Artificial Intelligence - Fuzzy Computing (1)

1. Fuzzy Theoretical Basis

1. Introduce what is fuzzy:

One seed certainly cannot form a pile, nor can two seeds... But people admit that 100 million seeds can definitely form a pile, so where is the limit? Is it possible to say that 325647 seeds are not a heap but 325648 seeds constitute a heap? ——E. Borel

In the above question, a bunch of definitions are vague~

2. Introducing Zadeh

Zadeh (Zadeh, LA; February 1921-September 6, 2017), an American automatic control expert, "the father of fuzzy sets", a member of the American Academy of Engineering Sciences, a foreign member of the Russian Academy of Natural Sciences, and an international European Union established in 1994 . One of the five founding academicians of the Sub-Academy of Sciences . Born in Baku , Soviet Union . Received Ph.D. in Electrical Engineering from Columbia University in 1949 . He was a professor in the Department of Electrical Engineering and Computer Science at the University of California, Berkeley . Recipient of the Institute of Electrical and Electronics Engineers (IEEE) Education Medal for pioneering work in the development of fuzzy set theory.

In 1965, he published a seminal paper entitled "Fuzzy Sets". The ideas of fuzzy sets and fuzzy logic have influenced scholars all over the world. According to Microsoft Academic, his work on fuzzy sets has been cited nearly 33,800 times.

Two, fuzzy

1. Fuzzy mathematics

The fuzzy English is Fuzzy, which has the meaning of "unclear" and "unclear boundaries". And mathematics is a very rigorous, very precise thing. Fuzzy mathematics is used to describe, study, and deal with the fuzzy features (that is, fuzzy concepts) of things. "Fuzzy" refers to its research object, and "mathematics" refers to its research method.

2. Vague concepts

In natural language, some concepts commonly to describe the characteristics of things are vague . For example, in the column of health status, fill in "good, relatively good, good, etc.", as for what kind of health is good and what kind of health is good, it is difficult to define exactly. Another example is to divide people into "young people, middle-aged people, and old people" according to their ages (grasp the main characteristics of things from a macro perspective and facilitate handling, and artificially blur them). For another example, in the control of water tank liquid level, temperature, etc., experienced operators will operate valves (open large, open small) according to the size of the controlled volume (high, too high, low).

3. Description of Fuzzy Concepts by General Mathematics

Taking age as an example, the traditional method is to define some domain values. Use y to represent age, y<40 is "young", 40<=y<60 is "middle-aged", and y>=60 is "old". This method is simple, but too absolute . In fact, people gradually progress from youth to middle age and then to old age as they grow older. There is no clear boundary between these concepts. The basis of traditional mathematics is set theory. The boundaries of these sets must be clear. An object must either belong to it or not belong to it. Traditional mathematics cannot describe and deal with this fuzzy concept without clear boundaries, so fuzzy mathematics is applied.

4. Application fields of fuzzy technology

Subway locomotives, robots, process control, fault diagnosis, traffic management, medical diagnosis, voice recognition, image processing, market forecasting and other fields.

3. Fuzzy sets and their operations

1. Fuzzy Sets

Given a universe of discourse U, any mapping μA from U to [0,1] closed interval

μA: U belongs to [0,1]

u belongs to μA(u)

Both determine a fuzzy subset A on U, called fuzzy set for short. μA is called the membership function of fuzzy set A (Membership Function). If the elements in the domain of discourse are represented by x, then μA(x) is called the degree of membership that x belongs to A.

2. Nature

The membership function reflects the degree to which elements in the domain of discourse belong to the set. When μA(x) is close to 1, it means that the degree of x belonging to A is high; when μA(x) is close to 0, it means that the degree of x belonging to A is low.

For example: using the fuzzy sets A, B, and C on the domain of discourse [1, 100] to represent "young, middle-aged, and old", the membership functions of A, B, and C μA(x), μB(x), μC(x )as the picture shows.

Explanation with fuzzy set is: 30-year-old youth is 0.75. 40-year-old is not too young (0.25), relatively close to middle age, but not too middle-aged (0.5), 50 years old is exactly Middle-aged (1), about to "old age".

3. Classification of fuzzy sets

(1) Taiwan set

The set As of the fuzzy set A is an ordinary set, which is composed of all u in the universe U satisfying μA(u)>0. Right now

(2) Single point fuzzy set

The set of fuzzy set A has only one element u0, and μA(u0)=1, then A is a single-point fuzzy set, expressed as:

(3) Convex fuzzy set

If A is a fuzzy subset with the real number R as the domain of discourse, its membership function is μA(x), if for any real number a<x<b, there is

The essence of a convex fuzzy set is that the membership function has a single peak characteristic.

Fourth, the representation of fuzzy sets

1. Discrete domain of discourse

(1) Zadeh notation

(2) Ordinal even representation

(3) Vector representation

2. Continuous domain of discourse

Zadeh notation is:

5. Basic operations of fuzzy sets

Assuming two fuzzy subsets A and B on the domain of universe U, the intersection, union and complement operations between them are defined as follows.

1. F Intersection