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FUZZY LOGIC
Sanal R.P.
Published in Science India, Vol. 5, No.5 May 2002
So far as the laws of Mathematics refer to reality, they are not certain. And so far as they are certain, they do not refer to reality. -
Albert Einstein, geometry and Experience
Everything is a matter of degree.- Anonymous
Today's smart washing machines sense the quality and quantity of dirt in the clothes, the weight of the load and the type of fabric. It then adjusts the wash cycle, temperature and detergent level accordingly. A television set automatically adjusts the volume as the ambient noise in the room increases or decreases and alters it's brightness as the intensity of the light in the room changes. These are some of the applications that have benefited through the use of fuzzy logic. In 1965, Lofti A. Zadeh, a computer scientist at the University of California, proposed a mathematical way of looking at vagueness that a computer could deal with. He called the new approach Fuzzy Logic. Although Fuzzy Logic was invented in USA, the rapid growth of this technology has started from Japan, which has more than 3000 products ranging from rice cookers to subways, use fuzzy logic. Fuzzy has become a key word of marketing. Electronic articles without fuzzy component gradually turn out to be dead stock. Then what is this Fuzzy Logic ?
The basic idea is simple. The fuzzy principle states that everything is a matter of degree. It does not view the world through Boolean eyes. The operation of present day computers is based on simple yes/no logic, which is widely different from the information processing inherent to human thinking. Abstraction and thinking in analogies is only rendered possible by the flexibility of "human logic". Fuzzy logic has developed a mathematical model to implement this human logic in system solutions. So, Fuzzy logic is the superset of Boolean logic that has been extended to handle the concept of partial
 truth - truth values between "completely true" and "completely false".
Fuzzy things have vague boundaries with their opposites, with nothings. The more a thing resembles it's
opposite, the fuzzier it is. The Yin-yang symbol is the emblem of fuzziness. It stands for the world of opposites. Here yin equals or balances yang.
The precision of mathematics owes it's success in large part to the efforts of Aristotle and the philosophers who preceded him, in their efforts to devise a concise theory of logic and later mathematics, the so-called "Laws of Thought" were posited. One of these, the "Law of the Excluded Middle", states that every proposition must be True or False. Even when Parminedes proposed the first version of this law (around 400 B.C.) there were strong and immediate objections: for example, Heraclitus proposed that things could be simultaneously True and not True. It was Plato who laid the foundation for what would become Fuzzy logic, indicating that there was a third region (beyond True and False) where these opposites "tumbled out". Modern philosophers notably Hegel, Marx and Engels echoed his sentiments but it was Lukasiewicz who first proposed a systematic alternative to the bi-valued logic of Aristotle.
In the early 1900s, Lukasiewicz described a three-valued logic along with the mathematics to accompany it. The third value he proposed can best be translated as the term "possible", and he assigned it a numeric value between True and False. Eventually, he proposed an entire notion and axiomatic system from which he hoped to derive modern mathematics.
Later, he explored four-valued logics, five-valued logics and then declared that in principle that there was nothing to prevent the derivation of an infinite-valued logic. Lukasiewicz felt that three and infinite-valued logics were the most intriguing, but he ultimately settled on four-valued logics because it seemed to be the most easily adaptable to Artistotelian logic.
It was not relatively recently that the notion of an infinite-valued logic took hold. In 1965, Lotfi A. Zadeh published his seminal work "Fuzzy Sets" which described the mathematics of Fuzzy Set theory and by extension Fuzzy Logic.
Fuzzy Sets
The characteristic function of a crisp set is to assign a value of either 1 or 0 to each individual in the universal set, thereby discriminating between members and non-members of the crisp set under consideration. This function can be generalized such that the values such that the values assigned to the elements of the universal set fall within a specified range and indicate the membership grade of these elements in the set in question. A larger value denotes a higher degree of set membership. Such function is called membership function and the set defined by it is known as Fuzzy Set.
