Perspective John Yen, Senior Member, IEEE A BSTRACT Traditionaly, fuzzy logic has been viewed in the AI community as an approach for managing uncertainty. In the1990's, however, fuzzy logic has emerged as a paradigm for approximating a functional mapping. This complemen-tary mordern view about the technology offers new insights about the foundation of fuzzy logic as well as new chal-lenges regarding the identification of fuzzy models. In this paper, we will first review some of the major milestones inthe history of developing fuzzy logic technology. After a short summary of major concepts in fuzzy logic, we dis-cuss a mordern view about the foundation of two types of fuzzy rules. Finally, we review some of the research inaddressing various challenges regarding automated identification of fuzzy rule-based models. 1 I NTRODUCTION Fuzzy logic (FL) has been a somewhat controversial technology since its birth. However, the large number of successful industrial fuzzy logic applications in 1990's, especially those developed in Japan, has generated an increas-ing interest in FL. Fuzzy logic and artificial intelligence (AI) have at least one common objective: to develop compu-tational methods that can perform reasoning and problem solving tasks that require human intelligence. However,fuzzy logic has an additional objective: to explore an effective tradeoff between precision and the cost in developingan approximate model of a complex system or function. While the issue of the cost was not considered an importantissue in early AI work, it has become important in the past decade due to an increasing interest in resource-con-strained intelligent agents. FIGURE 1. The Cost-Precision Trade-off   impreciseUtility preciseCost  October 20, 19982 Conceptually, we can use Figure 1 to depict this trade-off for many systems. The horizontal axis represents thedegree of precision of the system, while the vertical axis serves the dual-purpose of representing both the cost and thedegree of utility. As the precision of a system increases, the cost for developing the system also increases, typically inan exponential manner. On the other hand, the utility (i.e., usefulness) of the system does not increase proportionallyas its precision increases -- it usually saturates after a certain point. This insight about the trade-off between precision,cost, and utility inspired Zadeh and his followers to exploit the shaded area in Figure 1, which resulted in a new para-digm for developing approximate solutions that are both cost-effective and highly useful.Traditionally, fuzzy logic has been viewed as a theory for dealing with uncertainty about complex systems. Amodern complementary perspective is, however, to view fuzzy logic as an approximation theory. This perspective onfuzzy logic brings to the surface the underpinning of the theory described above - the cost-precision trade-off. Indeed,providing a cost-effective solution to a wide range of real world problems is the primary reason that fuzzy logic hasfound so many successful applications in industry to date. Understanding this driving force of the success of fuzzylogic will prevent us from falling into the trap of debating “whether fuzzy logic can accomplish what X can notaccomplish” where X is an alternative technology such as probability theory, control theory, etc. Such a debate is usu-ally not fruitful because it ignores one important issue -- cost. A better question to ask is “What is the differencebetween the cost of a fuzzy logic approach and the cost of an approach based on X to accomplish a certain task?” 2H ISTORICALDEVELOPMENTOFTHETECHNOLOGY In early 1960s, Lotfi A. Zadeh, a professor at University of California at Berkley well respected for his contribu-tions to the development of system theories, began to feel that traditional systems analysis techniques were too pre-cise for many complex real-world problems. The idea of grade of membership, which is the concept that became thebackbone of fuzzy set theory, occurred to him in 1964 [21], which lead to the publication of his seminal paper onfuzzy sets in 1965 and the birth of fuzzy logic technology [35]. The concept of fuzzy sets and fuzzy logic encoun-tered sharp criticism from the academic community; however, scholars and scientists around the world -- rangingfrom psychology, sociology, philosophy and economics to natural sciences and engineering -- became Zadeh’s fol-lowers. B.R. Gaines and L.J. Kohout gave a detailed bibliography of the first decade of fuzzy logic research in [10].Fuzzy Logic research in Japan started with two small university research groups established in late 1970s: onewas lead by T. Terano and H. Shibata in Tokyo, and the other lead by K. Tanaka and K. Asai in Kanasai. Like fuzzylogic researchers in the U.S., these researchers encountered an “anti-fuzzy” atmosphere in Japan during those earlydays. However, their persistence and hard work would prove to be worthwhile a decade later. These Japaneseresearchers, their students, and the students of their students would make many important contributions to the theoryas well as to the applications of fuzzy logic [11].  October 20, 19983 In 1974, S. Assilian and E. H. Mamdani in United Kingdom developed the first fuzzy logic controller, whichwas for controlling a steam generator [19]. In 1976, Blue Circle Cement and SIRA in Denmark developed a cementkiln controller -- which is the first industrial application of fuzzy logic. The system went to operation in 1982.In the 1980’s, several important industrial applications of fuzzy logic was launched successfully in Japan. Aftereight years of persistent research, development, and deployment efforts, Yasunobu and his colleagues at Hitachi put afuzzy logic-based automatic train operation control system into operation in Sendai city’s subway system in 1987[29]. Another early successful industrial application of fuzzy logic is a water-treatment system developed by FujiElectric. These and other applications motivated many Japanese engineers to investigate a wide range of novelfuzzy logic applications. This