Background of fuzzy math? And its related applications?

Modern mathematics is based on set theory. The importance of set theory, on one side, lies in the fact that it extends the abstraction power of mathematics to the depths of human cognitive processes. A set of objects identifies a set of properties, and one can state the concept (connotation) by stating the properties, or one can state it by specifying the objects. The totality of those objects that conform to the concept is called the denotation of the concept, and the denotation is in fact the set. In this sense, the set can express the concept, and the set theory of relations and operations can express the judgment and reasoning, all the reality of the theoretical system may be included in the mathematical framework of the set description.

However, the development of mathematics is also staged. Classical set theory can only limit its expressive power to those concepts and things that have a clear extension, and it clearly limits: each set must be composed of explicit elements, and the affiliation of the elements to the set must be clear, never ambiguous. For those concepts and things whose extension is not clear, classical set theory is not to reflect for the time being, belongs to the category to be developed.

Over a long period of time, exact and stochastic mathematics in describing the laws of motion of many things in the natural world, obtaining significant results. However, there is still a large number of fuzzy phenomena prevailing in the objective world. People used to avoid it, but, due to the increasing complexity of the systems confronted by modern science and technology, ambiguity always appears along with complexity.

Disciplines, especially the humanities, social sciences and other "soft sciences" of the mathematical, quantitative tendency of the fuzzy mathematical treatment of the problem pushed to the center. More importantly, with the rapid development of electronic computers, cybernetics, systems science, to make the computer can be like the human brain has the ability to recognize complex things, it is necessary to study and deal with fuzzy.

We study the behavior of human systems, or deal with complex systems comparable to the behavior of human systems, such as aerospace systems, human brain systems, social systems, etc., there are many parameters and variables, a variety of factors are intertwined with each other, the system is very complex, and its ambiguity is also very obvious. From the cognitive aspect, ambiguity refers to the uncertainty of the conceptual extension, which results in the uncertainty of judgment.

In daily life, often encounter many fuzzy things, there is no clear quantitative boundaries, to use some fuzzy words and phrases to describe, describe. For example, relatively young, tall, big fat, good, beautiful, good, hot, far ....... These concepts are not to be expressed simply as yes, no or numbers. There is often a lot of ambiguity in people's work experience as well. For example, to determine whether a furnace of molten steel has been refined, in addition to knowing precise information such as the temperature, composition ratio and smelting time of the molten steel, it is also necessary to refer to fuzzy information such as the color of the molten steel, the boiling condition, and so on. Therefore, fuzzy mathematics is needed in addition to the computational mathematics involving errors that have been available for a long time.

Compared with computers, generally speaking, the human brain has the ability to deal with fuzzy information, good at judging and dealing with fuzzy phenomena. However, computers are poor at recognizing fuzzy phenomena. In order to improve the ability of computers to recognize fuzzy phenomena, it is necessary to design the fuzzy language commonly used by people into instructions and procedures that can be accepted by machines so that the machines can be as concise and flexible as the human brain to make the corresponding judgments, thus improving the efficiency of automatic recognition and control of fuzzy phenomena. In this way, there is a need to find a mathematical tool for describing and processing fuzzy information, which pushes mathematicians to study fuzzy mathematics in depth. Therefore, the emergence of fuzzy mathematics is the inevitability of the development of science and technology and mathematics. The calculation speed and storage capacity of modern computers have reached an almost unparalleled level, which can not only solve complex mathematical problems, but also participate in the control of space shuttles and so on. Since the computer has such power, why is it sometimes inferior to the human brain in judgment and reasoning? Professor Zadeh (Zadeh) of the University of California, the United States of America, carefully studied this issue, so that she in the scientific research work often circled around with the "human brain thinking", "large systems" and "computer" contradictions. In 1965, he published his paper "Fuzzy Set Theory", in which he used the concept of "affiliation function" to describe intermediate transitions in the differences of phenomena, thus breaking through the absolute relationship of belonging or not belonging in classical set theory. This pioneering work of Prof. Zadeh marks the birth of the discipline of fuzzy mathematics.

The research content of fuzzy mathematics has the following three main aspects:

First, the study of the theory of fuzzy mathematics, and its relationship with exact and random mathematics.

