1. Fuzzy Logic

2. WHAT IS FUZZY LOGIC? Definition of fuzzy Fuzzy – “not clear, distinct, or precise; blurred” Definition of fuzzy logic A form of knowledge representation suitable for notions that cannot be defined precisely, but which depend upon their contexts.

3. Fuzziness Vs. Vagueness “I will be back sometime” FuzzyVague “I will be back in a few minutes” Fuzzy 3

4. TRADITIONAL REPRESENTATION OF LOGIC SlowFast Speed = 0Speed = 1 bool speed; get the speed if ( speed == 0) { // speed is slow } else { // speed is fast }

5. FUZZY LOGIC REPRESENTATION Every problem must represent in terms of fuzzy sets. Slowest Fastest Slow Fast [ 0.0 – 0.25 ] [ 0.25 – 0.50 ] [ 0.50 – 0.75 ] [ 0.75 – 1.00 ]

6. FUZZY LOGIC REPRESENTATION CONT. SlowestFastest float speed; get the speed if ((speed >= 0.0)&&(speed < 0.25)) { // speed is slowest } else if ((speed >= 0.25)&&(speed < 0.5)) { // speed is slow } else if ((speed >= 0.5)&&(speed < 0.75)) { // speed is fast } else // speed >= 0.75 && speed < 1.0 { // speed is fastest } SlowFast

7. ORIGINS OF FUZZY LOGIC Lotfi Asker Zadeh ( 1965 ) First to publish ideas of fuzzy logic. Professor Toshire Terano ( 1972 ) Organized the world's first working group on fuzzy systems. F.L. Smidth & Co. ( 1980 ) First to market fuzzy expert systems.

8. TEMPERATURE CONTROLLER The problem Change the speed of a heater fan, based on the room temperature and humidity. A temperature control system has four settings Cold, Cool, Warm, and Hot Humidity can be defined by: Low, Medium, and High Using this we can define the fuzzy set.

9. Introduction Fuzzy Logic is used to provide mathematical rules and functions which permitted natural language queries. Fuzzy logic provides a means of calculating intermediate values between absolute true and absolute false with resulting values ranging between 0.0 and 1.0. With fuzzy logic, it is possible to calculate the degree to which an item is a member.

10. For example, if a person is.83 of tallness, they are “rather tall”. Fuzzy logic calculates the shades of gray between black/white and true/false. Fuzzy logic is a super set of conventional (or Boolean) logic and contains similarities and differences with Boolean logic. Fuzzy logic is similar to Boolean logic, in that Boolean logic results are returned by fuzzy logic operations when all fuzzy memberships are restricted to 0 and 1. Fuzzy logic differs from Boolean logic in that it is permissive of natural language queries and is more like human thinking; it is based on degrees of truth.

11. Fuzzy LogicBoolean Logic

12. Fuzzy logic may appear similar to probability and statistics as well. Although, fuzzy logic is different than probability even though the results appear similar. The probability statement, " There is a 70% chance that Bill is tall" supposes that Bill is either tall or he is not. There is a 70% chance that we know which set Bill belongs. The fuzzy logic statement, " Bill's degree of membership in the set of tall people is.80 " supposes that Bill is rather tall. The fuzzy logic answer determines not only the set which Bill belongs, but also to what degree he is a member.

13. Fuzzy logic deals with the degree of membership. Fuzzy logic has been applied in many areas; it is used in a variety of ways. Household appliances such as dishwashers and washing machines use fuzzy logic to determine the optimal amount of soap and the correct water pressure for dishes and clothes. Fuzzy logic is even used in self-focusing cameras. Expert systems, such as decision-support and meteorological systems, use fuzzy logic.

14. History Fuzzy Logic deals with those imprecise conditions about which a true/false value cannot be determined. Much of this has to do with the vagueness and ambiguity that can be found in everyday life. For example, the question: Is it HOT outside? These are often labelled as subjective responses, where no one answer is exact. Subjective responses are relative to an individual's experience and knowledge. