Fuzzy Logic

Abstract

In 1965 the Berkeley systems theorist Lotfi Zadeh published a paper arguing that sets can have partial members: a man of 178 cm is “tall” to degree 0.7, not yes or no. Western academia treated the idea somewhere between indifference and open contempt. Japanese engineers put it in cement kilns, a subway, and by 1990 in washing machines, rice cookers, and camcorders, until “fuzzy” won a national buzzword award and became a marketing cliché. The boom collapsed in the mid-1990s; fuzzy controllers still run inside appliances and the Sendai trains today.

Lotfi Zadeh, 2005. Image: BBR100, CC BY-SA 4.0, via Wikimedia Commons.

Lotfi Zadeh

Lotfi Zadeh was born on 4 February 1921 in Baku, then the capital of Soviet Azerbaijan, to an Iranian-Azerbaijani journalist father and a Russian-Jewish pediatrician mother. The family moved to Tehran in 1931, where Zadeh attended the Alborz school and took a degree in electrical engineering from the University of Tehran in 1942. He emigrated to the United States in 1944, earned a master’s degree at MIT in 1946 and a PhD at Columbia in 1949, and taught at Columbia through the 1950s. His early work sat squarely in the mainstream: in 1952 he and John Ragazzini introduced the z-transform formulation that became a standard tool of sampled-data and digital signal processing. In 1959 he moved to Berkeley, chaired the electrical engineering department from 1963 to 1968, and drove its transformation into the Department of Electrical Engineering and Computer Sciences (EECS).

That establishment pedigree matters for what followed. When Zadeh began questioning the foundations of classical modeling, he was a department chair with impeccable credentials in exactly the mathematics he was about to relax.

The 1965 Paper

Classical set theory, the foundation beneath Boolean logic, forces every question of membership to a yes or a no: an element is in the set or it is not. Zadeh’s June 1965 paper “Fuzzy Sets” (Information and Control, vol. 8, pp. 338–353) proposed sets with graded membership, a number between 0 and 1 expressing how much an element belongs. “Tall people,” “warm rooms,” and “fast cars” have no natural boundary, yet humans reason with such categories constantly and well. Zadeh defined intersection, union, and complement for these sets (minimum, maximum, and one-minus, in his original formulation), giving vague categories an exact calculus. The idea extended the many-valued logics that Jan Łukasiewicz had explored in the 1920s from truth values to sets, and from there to practical engineering.

The follow-up that made fuzzy logic usable came in 1973, when Zadeh introduced the linguistic variable: a variable whose values are words. A rule like “IF the temperature is slightly too low, THEN increase heating slightly” becomes a computable object, with “slightly too low” defined as a fuzzy set over the thermometer scale. A handful of such rules, blended by degree of match, yields a smooth control surface. This is the mechanism inside nearly every fuzzy application since: precise interpolation between rules stated in words.

The 1965 paper became one of the most cited in the history of computing; by early 2021 it had accumulated more than 115,000 citations.

The Backlash

The American response went beyond indifference: parts of academia campaigned against the idea. Rudolf Kalman, whose filter had become the gold standard of estimation theory, said in 1972 that “fuzzification is a kind of scientific permissiveness” which “tends to result in socially appealing slogans unaccompanied by the discipline of hard scientific work and patient observation.” William Kahan, Zadeh’s Berkeley colleague and the principal architect of the IEEE floating-point standard, was blunter: “Fuzzy theory is wrong, wrong, and pernicious,” and, on another occasion, “the cocaine of science.” Statisticians objected that subjective probability already handled graded uncertainty. US research funding for the field stayed thin for two decades.

The two camps were arguing past each other. Kalman’s methods assume a precise mathematical model of the system being controlled. Fuzzy control was aimed at systems for which no such model exists, but for which experienced human operators demonstrably do the job.

Steam Engines and Cement Kilns

The first person to close the loop was Ebrahim “Abe” Mamdani at Queen Mary College, London. In 1974 he and his doctoral student Sedrak Assilian built a fuzzy controller for a laboratory steam engine, translating a human operator’s heuristic rules directly into Zadeh’s formalism; their 1975 paper “An Experiment in Linguistic Synthesis with a Fuzzy Logic Controller” showed the approach worked on real hardware. The rule-based architecture they used, “Mamdani inference,” became the field’s standard.

Industry followed where the modeling problem was worst. A rotary cement kiln is a slow, nonlinear process that resists differential equations but yields to an experienced operator’s rules of thumb, which made it the ideal first customer. Engineers Lauritz Peter Holmblad and Jens-Jørgen Østergaard at the Danish cement-plant builder F.L. Smidth began closed-loop fuzzy control experiments on a production kiln in 1978, and in 1980 the company launched the first commercial fuzzy control system for kiln automation.

