Term Paper on Fuzzy Theory | Artificial Intelligence

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Term Paper on Fuzzy Theory

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Term Paper # 1. Invention of Fuzzy Theory:

Fuzzy theory began with a paper on ‘fuzzy sets’ which was published in the journal Information and Control (1965) by Prof. L. A. Zadeh of University of California, Berkeley, USA, in 1965. In his paper Prof. Zadeh named fuzzy sets as those sets whose boundary is not clear, such as “a set of beautiful women”, “a set of tall men” and “a set of big numbers”. He pointed out that fuzzy sets play an important role in human reasoning for pattern recognition, which is elementary capability, communication of semantics and especially abstraction. He expanded his assertion into a mathematical theory.

In the conventional set defined in mathematics it must be clear whether any element is ‘in’ or ‘out’. For example, a “set of men” or “a set of integers “. If it is not clear, it cannot be called a set and an object of mathematics. In this sense, the “fuzzy set theory” is a very new proposal.

Why did Prof. Zadeh start to think about such fuzzy sets? He is one of the pioneers and authorities on modern control theory. Modern control theory is very modern, strict and precise. Modeling a complicated system in the field of control requires a lot of effort. Although a computer may be used for the processing we have to specify the details of a process step by step. For example, the control system of robots or rockets becomes complicated.

The modeling is extensive and the execution takes a long time. High speed computers cannot be useful because of the high computation time in the period of initial stage of computer analysis. Using another example, a computer can be used to predict the direction of a typhoon, but it takes so long that typhoon will have already passed.

Although a computer seems to be almighty, it is inferior to the human brain in certain types of processing. For example, consider a game of chess or go, a Japanese game. But to examine all possible solutions, even a current super computer would take as long time as the life of earth.

The target, becomes still more complicated, even though the computers are becoming faster. Clearly, there would be deadlock one day. In this way, Prof. Zadeh came to the conclusion that the conventional approach which requires specifying every minor detail was wrong. Thus, a technology in which the whole system can be roughly defined that is a ‘fuzzy’ thus fuzzy theory was proposed.

Limitation of Rigorous Computer Modeling:

Prof. Zadeh proposed fuzzy theory for another important reason.

Even if the complicated system’s problem were solved the following two main issues remains:

1. It is impossible to list all the conditions for a system, because we cannot predict what conditions will have an effect on the system. In practicality, the conditions which seem to have influence are chosen and used for control that is it is assumed that the system is controlled only by the selected conditions. What if an accident happens and conditions change.

2. The system parameters, which are conditions assumed to be important for the system, must be determined by some values. However, it is often the case that these values are unable to be fixed accurately. For instance, assume that a system does work at a certain high temperature, which unfortunately is not known, may be cost to achieve this temperature, maybe very high or requisite environment cannot be maintained, which is not uncommon. A computer requires this high temperature value to be determined accurately to make calculations.

Although, there are many local parameters but computer bases its reason on theories and formulas for analysis. The results, however, turn out to be wrong since computer cannot control the temporarily developed conditions. That is ambiguity has been overlooked by the computer, which may arise despite best calculations.

Term Paper # 2. Basis for Inventing Fuzzy Theory:

This is based on a true story. One day Prof. Zadeh (born in Iran) debated with his German friend the issue of whose wife was more beautiful. Since beauty depends on individual perception, the degree of beauty varies. No matter which theory was used to prove it their opinions would never be in agreement. If a questionnaire about “she is beautiful”, is filed wherein one has to alternatively decide, it is not reasonable to always answer either ‘yes’ or ‘no’. Many may fill in an intermediate answer.

This is the story about when Prof. Zadeh thought of the idea of fuzzy theory which deals with intermediate answer. Even if the statistics show that she is beautiful would we always agree with the findings.

How to Define “Middle Age”?

We will try to explain fuzzy without mathematical notations. Let us take an example of the word “middle age”. This-phrase-is quite common in our daily life but we do not know the exact years or how long the middle age lasts.

The answer to this depends upon an individual or a group of individuals. Assume that we have the age ranging from 35 to 55 years as middle ager. Then a man (or a woman) who was 34 years and did not belong to middle age become, suddenly middle aged on the day of his birthday. Similarly, just after the birthday of 55 years old he would no longer remain middle aged.

If the time of the day is also significant, we should also take into account the time of their birth. There are few people who remember the time of their birth (may be to get their horoscope prepared). It is a matter of common experience that no one wants such preciseness for middle age no matter how exact he is. Obviously, it is not necessary to pursue this preciseness.

