Mech.&Electri. Engineering Dept. of Zhengzhou University, Zhengzhou, 450052,PR China

Tel.:0049-421-218-2356/0049-179-1857778 Email: jie.wang@iat.uni-bremen.de (Germany)

As Visiting Scholar by Prof. Dr. A. Grдser

Institute of Automation, University of Bremen

Kufsteiner Str. NW1, 28359, Bremen, Germany

Abstract: Extension set theory and extension engineering methods are widely used in social and economical fields and have made a great amount of profit. There are also boundless perspectives for their applications on nature science and engineering. This paper presents the brief concepts about matter element, extension set, and matter element transformation. Some applying possibilities are discussed. A simple way of using extension set theory to control system syntheses is specially described.

Keywords: matter-element, extension-set, extension control

Important new mathematical concepts lead to normally great development in applied science and technique areas. For example, the classic control theories like linear and non-linear control systems, optimal and adaptive control theories are based on the classic sets theory named with Contor-so called Cantor set. The principal point of this kind of sets is that an element can either belong to a certain set or not. When the affiliation of an element is described quantitatively, then the number is either 1 or 0. As in 1965 L.A.Zahde^[1] published his paper titled with “Fuzzy Sets”, no anyone could expect that Fuzzy Sets theory is so widely used in many theory branches and even in our daily lives, such as in wash machines, microwave ovens, intelligent robots and automatic autos. It has gone about 20 years since Zadeh brought out the concept of Fuzzy Set to the time that it was paid extensively attention in theoretic and practical areas. Even to nowadays, 35years past, it is still not thoroughly paid to applications.

At very beginning of the year 1983, a Chinese researcher, Prof. Cai Wen, published his creative paper “The Extension Set and non-compatible Problems^[2]” and later a book named “Analysis of Matter Element^[3]”. This stimulated great researching enthusiasms inside of China on the fields of economics, sociology, business management, production design and market plan, and so on. Some applied accomplishments are also obtained^[4]. Because of lacking of systematic and analytic tools for using the theory, many difficulties were encountered when it was used to the fields of engineering especially the fields that need stark theoretic supports. On the other hand, due to the limitation of that the publications were written nearly only in Chinese and were short of many international outstanding scientists to take part in the researching works also resulted in the slower developing speed of Extension set theory and its applications. The writer of this paper doesn’t believe that with the more and more quickly advanced science and technology, the successful applications of extension set theory, when it is really a very good mathematical branch in respective, will be delayed another 40 years.

In this paper, the main concepts about extension set will be roughly described and the probability of using this theory to system control will be discussed.

As well known, when classic set(Cantor set) theory and classic mathematics based on Cantor set theory and represented by functional theories were extremely expanding, Cantor set faced two sides of challenges from inside and outside respectively. The inner challenge came mainly from the Russell’s Set Absurdity, it brought out so called “the third crisis of mathematics”. Studies on this absurdity yielded a series of positive results with deep thoughts. The external challenge lied in that the Cantor set can not describe the fuzzy characteristics of matters and things. This defect made great number of things those have fuzzy natures could not be treated with Cantor set theory. Fuzzy set theory, that was grounded by Zadeh in 1965 and can be used to solve fuzzy problems, is a breaking through for traditional concepts of Cantor set.

With affiliation functions, fuzzy set theory combines elements and sets together. The value of the affiliation function is a degree of belief that an element belongs to a pointed set. When all elements, their affiliation function values for a pointed set are not zero, could be enumerated or clearly divided into two different parts, the boundary of the set is determined. For fuzzy theories, it is not possible to transfer the elements with zero affiliation function values to being non-zero valued elements. The brought out of extension set theory changed the concept of classic set qualitatively. It made the set having new and anti-regular characteristics, such as dynamics, multi-layered, unified between quantities and qualities and variable inner structure of elements. According to Lebnizi(a German mathematician ), the essence of mathematics is not the objects it treats, but the method it adopts. For a given domain U and a given character P, the process of constructing a set is mainly that people recognize the relationship between the element and the character P and then classify the elements in accordance with certain principles. Recognizing and classifying the objects bases of different principles, Cantor set, fuzzy set and extension set could be achieved respectively. Cantor set theory describes the determined concepts of “yes(true)” or “no(false)” in the real world with formal logic. Fuzzy set studies the fuzzy relationship of affiliate degree of things those belong to a fuzzy set by using of fuzzy logic. And the extension set theory treats the variability of things by means of extension logic. It developed the qualitative description for “yes(true)” and “no(false)” to quantitative description and also to the variation procedures of “from yes to no” and “from no to yes”.