The Sendai Subway

A Sendai Subway 1000 series train, whose fuzzy-logic ATO has controlled acceleration and braking since 1987. Image: Jet-0 at the Japanese Wikipedia, CC BY 3.0, via Wikimedia Commons.

The demonstration that converted an industry came from Hitachi. Starting around 1979, engineer Seiji Yasunobu developed a predictive fuzzy controller for automatic train operation, encoding the judgments of veteran drivers about when to notch power and when to coast; by 1985 he and Soji Miyamoto had shown its feasibility in extensive simulation. On 15 July 1987 the Sendai Subway’s Namboku Line opened with the system in command of acceleration, braking, and station stops along its 14.8 km and 17 stations, the first public railway in the world to run on fuzzy control. The trains rode noticeably more smoothly than human-driven ones and used about 10 percent less energy. The same year, Takeshi Yamakawa balanced an inverted pendulum using dedicated fuzzy-logic chips, a laboratory stunt that suggested the technique could be made fast and cheap in silicon.

For Japanese manufacturers the subway was proof in steel: a mathematically disreputable American idea was carrying commuters through Sendai with measurable gains. The follow-the-leader dynamic that had powered earlier waves of Japan’s computing industry took over from there.

The Fuzzy Boom

In February 1990 Matsushita put on sale a fully automatic washing machine called Aisaigo Day Fuzzy (“aisaigo” translates roughly as “beloved-wife model”). Its sensors estimated load size and how dirty the water was; fuzzy rules then chose the wash program, adjusting in the graded increments (“a little weaker”) that fixed programs could not express. It sold well, and within the year fuzzy control appeared in vacuum cleaners that sensed dust, rice cookers that adjusted heat, camcorders with image stabilization, Hitachi washing machines, Mitsubishi air conditioners, and a Canon autofocus camera that ran on 12 inputs and 13 rules. Nissan tested fuzzy automatic transmissions and anti-lock brakes, an early instance of the computerization of the car. Time reported in September 1989, with some alarm, that Japan was pursuing more than 100 fuzzy applications while American researchers scoffed.

The word itself became a phenomenon. In Japanese marketing “ファジィ” (fuzzy) came to signal a machine with human-like judgment, and in 1990 it won the gold prize in the new-word category of Japan’s New Words and Buzzwords Award (Shingo Ryūkōgo Taishō). MITI, the ministry that had launched the Fifth Generation computing project (see Expert Systems and the First AI Winter), opened the Laboratory for International Fuzzy Engineering Research (LIFE) in Yokohama in 1988, a cooperative venture of 48 companies with roughly $34 million committed over six years. A 1991 trade-press survey passed along venture-capital projections of a fuzzy industry worth $2–3 billion a year. Belated Western recognition arrived for Zadeh personally: the IEEE awarded him the Hamming Medal in 1992 and its highest distinction, the Medal of Honor, in 1995. That same year, Maytag introduced a fuzzy-controlled dishwasher, five years behind Osaka.

Dead End: The Hype That Wore Out

As a technology, fuzzy control never died. As a movement promising a new kind of machine intelligence, it fell apart within half a decade, for reasons worth separating.

The word wore out first. By 1991 “fuzzy” was printed on so many Japanese appliances that it stopped discriminating between products, and marketers began stacking fresh qualifiers on top (“neuro-fuzzy” rice cookers). In colloquial Japanese the loanword drifted into meaning simply “vague,” and the boom’s own vocabulary became a mild joke. LIFE closed quietly when its six-year funding term ran out in the mid-1990s, leaving useful engineering results but nothing resembling the fuzzy computer some proponents had promised.

The intellectual claims deflated next. Control theorists observed that a Mamdani controller is, mathematically, a smooth interpolation between operating points, something achievable by gain scheduling or a well-tuned nonlinear PID loop; the fuzzy layer was often a convenient notation for encoding expert rules cheaply rather than a new capability, and where formal stability guarantees were required, classical methods kept the job. Mainstream AI, meanwhile, went statistical: through the 1990s, neural networks and machine learning made hand-written rule bases look like the bottleneck, not the breakthrough. The field absorbed the lesson; Jyh-Shing Roger Jang’s ANFIS (1993) trained fuzzy rule parameters with neural-network methods, and Zadeh himself folded fuzzy logic into the broader banner of “soft computing.”

What remained is the niche the technology had genuinely earned: encoding an experienced operator’s judgment into a cheap embedded controller for a process nobody can model exactly. Rice cookers, washing machines, air conditioners, camera stabilizers, and transmission controllers still run fuzzy rules today, mostly unadvertised; the Sendai trains never stopped; IEEE Transactions on Fuzzy Systems, founded in 1993, still publishes. Zadeh died in Berkeley on 6 September 2017, aged 96, and was buried with state honors in Baku. The hype died of overexposure; the controllers kept shipping.