Middle age, for the present, has been defined to be between 35 and 55 years. These numerical representations are conventional assumptions only good for computers. Computers force us to use the mathematical model.

This means something which is actually continuous and ill-defined but has been forced to be well-defined. That is the analog information has been represented digitally in order to deal with computers. Fig. 13.1, shows the definition of middle age. x-axis shows ages or a period of time, the y-axis shows ‘yes’ (1) or ‘no’ (0).

One more interpretation is that it is a name of a set, a collection ‘of objects. For example, the set of “middle aged” is the numbers 35, 36, …, 55. Any intermediate number in this set belongs to “middle-aged”, all others do not belong. All standard theories can be explained by means of set theory. Binary logic is the principle of set theory.

Term Paper # 3. Meaning of Fuzzy Theory:

The definition of middle age given in Fig. 13.1, sounds a bit unnatural because the boundaries are loosely defined. For this sake, we shall consider the degree of middle age. For example, 35 years old is middle age with a degree of 0.6 and 36 years with a degree of 0.65, the degree changes slightly with the age.

Fig. 13.2, shows how this works, in this definition there are no extreme steps to middle age whereby one becomes old with just one year. Even 25 years is considered young though the degree is not high. This is an extended interpretation from set theory. Prof. Zadeh called this “fuzzy theory”.

Fuzzy means soft with no definite boundary. A fuzzy set is allowed to have an intermediate membership, which is neither 0 nor 1 as in the conventional set theory. A membership of 1 represents that an element (age) definitely belongs to a set (middle-aged), 0 represents that it does not. A membership of 0.5 shows that it belongs with a half degree.

In a way, a fuzzy set can include a standard set as a special case. This is the reason that a fuzzy set is a generalised form of a standard set. Hence, by having membership restricted within 0 or 1, any theory based on fuzzy theory provides the same result as that derived from conventional theory. In other words, the fuzzy theory needs to be developed so that it can be consistent with the conventional theory.

Now, we are ready to discuss the semantics and syntax of fuzzy theory. For example, “middle aged” is the label or name of a fuzzy set by which a behaviour of the contents is represented in Fig. 13.2.

If we consider the example conversation, the words we speak correspond to the label, what it means in the contents. In fuzzy theory, the label is said to be a fuzzy set, the figure IS a membership function of the fuzzy set. Generally, a fuzzy set is considered to have a label and a membership function together. If the Fig. 13.2, is extended over a set of ages (call it input space) from 1 to 100, we call the whole set (1, 2, …, 100) the universe of discourse. The fuzzy set of “middle-aged” is defined over the universe of discourse.

Term Paper # 4. Notations of Fuzzy Theory:

So what we come to understand is that a fuzzy set has soft boundaries and a standard set (crisp set) has crisp boundaries. The member function in Fig. 13.2, can be represented by a table, which is particularly useful for computers. For example, consider a dice whose values are 1, 2, …, 6.

 

The set of even numbers can be mathematically defined to be consisting of three numbers, 2, 4 and 6. Then let us consider a set of large numbers (which has a fuzzy sense) defined by 4, 5 and 6. It sounds somewhat strange. So, we should have a fuzzy set represent such intermediate value using a membership function. Table 13.1., gives the answer.

In the table the second column gives the even numbers and the third column shows the large numbers. For example, the number 5 belongs to the fuzzy set large numbers with a degree/membership function µ_B(x) of 0.8. A membership function is a curve which defines how each point in the input space is mapped to a membership value (degree) between 0 and 1.

A few important characteristics of the membership function are:

1. Fuzzy sets describe vague concepts (intelligent, beautiful, weekends).

2. A fuzzy set admits the possibility of partial membership in it (Friday is a sort of weekend).

3. The degree an object belongs to a fuzzy set is denoted by a membership value between 0 and 1 (Friday is a weekend to the degree 0.8 for example).

4. Usually the membership grades are defined in the closed interval (0, 1) and some possible membership grade is 1.

5. The input space is some times referred to as the universe discourse number.

6. If an element in a fuzzy set attains the maximum possible grade, then set is called normalized.

7. A membership function associated with a even fuzzy set maps an input value to its appropriate membership value.

Term Paper # 5. Operations in Fuzzy Theory:

Let us recall the example of the dice and choose our favourite number of all the numbers on a dice. Table 13.2., shows a fuzzy set of our favourite numbers. The degrees of membership are given, as we like (favourite) in second column. Next, let us think about disliked numbers. It is the opposite case of favourites. There is a complimentary relationship between them. Thus, the fuzzy set of disliked numbers can be determined by the fuzzy set of favourite numbers and shown in the third column of the table.