lead to the fuzzy boom.The fuzzy boom in Japan was a result of close collaboration and technology transfer between universities andindustries. Two large-scale national research projects were established by two Japanese government agencies in1987: the better known of the two is the Laboratory for International Fuzzy Engineering Research (LIFE). In lateJanuary 1990, Matsushita Electric Industrial Co. named their newly developed fuzzy controlled automatic washingmachine “Asai-go (beloved wife) Day Fuzzy” and launched a major commercial campaign for the “fuzzy” product.This campaign turns out to be a successfully marketing effort not only for the product, but also for the fuzzy logictechnology. A foreign word pronounced “fuzzy” was thus introduced to Japan with a new meaning -- intelligence.Many other home electronics companies followed Panasonic’s approach and introduced fuzzy vacuum cleaners,fuzzy rice cookers, fuzzy refrigerators, fuzzy camcorders (for stablizing the image under hand jittering), camera (forsmart auto-focus) and others. This resulted in a fuzzy vogue in Japan. As a result, the consumers in Japan all recog-nized the Japanese word “fuzzy”, which won the gold prize for the new word in 1990 [11]. This fuzzy boom in Japantriggered a broad and serious interest in this technology in Korea, Europe, and, to a lesser extent, in the United States,where fuzzy logic was invented.Fuzzy logic has also found its applications in the financial area. The first financial trading system using fuzzylogic was Yamaichi Fuzzy Fund. It handles 65 industries and a majority of the stocks listed on Nikkei Dow and con-sists of approximately 800 fuzzy rules. Rules are determined monthly by a group of experts and modified by seniorbusiness analysts as necessary. The system was tested for two years, and its performance in terms of the return andgrowth exceeds the Nikkei Average by over 20%. While in testing, the system recommended “sell” 18 days before theBlack Monday in 1987. The system went to commercial operations in 1988.The first special-purpose VLSI chip for performing fuzzy logic inferences was developed by M. Togai and H.Watanabe in 1986 [27]. These special-purpose VLSI chips can enhance the performance of fuzzy rule-based systemsfor real-time applications. Several companies were formed (e.g., Togai Infralogic, APTRONIX, INFORM) wereformed to commercialize hardware and software tools for developing fuzzy systems. Vendors of conventional con-trol design software also started introducing add-on toolbox for designing fuzzy systems. The Fuzzy Logic Toolbox for MATLAB, for instance, was introduced as an add-on component to MATLAB in 1994.  October 20, 19984 2.1 Learning of Fuzzy Knowledge The development of fuzzy systems in early days required the manual tuning of the system parameters based onobserving the system performance. This drawback has become one of the major criticisms toward fuzzy logic. Eventhough Mamdani and Baaklini introduced self-adaptive fuzzy logic control as early as 1975, the most common cita-tion to the first work in this area is a paper by T. J. Procyk and E. H. Mamdani published in 1979 [22]. This was fol-lowed by Japanese researchers in the 1980’s. T. Takagi and his advisor M. Sugeno together took an important step bydeveloping the first approach for constructing (not tuning) fuzzy rules using training data [26]. Their approachlearned fuzzy rules for controlling a toy vehicle by observing how a human operator controlled the vehicle. Eventhough this important work did not gain as much immediate attention as it did later, it laid the foundation for animportant subarea in fuzzy logic, which is later referred to as  fuzzy model identification  in the 1990’s.Another trend that contributed to research in fuzzy model identification is the increasing visibility of neural net-work research in the late 1980’s. Because of certain similarities between neural networks and fuzzy logic, researchersbegan to investigate ways to combine the two technologies. The most important outcome of this trend is the develop-ment of various techniques for identifying the parameters in a fuzzy system using neural network learning techniques.A system built this way is called a neuro-fuzzy system [15, 18].The 1990’s is an era of new computational paradigms. In addition to fuzzy logic and neural networks, a thirdnon-conventional computational paradigm also became popular -- evolutionary computing , which includes geneticalgorithms, evolutionary strategies, and  evolutionary programming . Genetic algorithms(GA) and evolutionary strate-gies are optimization techniques that attempt to avoid being easily trapped in local minima by simultaneously explor-ing multiple points in the search space and by generating new points based on the Darwinian theory of evolution --survival of the fittest. The popularity of GA in the 1990’s inspired the use of GA for optimizing parameters in fuzzysystems [13]. Various synergistic combinations of neural networks, genetic algorithms, and fuzzy logic help people toview them as complementary. To distinguish these paradigms from the conventional methodologies based on preciseformulations, Zadeh introduced the term soft computing in early 1990’s [41]. 3F UZZY S ETS ,P OSSIBILITY D ISTRIBUTIONS , AND C OMPOSITIONAL R ULEOF I NFERENCE Fuzzy sets, linguistic variables, and possibility distributions are three core concepts in fuzzy logic. A fuzzy setis a generalization to classical set to allow objects to take partial membership in vague concepts (i.e., fuzzy sets) [35].The degree an object belongs to a fuzzy set, which is a real number between 0 and 1, is called the membership valuein the set. The meaning of a fuzzy set, is thus characterized by a membership function  that maps elements of a uni-verse of discourse to their corresponding membership values. The membership function of a fuzzy set  A  is denoted as. In addition to membership functions, a fuzzy set is also associated with a linguistically meaningful term (e.g., µ  A
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