**Charles Elkan,
an assistant professor of computer science and engineering at the University of
California at San Diego, offers the following definition:**

"Fuzzy logic is a generalization of standard logic, in which a concept can possess a degree of truth anywhere between 0.0 and 1.0. Standard logic applies only to concepts that are completely true (having degree of truth 1.0) or completely false (having degree of truth 0.0). Fuzzy logic is supposed to be used for reasoning about inherently vague concepts, such as 'tallness.' For example, we might say that 'President Clinton is tall,' with degree of truth of 0.9.

"It turns out that the useful applications of fuzzy logic are not in high-level artificial intelligence but rather in lower-level machine control, especially in consumer products. Usually, fuzzy controllers are implemented as software running on standard microprocessors. A few special-purpose microprocessors have been built that do fuzzy operations directly in hardware, but even these use digital binary (0 or 1) signals at the lowest hardware level. There are some research prototypes of computer chips that use analog signals at the lowest level, but these chips simulate the operation of neurons rather than fuzzy logic."

**Shlomo
Zilberstein, an assistant
professor in the computer science department at the University
of Massachusetts
at Amherst, provides additional information and more fuzzy
analysis of the
U.S. president:**

"Fuzzy logic is a technique for representing and manipulating uncertain information. In the more traditional propositional logic, each fact or proposition, such as 'it will rain tomorrow,' must be either true or false. Yet much of the information that people use about the world involves some degree of uncertainty. Like probability theory, fuzzy logic attaches numeric values between 0 and 1 to each proposition in order to represent uncertainty. But whereas probability theory measures how likely the proposition is to be correct, fuzzy logic measures the degree to which the proposition is correct. For example, the proposition 'President Clinton is young' may have a degree of correctness 0.8.

"The important distinction between probabilistic information and fuzzy logic is that there is no uncertainty about the age of the president but rather about the degree to which he matches the category 'young.' Many terms, such as 'tall,' 'rich,' 'famous' or 'dark,' are valid only to a certain degree when applied to a particular individual or situation. Fuzzy logic tries to measure that degree and to allow computers to manipulate such information.

"Fuzzy logic was formulated by Lotfi Zadeh of the University of California at Berkeley in the mid-1960s, based on earlier work in the area of fuzzy set theory. Zadeh also formulated the notion of fuzzy control that allows a small set of 'intuitive rules' to be used in order to control the operation of electronic devices. In the 1980s fuzzy control became a huge industry in Japan and other countries where it was integrated into home appliances such as vacuum cleaners, microwave ovens and video cameras. Such appliances could adapt automatically to different conditions; for instance, a vacuum cleaner would apply more suction to an especially dirty area. One of the benefits of fuzzy control is that it can be easily implemented on a standard computer.

"Despite its commercial success, fuzzy logic remains a controversial idea within the artificial-intelligence community. Many researchers question the consistency and validity of the methods used to 'reason' with fuzzy logic.

**Jacoby Carter of the National Biological Service's National
Wetlands
Research Center in Lafayette, La., clarifies the difference
between fuzzy
and traditional logic; he also offers a more upbeat assessment
of the
potential of fuzzy logic for artificial intelligence (AI):**

"Traditional logic theory, sometimes called 'crisp logic,' uses three logic operations--AND, OR and NOT--and returns either a 0 or 1. Similarly, traditional set theory, or 'crisp set theory,' assigns to objects either membership or nonmembership in a class or group that has been assigned strict mathematical boundaries so that, for example, 80 degrees Fahrenheit is warm and 81 degrees F is hot. In fuzzy logic, the three operations AND, OR and NOT return a degree of membership that is a number between 0 and 1.

"Fuzzy set theory has been used in commercial applications of expert systems and control devices for trains and elevators; it has also been combined with neural nets to control the manufacture of semiconductors. By incorporating fuzzy logic and fuzzy sets in production systems, significant improvements have been gained in many AI systems. This approach has been particularly successful with ambiguous data sets or when the rules are imperfectly known."

**Heidar A.
Malki, an assistant
professor in the College of Technology at the University of
Houston,
provided further perspective on the likely applications of fuzzy
logic:**

"Increasingly, people in industry and academia are exploring the benefits of of fuzzy logic and its related technologies. Fuzzy logic can be used for situations in which conventional logic technologies are not effective, such as systems and devices that cannot be precisely described by mathematical models, those that have significant uncertainties or contradictory conditions, and linguistically controlled devices or systems. As Lotfi Zadeh once stated, fuzzy logic is not going to replace conventional logic (computers) or methodologies, rather it will supplement them in circumstances where conventional approaches fail to solve a problem effectively.

"In recent years, there has been a growing interest in fuzzy logic, both in industry and academia. Current applications include modeling, evaluation, optimization, decision making, control, diagnosis and information. In particular, fuzzy logic is best suited for control-systems fields. For instance, fuzzy logic has been applied in areas such as breakdown prediction of nuclear reactors in Europe, earthquake forecasting in China, and subway control in Japan.

"One prominent application of fuzzy logic is in the anti-lock braking system found in many modern automobiles. The control rules that describe an anti-lock braking system may consist of parameters such as the car's speed, the brake pressure, the brake temperature, the interval between applications of the brakes and the angle of the car's lateral motion to its forward motion. The range of values of these parameters are all continuous, open to interpretation by a design engineer. One such rule in an anti-lock braking system could be:

```
IF brake temperature is 'warm' AND speed is 'not very
fast,' then brake
pressure is 'slightly decreased.'
```

"The temperature might have a range of states such as cold, cool, warm and hot; the range of these linguistic terms can be precisely determined by defining membership functions by an expert.

"There are many consumer products that use fuzzy logic in their operation. There are also many fuzzy logic chips (processors) that are built to do special tasks without using conventional computers. The outlook for fuzzy logic is therefore very promising."

Not everybody can ignore the humorous potential in a concept such as fuzzy logic. Jim Diederich, a professor of mathematics at the University of California at Davis, is working on the applications of fuzzy logic in biological systems. He recently tried out fuzzy logic techniques on one specialized set of biological systems--his students--when he proposed the following rules for one of his courses

Special Topics in Mathematics Math 180-01

Fuzzy Sets, Numbers and Logic

**Course Information**

- A midterm will be given around mid term.
- The final will be given around final time.
- Homework will be assigned fairly regularly.
- The midterm and final each will normally count as a substantial part of the grade.
- The homework will not be insignificant in counting as part of the grade.
- An excellent final will result in a somewhat excellent grade.
- Solid work in two of the three areas, midterm, final and homework, will result in a solid grade.
- Good homework will offset poor exams somewhat.
- Your grade will be a fuzzified linguistic bureaucratic terminological value.
- If you don't understand this by the end of the quarter, your grade will reflect it.