An operation of Negation on a fuzzy set is defined by the membership function of the set. We may note that in the extreme case when, a degree is given 1 or 0 a standard ‘yes’ and ‘no’ are in complement relationship, as in binary logic. We should give a new definition of these operations so that it can include the conventional theory of crisp set also.

Next, let us consider a number which we like and dislike. It does not make sense in conventional theory and only false (0) is assigned to this contradictory truth. But in fuzzy theory, however, this can make sense. The meaning depends on what we interpret as the semantics of and. We by now know that a preciseness of membership functions is not important, thus simple definition is sufficient to manipulate membership functions.

The simpler the better. In general, a smaller degree is chosen as a result of ‘and’. According to this definition, the degree of ‘likeness’ and ‘dislike ness’ are given in column four of table 13.2. We can get a answer! The result can be considered as a degree of consistency or a degree of how roughly defined it is.

After all, we see dice number ‘5’ satisfies a condition of a number which is preferred and not preferred, the best of all numbers. We sometimes say we like but dislike in our daily conversation, and it definitely makes sense. But in the world of logic, it does not make sense.

In the table 13.2, ‘V shows a completely favourite number and ‘4’ means a definitely disliked number. In these cases the fuzzy set of numbers which we like but dislike provides membership value of 0. In other words, it is completely contradictory. While fuzzy set representation provides the intermediate feeling in which we have no contradiction any more.

In fuzzy representation, compared to binary logic, we can feel free to represent our feeling rather real feelings. The law of excluded middle, of binary logic does net hold any more. This fuzzy logic is necessary where we can take intermediate values.

Conventional set theory and binary logic have three elementary operations:

i. Complement set (means negation),

ii. Intersection (means and),

iii. Union (means or).

In fuzzy theory, we have defined 1 minus the mean a negation and taken the smaller to mean and. Then, how shall we define or. Taking a maximum seems to be the best definition of or, the same as of binary logic (there are many ways of defining these operations).

(a) C — favourite numbers, (liked numbers)

(b) D — dislike numbers (not favourite),

(c) E — both favourite AND not favourite numbers.

(a) F — favourite OR not favourite,

(b) G — large AND favourite numbers,

(c) H — large OR not favourite numbers.

The fuzzy set of numbers which are favourite or not favourite in table 13.2, has a membership function shown in second column of table 13.3. In binary logic, since any truth is either yes or no such a statement is not always true, that is the degree of membership is always 1, which is called tautology. In fuzzy logic, as we have seen, there are meaningful values other than 1.

Let us take another example of ‘AND’ and ‘OR’. Table 13.3, (b)-(c) shows fuzzy sets G and H which are defined with fuzzy sets in Table 13.1., and Table 13.2 large (Table 13.1) AND favourite numbers (Table 13.2), and large table 13.1 not favourite number respectively. The operations of fuzzy sets AND, OR and NOT are illustrated in Fig. 13.3. Fuzzy set A and B, the intersection “A and B”, the union “A and B” and the negation “not A” are indicated.

The following properties of negation are worth nothing:

Negation of negation of A becomes A itself. Negation of ‘A, B’ becomes either ‘A’

These properties hold good in fuzzy logic as in binary logic. For example, fuzzy set E in table 13.2., “favourite and disliked number” is identical to the negation of fuzzy set F in table 13.3., “not favourite or not disliked numbers”, that is 1 minus membership of E is equal to membership of F. This makes sense because negation of what is favourite and non-favourite is equivalent to non- favourite or non-non-favourite (favourite).

Fuzzy sets and fuzzy operators are subjects and verbs of fuzzy logic. The if-then rule statements are used to formulate the conditional statements which comprise fuzzy logic.

Calculations Rules for Fuzzy Sets:

Simple calculating rules can be derived from the definitions of union, intersection and complement. From union we obtain the following arbitrary fuzzy sets A, B, C.

The same distribution laws which apply to ordinary sets also apply to fuzzy sets

Calculating rules for complements:

Not all calculating rules which apply to ordinary sets are automatically applied to fuzzy sets. Due to characteristics of fuzzy sets, the following possibilities exist.

Relationship between Fuzzy Sets:

Let us recall our old friends Arisha and Bobby who take a drive on a highway. Alice says, “Please drive car at a safe speed”. Bobby answers ‘sure’ and he drives the car at a safe speed as far as he believes.

Fuzziness is involved in our daily conversation such as the above. Fuzziness helps in making the communication easier. Let us suppose that Arisha expects about 90 km/hr to mean safe speed and Bobby thinks about 70 km/hr. In fuzzy theory, this situation can be illustrated in Fig. 13.4.