As is showed in table1, itself of the extension set theory is quite a great system and good expanded. But the practical method of applying this theory to engineering field is still under developed. Applying it to system control is said to be having very boundless perspective. Chinese scientists have made the word “extenics” to stand for the new subject of mathematics. Extenics has three typical merits as following:

First of all, it brought out the basic thought and method for transforming non-potable problems to potable problems. And second, with it the matter element theory was grounded and based on the extenability of matter element it has strongly provided tools to solve conflict problems. The last one is that it has constructed the extension set theory. To describe the extension domain and elements with critical values quantitatively and then the quantitative tool of extension set theory—Extension mathematics will be constructed.

In the objective world, all things are unified systems of qualities and quantities, so the variations of them in qualities and in quantities are tightly combined together and interact each other. For this reason, extenics introduces a concept of matter elements.

With an ordered three elements group to describe the basic unit of an object and it is named as a matter element. Herestands for the objective, the name of character of the

object and the quantity of about . These ,andare called three essential factors of a matter element. Here reflects the relationship between the quantity and quality of the an object. If an object has multiple characters or more than one of them are need to be listed out, then a multi-dimensional(for example, n-dimensions)matter element can be defined as:

.

When an object is dynamic or its dynamical character must be studied, then a dynamical matter element could be defined as .The dynamical matter element expresses the variations of the object with time.

The key point of treating conflict problems is to study the principal natures of matter elements. A researcher must be creative and jump out from customs and try to expand the thought when one is dealing with the conflict problems by using of matter element methods. The extenability includes characteristics of diffusion, conjugation, interaction, containing and extension. These were discussed in reference[5].

(1) Divergence natures of matter elements

Nature 1: One object may have multiple characters, it is simplified as single object multiple characters and noted as , and . Here the sign stands for extenable.

Example: Let =a piece of paper, and , , . Then we have a matter element

.

When a piece of paper is to be used, one can not only think about the normal usage of it, but also the other characters should be calculated, for example, some papers are overlapped and folded up to a box. Another example is that suppose one has been asked to arrange four equilateral triangles with six sticks of match. When an unsuitable condition is added to the equilateral triangles then the problem comes to be a conflict one: .

Nature 2: By a single character there may exist many objects, it is simplified as multiple objects single character and noted as , and . An example to use this nature is that an American purchaser named Mailce was pointed to buy fire-resistant boards for floor decoration but this kind of materials were sold out in the market. What he did was to substitute the boards with fire-resistant papers. The demand was satisfied and the cost was greatly reduced, too. To describe this question with matter element theory, so we have a normal matter element

and.

So long as and , the substitution is reasonable. Later Mailce developed this thought to “Value Engineering Techniques”, so we could say that the basis of “Value Engineering Techniques” happened to be the extenability of matter elements.

Nature 3: For different parameters an object about a character could has different quantities. It is called simply as single object single character with multiple quantities and could be formulated as, here. An example to support this nature is that by man made satellites more and more channels and signals need to be emitted but too many transmitters would make the satellite overloaded. To overcome this difficulty, a scheme of “divide time to fit multi-channel” was designed. To describe this question with matter element theory, it can be written as: , here

and 17 channels with only one transmitter is pre-supposed. There are still some other natures and these can be found in reference[5].