For example, if a doctor does not have a precise threshold in mind when evaluating whether a patient suffers from "high fever", How then does he work? The doctor compares the two prototypes- the perfect high fever patient and the perfect balanced temperature patient and evaluates where his patient ranks in between the two. How this can be modelled mathematically? Conventional set theory identifies each patient as either a member or non-member of the set of all patients with high fever. Following Figure 1 shows the set of patients with high fever (black area), where high fever is defined as temperature higher than 102oF .
In Figure 2, the shades of grey indicates the degree to which the body temperature belongs to the set of high fever. This degree is called the degree of membership, µHF (x) of element x e X to the set of high fever (HF). Here the degree of membership in a set becomes the degree of truth of a statement. For example, the expression "the patient has a high fever" would be true to the degree of 0.70 for a temperature of 105oF. The primary building block of any Fuzzy logic systems is the so-called linguistic variables. The Figure 3 shows membership functions for all the terms of linguistic variables plotted
together.
Figure 3: The linguistic variable "the body temperature" translates real temperature values into linguistic values
Figure 4. A typical multiple input-output fuzzy system
Fuzzy/Neural Technology
Fuzzy logic is the technology that mimics the human decision making process on a very high abstraction level of natural language. On the contrary, neural nets try to copy the way a human brain works on the lowest level. The combination of Fuzzy logic and neural net technology is called NeuroFuzzy. A combination of explicit knowledge representation of Fuzzy logic (with simple IF-THEN relations) with the learning power of neural nets results in NeuroFuzzy.
Fuzzy Chips
At present, simple everyday microprocessors are used and Fuzzy logic software were designed for specific applications. However AT & T's Bell Laboratory has designed the reality of dedicated Fuzzy logic processor chips. Also there is much talk of combining fuzzy logic with neural network computing in efforts to mimic the human brain responses. Prospects for a fuzzy future appear bright and exciting
Fuzzy products use both microprocessors that run fuzzy inference algorithms and senses that measure changing input conditions. Fuzzy chips are microprocessors designed to store and process fuzzy rules. In 1995 Masaki Tojai and Hiroyuki Watanabe, ten working at AT & T Bell Laboratory, built the first digital fuzzy chip which processed 16 simple rules in 12.5 micro seconds, a rate of 0.8 million fuzzy logical inferences per second.
Applications using Fuzzy Logic
Cement kiln control : this was the world's first industrial application of Fuzzy logic theory. Cement kiln control is an operation which requires that an operator monitor four internal states of a kiln, controls four sets of operations and dynamically manage forty or fifty "rules of thumb" about their interrelationships, all with the goal of controlling a highly complex set of chemical interactions.
Home appliances: Fuzzy logic finds application in a wide range of home electrical appliances such as refrigerators, vacuum cleaners, washing machines, rice cookers and air-conditioners. Matsushita's canister vacuum cleaner senses what type of floor surface it is cleaning, whether carpets or smooth surface. Coupled with a dust quality sensor, suction power of the system is varied accordingly, thereby saving energy.
Figure 5. Schematic diagram of a NeuroFuzzy Washing machine
Video Camera: In a video camera, Fuzzy Logic applications include automatic focusing, automatic exposure, automatic white balancing and image stabilization. The automatic technique utilizes approximate measure of sharpness in fuzzy rules to control the motor speed and improve the focusing quality.
Automobile: Fuzzy control system is implemented in advanced car's fuel injection, transmission and brake systems. The transmission system provides a smoother ride with a more human like shifting pattern and thereby reduces wear and tear.
In medical field, expert systems using fuzzy inference help doctors to diagnose diabetics and prostate cancer. Other applications include automatic road traffic signals for traffic management, fuzzy expert systems, elevators, anti-lock break systems, collision avoidance systems,
etc.
Common sense human thinking and judgment are the lures of Fuzzy logic and are only one of the whole range of emerging technologies which are at the forefront of the user friendly revolution. Many systems may be modeled, simulated and even replicated with the help of fuzzy systems, not the least of which is human reasoning itself.
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