Chad based his work on the set theory of exact mathematics and considered modifying and generalizing the concept of set in mathematics. He proposed the use of "fuzzy sets" as a mathematical model for representing fuzzy things. And in the "fuzzy set" and gradually establish the operation, transformation law, to carry out the relevant theoretical research, it is possible to construct a study of the real world in a large number of fuzzy mathematical basis, able to seem quite complex fuzzy system quantitative description and processing of mathematical methods.

In the fuzzy set, a given range of elements to its affiliation is not necessarily only "yes" or "no" two cases, but with a real number between 0 and 1 to indicate the degree of affiliation, there is also an intermediate transition state. For example, "the elderly" is a fuzzy concept, 70 years old certainly belongs to the elderly, it is subordinate to the degree of 1, 40 years old is certainly not the elderly, it is subordinate to the degree of 0, in accordance with the formula given by Chad, 55 years old belongs to the "old" degree of 0.5, that is, "half old", "half old". According to the formula given by Chad, 55 years old belongs to the "old" degree of 0.5, i.e. "half old", and 60 years old belongs to the "old" degree of 0.8. Chad believes that specifying the set of affiliation of each element is equivalent to specifying a set. When affiliated with a value between 0 and 1, it is a fuzzy set.

Second, the study of fuzzy linguistics and fuzzy logic.

Human natural language is fuzzy, and people often accept fuzzy language and fuzzy information, and can make correct identification and judgment.

In order to realize the use of natural language with the computer to have a direct dialogue, it is necessary to refine human language and thinking process into a mathematical model, in order to give the computer input instructions to establish the appropriate fuzzy mathematical model, which is the key to the use of mathematical methods. Chad used the fuzzy set theory to establish a mathematical model of fuzzy language to quantify and formalize human language.

If we set the value of the subordination function of a grammatical standard sentence to 1, then other sentences that are close to the meaning of the word and that express similar ideas can be characterized by a continuous number between 0 and 1 to indicate the degree of subordination to the "correct sentence". In this way, the fuzzy language is quantitatively described, and a set of arithmetic and transformation rules are defined. At present, the fuzzy language is still very immature, linguists are studying in depth.

People's thinking activities often require the concept of certainty and precision, using formal logic of the row in the law, that is: either true or false, and then judgment and reasoning, to reach a conclusion. Existing computers are based on binary logic, which plays a great role in dealing with the certainty of objective things, but does not have the ability to deal with the uncertainty or ambiguity of things and concepts.

In order for computers to be able to simulate the characteristics of the advanced intelligence of the human brain, it is necessary to transfer computers to the basis of multi-valued logic and study fuzzy logic. At present, fuzzy logic is still very immature and requires continued research.

Third, the study of the application of fuzzy mathematics.

Fuzzy mathematics is to the uncertainty of things as its object of study. The emergence of fuzzy sets of mathematics to adapt to the needs of describing complex things, Chad's merit lies in the theory of fuzzy sets to find a solution to the object of ambiguity to be exact, so that the study of deterministic objects of mathematics and uncertainty in the object of mathematics to communicate, the past precise mathematics, random mathematics to feel the inadequacies of the description, can be made up. In fuzzy mathematics, there are fuzzy topology, fuzzy group theory, fuzzy graph theory, fuzzy probability, fuzzy linguistics, fuzzy logic and other branches. Fuzzy mathematics is an emerging discipline, which has been initially applied to various aspects of fuzzy control, fuzzy identification, fuzzy cluster analysis, fuzzy decision making, fuzzy judging, system theory, information retrieval, medicine, biology and so on. There have been specific research results in meteorology, structural mechanics, control, psychology and so on. However, the most important application area of fuzzy mathematics is computer functions, and many people believe that it is closely related to the development of a new generation of computers.

At present, the world's developed countries are actively researching and trial production of fuzzy computers with intelligence, in 1986, Japan's Dr. Yamakawa Lie first trial production of successful fuzzy inference machine, which inference speed is 10 million times / sec. 1988, China's Prof. Wang Peizhuang guidance of a number of PhDs have also developed a successful fuzzy inference machine --- discrete component prototype, which is the first time in the world to develop a fuzzy inference machine. -Discrete component prototype, its inference speed is 15 million times / sec. This shows that China has taken an important step in breaking through the fuzzy information processing difficulties.

Fuzzy mathematics is still far from mature, there are still different opinions and views on it, to be tested in practice.