(2) Conjugation of matter elements

Just like the domain definition for complex numbers, matter elements include real and imaginary parts, too. For a given object , it could be written as , where the is the real part of and the imaginary part of it. A kind of product as a matter element has two sides of valuation. One is the product itself, the real part of it. The effective of the brand of the product and the reputation of the producer belong to the imaginary part of the matter element. There are many successful examples of using the imaginary parts of matter elements to civil decision makings and even in military directions[5,6].

(3) Interaction of matter elements

When there exist certain interactive dependencies between objects about the quantities of some characters, it will be called interactions of matter elements. For a given matter element , all the about objects or about characters interactive dependant matter elements are called a interactive dependant network. A varying of a quantity about a character of an object in a net may yields relative varying in quantity of the same character of another object in the same net. This character may be used in market planing. A kind of food for children, its price was 6 yuan/sack. For being more attractive to the customers, the company added small playthings with price of 0.5 yuan/piece in the sacks and the new price for a sack is 10 yuan/sack. And then the sales volume greatly increased and the company got a big mount of profits. This is an example of taking up transform using the interactive net nature of matter elements.

(4) Implication of matter elements

If and then certainly , this called implies , notified as . Here @ stands for existence. The relation between and is called Implication. References[6,7] discussed implicative systems of matter elements and their applications thoroughly.

(5) Extension of matter elements

The extension abilities of matter elements describe the combinations, decompositions and substitutions abilities between different matter elements. More deeply discussions about extension characters of matter elements see references[3],[5] and [8].

After introducing the concept of extenability to matter elements, transformations on objects, characters of objects and quantities of characters could be as specially designed calculations (operations) applied to matter elements. These calculations can handle the transforming of quantities and also the qualities of objects.

The extenabilities of matter elements pointed out the main thought methods of solving practical problems and the methods, policies and knacks to solute problems could be described with a series of matter element transformations. The are four basic transformations of matter elements: substitution, resolution, addition/subtraction and expansion and also four basic calculations for matter elements: and, or, multiplication and inversion^[5]. Some researchers studied the philosophical subjects of matter elements and extenics^[9,10].

Set is a mathematical method for thought of brains to recognize and/or to classify the objective things. This model of recognition and classification for objects must be diversified, for the real world is in variation. Cantor set describes the determined objects and fuzzy set handles the fuzzy characters of determined objects. For description of transformation of object with character to objectwith character , a new concept of set must be constructed and it will be the set basement for treating non-potable problems.

The concepts of extension set could be described with following two definitions. The first one may used as a quantified tool to treat qualitative and quantitative changes of objects and the second could be used to describe the variations of an object under a pointed transformation.

Definition 1: Suppose as a discussing domain and a reflection from to the real domain . Then is called an extension set on domain and the correlation function of . Here the is the relating degree of element about . The , and are called positive, negative domains and zero boundary of respectively. It is evident that if , then and .

Example: Suppose are all the processed machine spare parts and the up to standard diameter of the parts is =. For any with a diameter , build a function as , it represents the degree of a spare part belongs to qualified products. Then the extension set on is

The positive domain of is

and it represents all of the qualified parts. The negative domain of is

and it represents the parts that not belong to the set of qualified parts. The differences of values of showed the different degrees of that the parts are not qualified. The zero boundary of is

is the boundary of qualified and not qualified spare parts.

Definition 2: Suppose is a extension set on the domain and is the transform of . Notify , and . So the domains

and are respectively called as positive extenable domain and negative extenable domain of about transform . Here it is pointed that when .

This definition implies that it is possible to transform a part of the negative extenable domain to positive extenable domain by means of certain suitable transformations.

Classic set theory studies the accuracy of objects and fuzzy set theory is keen on the fuzzy characters of objects. Variability of objects is the main attention point of extension set theory. Their principal differences are that (1) Extension domain describes the process of varying of an element from not having the property to having the . This is a qualitative change. (2) Zero boundary describes the “dividing value” of qualitative variation of an element. (3)When the elements of a extension set are matter elements, this extension set will become a matter elements extension set.

