# fuzzy

**..introduction, or what is fuzzy thinking?**

Experts usually rely on common sense when they solve problems. They also use vague and ambiguous terms. For example, an expert might say, ‘Though the power transformer is slightly overloaded, I can keep this load for a while’. Other experts have no difficulties with understanding and interpreting this statement because they have the background to hearing problems described like this.

However, a knowledge engineer would have difficulties providing a computer with the same level of understanding. How can we represent expert knowledge that uses vague and ambiguous terms in a computer? Can it be done at all?

This chapter attempts to answer these questions by exploring the fuzzy set theory (or fuzzy logic). We review the philosophical ideas behind fuzzy logic, study its apparatus and then consider how fuzzy logic is used in fuzzy expert systems.

Let us begin with a trivial, but still basic and essential, statement: fuzzy logic is not logic that is fuzzy, but logic that is used to describe fuzziness. Fuzzy logic is the theory of fuzzy sets, sets that calibrate vagueness. Fuzzy logic is based on the idea that all things admit of degrees. Temperature, height, speed, distance, beauty – all come on a sliding scale. The motor is running really hot. Tom is a very tall guy. Electric cars are not very fast. High-performance drives require very rapid dynamics and precise regulation. Hobart is quite a short distance from Melbourne. Sydney is a beautiful city. Such a sliding scale often makes it impossible to distinguish members of a class from non-members. When does a hill become a mountain?

Boolean or conventional logic uses sharp distinctions. It forces us to draw lines between members of a class and non-members. It makes us draw lines in the sand. For instance, we may say, ‘The maximum range of an electric vehicle is short’, regarding a range of 300km or less as short, and a range greater than 300km as long. By this standard, any electric vehicle that can cover a distance of 301km (or 300km and 500m or even 300km and 1m) would be described as long-range. Similarly, we say Tom is tall because his height is 181 cm. If we drew a line at 180 cm, we would find that David, who is 179 cm, is small. Is David really a small man or have we just drawn an arbitrary line in the sand? Fuzzy logic makes it possible to avoid such absurdities. Fuzzy logic reflects how people think. It attempts to model our sense of words, our decision making and our common sense. As a result, it is leading to new, more human, intelligent systems.

Fuzzy, or multi-valued logic was introduced in the 1930s by Jan Lukasiewicz, a Polish logician and philosopher (Lukasiewicz, 1930). He studied the mathematical representation of fuzziness based on such terms as tall, old and hot. While classical logic operates with only two values 1 (true) and 0 (false), Lukasiewicz introduced logic that extended the range of truth values to all real numbers in the interval between 0 and 1. He used a number in this interval to represent the possibility that a given statement was true or false. For example, the possibility that a man 181cm tall is really tall might be set to a value of 0.86. It is likely that the man is tall. This work led to an inexact reasoning technique often called possibility theory.

Later, in 1937, Max Black, a philosopher, published a paper called ‘Vagueness: an exercise in logical analysis’ (Black, 1937). In this paper, he argued that a continuum implies degrees. Imagine, he said, a line of countless ‘chairs’. At one end is a Chippendale. Next to it is a near-Chippendale, in fact indistinguishable from the first item. Succeeding ‘chairs’ are less and less chair-like, until the line ends with a log. When does a chair become a log? The concept chair does not permit us to draw a clear line distinguishing chair from not-chair. Max Black also stated that if a continuum is discrete, a number can be allocated to each element. This number will indicate a degree. But the question is degree of what. Black used the number to show the percentage of people who would call an element in a line of ‘chairs’ a chair; in other words, he accepted vagueness as a matter of probability. However, Black’s most important contribution was in the paper’s appendix. Here he defined the first simple fuzzy set and outlined the basic ideas of fuzzy set operations. In 1965 Lotfi Zadeh, Professor and Head of the Electrical Engineering Department at the University of California at Berkeley, published his famous paper ‘Fuzzy sets’. In fact, Zadeh rediscovered fuzziness, identified and explored it, and promoted and fought for it.

Zadeh extended the work on possibility theory into a formal system of mathematical logic, and even more importantly, he introduced a new concept for applying natural language terms. This new logic for representing and manipulating fuzzy terms was called fuzzy logic, and Zadeh became the Master of fuzzy logic.

**..why fuzzy?**

As Zadeh said, the term is concrete, immediate and descriptive; we all know what it means. However, many people in the West were repelled by the word fuzzy, because it is usually used in a negative sense.Why fuzzy? As Zadeh said, the term is concrete, immediate and descriptive; we all know what it means. However, many people in the West were repelled by the word fuzzy,

because it is usually used in a negative sense.Why fuzzy? As Zadeh said, the term is concrete, immediate and descriptive; we all know what it means. However, many people in the West were repelled by the word fuzzy, because it is usually used in a negative sense.

**..why logic?**

Fuzziness rests on fuzzy set theory, and fuzzy logic is just a small part of that theory. However, Zadeh used the term fuzzy logic in a broader sense (Zadeh, 1965): Fuzzy logic is determined as a set of mathematical principles for knowledge representation based on degrees of membership rather than on crisp membership of classical binary logic.

Unlike two-valued Boolean logic, fuzzy logic is multi-valued. It deals with degrees of membership and degrees of truth. Fuzzy logic uses the continuum of logical values between 0 (completely false) and 1 (completely true). Instead of just black and white, it employs the spectrum of colours, accepting that things can be partly true and partly false at the same time. As can be seen in Classical binary logic now can be considered as a special case of multi-valued fuzzy logic.