Fuzzy Logic is not Fuzzy

“Fuzzy Logic is not fuzzy”

This is how Zadeh started his paper (Is there a need for fuzzy logic?) [2] that he wrote in 2008 in response to the attack of some mathematicians on the Fuzzy Logic, which was established by him in 1965 [1]. In their view, it is fuzzy, imprecise, and has no actual benefits.

Therefore, he started his response with this sentence, explaining that the fuzzy logic in not fuzzy by itself, but rather a very precise tool to represent the fuzzy concepts that we couldn’t represent well using the conventional logic. He also showed some of the applications that utilized this logic to solve some realistic problems.

We saw -in the previous article- the reasons for why the traditional logic was unable to represent many life concepts realistically, and we showed how important it is to use the new fuzzy logic, which is more flexible with the fuzzy concepts and data.

The problem emerged from having some information and statements that we were not able to judge whether they were “true” or “false”, yet we needed other options. For example, as we said: the statement “Zaid is tall” is fuzzy, since we cannot call it as “true” or “false” firmly for the reasons discussed in the previous article.

Now, we are going to deal with this statement differently according to the new logic.
We are used to deal with it as if there is -for instance- a set of “tall” people, and a set of “short” ones. So, when we look at someone, we decide where he belongs. If he is tall, we will put him in the set of “tall” people. On the other hand, if he is short, then he will belong to the “short” set, and then he won’t be in the “tall” set.

However, the concept of “height” -as we showed earlier- is a fuzzy concept, which makes it unrealistic to classify people like that. Therefore, instead of using the usual sets we are used to, we are going to use a new type of sets, that is the “Fuzzy Sets”.

For instance, we are going to deal with the set of “tall” people in a different way, where we will measure the height of Zaid and decide his “degree of membership” in the set of “tall” people. This means that he won’t completely belong to it, neither will he “not belong” at all, but there will be another option in between according -for instance- to his height and to the average of the heights in his area.

However, if we had the tallest man in the world, and we wanted to specify his degree of membership in this set, he would definitely belong completely. This guy would have a degree of membership in the set of “tall” people that equals 1 (i.e. 100%). On the other hand, the shortest man in the world does not belong to this set at all, so his degree of membership equals zero. Any other human has a degree of membership in the set as a value between zero and one. This means that we now have a fuzzy set.

Similarly, the set of “short” people can be dealt with as a fuzzy set.

This is for general sets, but what about the mathematical sets? For instance, what about the set of the numbers that are “slightly greater than 5”, or the set of numbers that are “about 7”? In fact, things are not that different for them.

Do you remember the fuzzy number “about 7” that we represented on the real line in the previous article using a foggy cloud surrounding it, which was very thick at exactly 7, and faded away as we move far from it in both directions?

test_1.png

Zadeh represented the thickness of the fog as a number between zero and one, where the thickness at exactly 7 equals 1, and decreases gradually as we move until it reaches zero if we reached a point that is very far from 7.

We may see this “fuzzy number” as a fuzzy set, not as the regular numbers that we are used to.

This means that the number “about 7” is a subset of the set of real numbers, that contains all the numbers that are close to 7. However, these numbers do not belong to this set with the same degree of membership, but the degree of membership of each number equals the thickness of the fog at the corresponding point.

Here, the number 7 belongs to the set of the numbers that are “about 7” with a degree of membership that equals 1, while the number one million is not “about 7” at all, so its degree of membership is 0.
What about the number 8? It is kind of “about 7”, so its degree of membership can be 0.8 for instance.
The number 9 is also close to 7, but it is further than 8, so its degree of membership in the set of the numbers that are “about 7” is less, and it can be 0.7 for example.
Like that, the degree of membership increases as we get closer to 7, and decreases as we go further.

But how can we represent this mathematically? How do I attach each number with its degree of membership in some fuzzy set?

Zadeh [1] represented numbers like “about 7” through a function \(\mu\), which’s domain is the set of real numbers \(\mathbb{R}\), and its range is the closed interval [0,1].

\(\mu : \mathbb{R} \rightarrow [0,1]\)

This function relates each real number on the real line with a number between zero and one, which indicated how close it is to 7.
For instance, \(\mu (7)=1\) since 7 is exactly equal to 7, \(\mu(7,000,000)=0\) since the entry is very far from 7, and \(\mu (10)=0.65\) since it is kind of far from 7, but it is closer than many other numbers, and so on…
The function \(\mu\) is called the membership function of the fuzzy number “about 7”, and we call \(\mu (x)\) the degree of membership of the number \(x \) in the fuzzy set. If we try to sketch this function, we will get a figure like:

2.png

In fact, fuzzy numbers were studied widely, and there is a special interest in some fuzzy numbers that have certain shapes for their membership functions. For example, there is the Triangular fuzzy number, the Trapezoidal fuzzy number, the s-shaped fuzzy number and the Gaussian fuzzy number.

3.png

Let’s go back to the first claim, that the fuzzy logic is a generalization of the usual logic. As a result, fuzzy sets generalize the usual sets (that are now called “crisp sets”). Then we must be able to deal with the crisp (unfuzzy) sets as a special case of the fuzzy sets, but how?

Let’s take, as an example, the set of the numbers in the interval [5,8], which is a crisp set. Can we find a membership function for it?

It is very easy, since this set is a fuzzy set in a special case. We can simply give a complete degree of membership (i.e. 1) for the numbers inside the given interval, and not give any degree of membership (i.e. 0) for the other numbers. In other words:

4.png

which gives when sketched:

5.png

What if the set was not connected? We will deal with it exactly the same.

As an example, take the set \(A=[1,2] \cup \{ 3 \} \cup [4,5]\). We can represent this set through the membership function:

6.png

which gives the figure:

7.png

This shows that it is really possible to represent crisp sets as special cases of the fuzzy sets, and this shows that fuzziness is truly a generalization of the conventional logic.

In conclusion, we find that the fuzzy logic is a tool that helps us to represent the fuzzy numbers in a 100% true mathematical form. This is due to that the fuzzy number is simply a function, which is constructed according to the aforementioned fuzzy philosophy. This is why Zadeh said: “Fuzzy logic is not fuzzy”.

Fuzziness is a revolution in the world of applied mathematics, and the evidence for that is the great number of the pure and applied researches that are related and depend on the fuzzy logic and fuzzy sets. However, it is just like anything new, it was not easily accepted in the mathematical society, but it will soon prevail, and it might enter schools within few years.

A question may arise in one’s mind: It seems that this is the same as probability and randomness! Don’t the numbers between zero and one represent the probability of the number being close to 7 in the given example?

What do you think? Share your thoughts with us in the comments, and follow us to find out the reasons for why people get confused about “fuzziness” and “randomness” in the next article. What are the similarities and the differences between them?

 

Marwa Tuffaha

marwat.math@gmail.com

 

References:

1- Zadeh, L.A., 1965. Fuzzy sets. Information and control, 8(3), pp.338-353.

2- Zadeh, L.A., 2008. Is there a need for fuzzy logic?. Information sciences, 178(13), pp.2751-2779.

 

إضافة تعليق جديد