The conflict degree of non-potable problems is measured by the qualifying degree of the objects about certain quantities. These quantities are sometimes numbered and sometimes not numbered. the non-numbered quantities must be numbered at first then can the relations between objects and quantities be studied on the real axis by means of extension set theory. The extension sets are depicted by incidence functions and therefore these function s must be established on real axes to make it possible of solving non-potable problems.

(1) Distance

In classic mathematics, the distance between two points and is defined as :

Now suppose is an arbitrary point on the real domain and an arbitrary interval on the real domain. So the formula

is defined as the distance between the point and the interval . When a point exists inside of an interval, then the classic mathematical distance between them is and according to the definition here, a negative value will be got. It is actually a measure of a degree for a point do not belongs to an interval.

(2) Position Value

In practice, the relations among a point and two other intervals must be also considered. For a electric machine with specified current of 20~50 amperes, actually when it is running under a current of a litter more than 50 (say 55amperes) or less than 20 (for example: 15 amperes) the specified current value will the machine not be damaged or unable to be started. These two interval form a nested interval. A distance between an arbitrary point and a nested interval will be called as a position value. Generally, if , and , then the position value of arbitrary point about the nested interval will be defined as:

It is evident that when there is no common terminal point between and , , there is this point then .The position value reflected the comprehensive distance between a point and a nested interval.

(3) Definition of Incidence Functions

Suppose , , and there is not any common terminal points of intervals. Let be an incidence function of point about the nested interval and . Following are some natures of it:

and ;

or ;

, and and ;

or ;

and and

.

(4) A Simple Incidence Function

Suppose , . The following function is called as a simple incidence function of point about the interval :

. It has also some natures as following:

At the point ,achieves its maximal value ;

and ;

and ;

or

(5) Elementary Incidence Function

When , , and there is not any common terminal points of intervals, the formula is called also as an elementary incidence function of point about the nested interval and . This function achieves its maximal value at the point of .

(6) Mass Degree Function

Mass degree function is a special example of elementary incidence function. When the interval is the maximal interval that inside of it the quality of an object will be kept not change, then the interval will be called “name keeping interval”. An incidence function about this kind of intervals is named as a mass degree function. The variations of its value accurately represents the process of changing from quantity variation to quality variation of an object. It has therefore a very important significance.

Example: By the process of making chemical dissolving liquid in a chemical laboratory, the best temperature range of water is between 40 ~70^oC. Let . Under the normal atmospheric pressure, the maximal temperature range in which water will keep its state is about 0~100^oC, namely, . Then the mass incidence function of this is:

It is clear that when takes the value 0 or 100, and when , . The value –1 explained that the temperature of water already exceeded the maximal interval of keeping the state of water. Sometimes this interval is named as “name keeping domain”^[5].

“Yes” and “no”, in the real world, could be transformed from each other and so do the problems with features of “potable” and “non-potable”. From studying on these transforming principles the matter element logic and inference regulations can be introduced. Reference [11] pointed that the extension logic is outside to the formal logic and dialectical logic. It includes matter element logic, things element logic and questions logic.

According to the studies on extenabilities of matter elements, a series of engineering methods based on extenics were already introduced to many applied fields. These methods are called extension methods. These mainly include: Matter elements extenability methods--Divergence tree, correlation net, divide and combine chain and Implication system; Matter transformation methods—Ground transform, composite transform and transform bridge^[12]; Evaluate methods-- Evaluation method for optimal degree and rhomb shape thought method.

A piece of information could be expressed by a matter element or a composition of matter elements. By using of the extenabilities of matter elements much more useful information can be derived from the original matter element. Generally, from a known information matter element a information matter element set

could be obtained through expanding the matter element . Extenable information describing all the possibilities of accomplishments through creative thought processes. Rhomb shape thought method^[13] is a kind of these creative thought process.