If Bobby drives at 70 km/hr, Arisha might request him, “speed up bit”. Both can be comfortable when they drive in the middle speed of the expectations. Even though their expected speeds are different since the expected speeds have some fuzziness, at the intersection of speed expectations of both can be met. If we use the computer to define a “safe speed” by a strict value, Arisha and Bobby will never reach a consensus.

Our communication cannot mature without such fuzziness or a span where both parties allow for each other. Language always has fuzziness. By representing fuzziness with fuzzy sets, we might be able to make computers to deal with the semantics of natural language.

Let us understand fuzzy sets in a closer way. Consider a simple relationship between the names of some diseases and their symptoms. Let diseases be a₁, a₂, a_3, and a₄ and b₁, b₂, b_3, and b₄ are certain symptoms.

Let us suppose that disease a₁ causes symptoms b₁ and b₃, disease a₂ has symptoms b₃ disease a₃ has symptoms b₂ and b₃ and disease a₄ has symptom b₄. This relationship between disease and symptoms can be expressed table 13.4, where 1 means the presence of a particular symptom and 0 means absence of symptom.

In the above table we have assumed a binary logic. In real life, symptoms may be clear or even unclear at times. Hence we should use an intermediate degree between 1 and 0 as elements of relation R, between the diseases and symptoms that is, we introduce fuzzy theory. Table 13.5. Gives membership functions. By having a universal sets of diseases U and symptoms V given by U = {a₁ a₂, a₃, a₄ } and V = {b₁ b₂, b₃ b₄,} respectively we call the table 13.5., a fuzzy relation R of U over V.

For the disease a₁ in table 13.5., we have fuzzy set B₁ over V, in Fig. 13.5. This is a fuzzy set of symptoms for the given disease a₁. Fig. 13.6., shows a fuzzy set A₂ over U, which shows possible disease with regard to symptom b₂.

Now let us see how the knowledge of these sets helps us to detect a disease, from the relationship table 13.5., given the symptoms (0.8, 1, 0.2, 0) respectively to the diseases a₁, a₂, a₃, a₄. Fig. 13.7., shows the fuzzy set, say B’ over V to depict the symptoms friend Chopra, has. From the table 13.5., of relationship between disease and symptoms, the disease Mr. Chopra is having can be found by means of fuzzy sets. How is this done?

First, we know symptom b₁ has a degree of 0.8 so we take minimum (0.8 and fuzzy set A₁) (table 13.6), which shows possible diseases given symptom b₁. The notation min (0.8, A₁) is used to mean choosing a smaller value. Simply speaking, this operation has fuzzy set A₁ truncated at 0.8. Similarly, rest of the results can be found. Finally, by choosing a maximum from possible values of disease a₁, we have the resulting value of 0.8. Similarly, the possibilities of diseases a₂, a₃ and a₄ are given by 0.2, 0.8 and 0.2. respectively.

Fig. 13.8., gives the resultant disease in the form of fuzzy set. Given fuzzy set B’ of symptoms over the set V and using the fuzzy relationship in table 13.6., we have a new fuzzy set to mean possibilities of disease related to B’. Conversely, we may determine a fuzzy set of symptoms from a given set of diseases using the relationship R.

Thus, if we know the fuzzy relations between two universal sets U and V, and if a fuzzy set X is given over U, we can manipulate fuzzy set Y over V and given fuzzy set Y over V, we can guess fuzzy set X over U.

Consistency of Fuzzy Sets:

In the middle-age fuzzy set the membership value of 30 years is 0.6. We can say that the consistency (or consensus) of 30 years old and middle-age is 0.6. In Fig. 13.3., we have two fuzzy sets, A and B. The intersection of A and B, ( shown by dotted line) computes the consistency of A and B, given by the highest membership value at the intersection.

The consistency is thus calculated by max-min operation. The reader should confirm that the fuzzy set of large and favourite dice numbers in (b) Table 13.3., is an intersection of two fuzzy sets large numbers and favourite numbers. It value 0.7 at dice 6 is given by their consistency. Similarly, the consistency of Arisha’s and Bobby’s safe speeds is 0.7 (80 km/hr).

The concept of consistency of fuzzy sets is one of the key ideas in fuzzy theory. Even if either/both of the two sets are crisp the max-min operation still works since 1 and 0 of crisp value are included in a special case of fuzzy sets. Thus, fuzzy theory is established, based on the fuzzy sets as are the mathematical theories based on set theory.

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