From the extenability of matter elements the extenability of systems can be further discussed. In reference[4] concepts about system matter element and construction matter element were built up. In this definition, the difference between real states and desired states of systems is treated as a non-potable problem. Their set models and relation models were founded and some basic regulations about extension systems were also introduced. By means of matter element transformation, the main steps for solving the non-potable problems of systems were designed.

The most frequently used method in dealing with decision problems is the transform bridge method. An imaginary example for using this method is how people in Hong Kong solved the so called “connections between right side drive and left side drive” ^[6]problem.

Artificial intelligence and control are newly opened areas of applications of extension engineering methods. And these areas are being said to have the most boundless prospects.

In using of extension methods to expert systems, the experiences of experts are organized in different surfaces and networks. By handling of non-potable problems, the searching range will be expanded step by step according to the degree of conflict of to be solved problem. As an example the chess playing process could be analyzed of being using an extension expert system. When more than one piece of chess are in danger, by normal optimal algorithms the smallest lose will be calculated and the piece with smallest lose will be abandoned. By means of an extension strategy, the already calculated smallest lose will be used to extend the set and search for a new chance among the other pieces those are momentarily not in danger but have an opportunity to start a new attack. But the price of this new attack must be even smaller than the calculated lose before. In reference[14] a technique extension expert system was built and be initially used to Chinese medical diagnose process.

By language processing and natural language comprehension, extension methods have had broad and deep applications^[16]. As well known, Chinese words are difficult to process and to input with a normal keyboard. So the two great tasks are to recognize Chinese handwritings and to input Chinese words without necessity of thinking always about the input regulations. Chinese Natural Science Foundation has supported a project of Chinese Handwriting Recognition in Qinghua university to develop a computer software named as OCR. In Kangxi Chinese Dictionary there are over 500,000 different words and by handwriting the possibility of variations is more boundless. An efficient method adopted is to combine the sentence meanings with words and by more difficult situations the more contents will be also used to help judging the meaning of a word. In actually the extension language algorithms are applied. Of course a great mount of priori learning and a big enough knowledge storage are necessary. It is similar with the Chinese input software “Zhi Neng Kuang Pin”(Crazily Input with Pinyin) developed by ChineseStarд Co. Ltd in Beijing. The searching and developing stuff there briefed the writer that they have used only E-Markov Chin algorithm helped with a learning of huge amount of language documents, that is approximately equal to the words quantity of a sum of 218 years of Peopleґs Daily. But they admitted also that there are still many bugs in the software and the rate of correct inputting achieved only 95%. They will apply extension logic and extension language set theory to enhance the rate of correct inputting.

Study on mechanic intelligent CAD systems based on extension knowledge model and decision model can not only represent the informational transform relations, but also make complicated dialectical inference and weaken the conflict factors of constraints. Reference [17] developed a set of intelligent CAD system theory and design method based on extension decision theory. A construction model of the system is build according to the demand of production designs.

For a response of a control process, speed , accuracy and stability are normally in the same position of important. But these three indexes are generally contradict each other and could not be satisfied on the same time. For solving this non-portable problem, the transient could be divided to several time pieces. In different time interval, the demands wanted are practically not the same. At very beginning it should be very quick and at last the accuracy will be most of all important. Divide the response process into some time intervals and that is correspond to divide the matter element of this transient

into some smaller matter elements and these matter elements each has a main criterion for a given time zone. This will guide the designer to have a thought that in each time zone the criterion could be satisfied with a certain specially synthesized controller. Then the variable structure control method yields. The writer has published a paper titled “A fast hybrid control algorithms to AC servo drive systems based on neural networks” and in this algorithm a very simple single neuron controller and a normal BP network controller are combined together for different control time zones. A non-disturbance weights transfer method is also presented.

By robots especially intelligent service robots control the process of decision making is very important. An electrically driven wheelchair may a good choice for people without ability of walking. But the demands of human on service robots include that the robot must walk around furniture and other obstacles. The idea model to satisfy this demand is a human like walking robot, as like as that developed in Honda Co. Ltd in project P3, which is named as “Humanoid Robot”. With much more degrees of freedom and complicated structure, a dynamic model of this kind of robot must be of very high order and seriously non-linear. Even though the errors about the modeling will result in more serious trajectory control errors. So a control strategy that is model free and high intelligent is necessary. There must be many non-portable problems in control. For example, it should “go” fast enough and keep dynamic stable by walking, it should avoid obstacles but keep the diameter of round going as small as possible, and so on. It is sure that extension decision and control strategies must be the best choices of all.

Extension set theory has been successfully applied in social and economical fields and yielded a great amount of profit. That is clear that most of such questions are difficult to build accurate and analytic models. The only good way is to treat without model. So it seems easier to use extension set theory. In engineering field, engineers are get used to mathematical descriptions for any questions. When there is not or is not possible to get an accurate and analytic model, people will be idealess. The writer believes that extension set theory and extension engineering methods are very good choice when model free questions are faced. Of course the theory and methods themselves must be continuously developed and a series of analytic theory for nature science and engineering are also very important.

The writer has personally contacted with Professor Cai Wen and discussed about how to apply extension set theory and extension engineering methods to practical problems, especially to artificial intelligence and control. He believes firmly that the successfully applications to these fields will not be very late.

The writer thanks Professor Dr. Axel Graeser for giving the writer a chance to briefly introduce the concepts to his colleges.

[1] L. A. Zadeh. Fuzzy sets. Information and Control, 8(3), June 1965, 338-353

[2] Cai Wen, The Extension Set and non-compatible Problems, Journal of Scientific Explore, No.1, 1983, 83-97

[3] Cai Wen, Analysis of Matter- Element, Guangzhou, Guangdong Higher Educational Publisher, 1987

[4] Cai Wen edited: From Analysis of Matter-Elements to Extension Sets, Beijing, Publisher of Scientific and Technologic Literature, 1995

[5] Cai Wen, Yang Chunyan and Lin Weichu, Applied Engineering Methods of Using Extension Set Theory, Bejing, Publisher of Science, 1997

[6] Cai Wen, Extension Set Theory and its Applications, Chinese Science Bulletin, No.7, 1999

[7] Zuo Jing, Gong Shaodong and Zuo Ning, Domain Limited Implicit Systems and Their Applications in Business Planing and Diagnosing, Systems Engineering—Theory&Practice, No.2,1988, 87-94

[8] David K., Cai Wen, Trating Non-Compotable Problem from Matter Element Analysis to Extenics, ACM SIGICE Bulletin, 22(3),1977, 1-9

[9] Li Heping, The Philosophical Thought of Extenics, Systems Engineering—Theory&Practice, No.1,1988, 118-120

[10]Zhang Yuxiang, The Philosophical Thought of Extenics and Matter Elements, Theory and Practice of System Engineering, No.1,1988, 113-115

[11] Tian Shuangliang, An Inference Model for Extenabilitis of Matter Elements Logic, Systems Engineering—Theory&Practice, N0.1, 1998, 121-123

[12] He Bin, Kang Zhirong, Transform Bridge Method, Systems Engineering—Theory&Practice, N0.2, 1998

[13] Cai Wen, Matter Element Model and Its Applications, Beijing, Publisher of Scientific and Technologic Literature, 1994

[14] Li Jian, Wang Xingyu, A Novel Expert System—Extension Expert System, Journal of Southern-East Chemistry College, 19(5),1993, 617-621

[15] Pan Dong, Jin Yihui, Exploration and Study on Extension Control, Control Theory and Applications, 3(13),1996, 305-311

[16] Chen Chaochun, Linfuhua, Research and Review on Extension Languages, Brain Truster and Matter Elements Analysis, No.3, 1986, 40—41

[17] Wang Wanliang, Zhao Yanwei, Exploration on Mechanic Intelligent CAD Systems Using Extension Decision Method, Systems Engineering—Theory&Practice, N0.2, 1998, 114-117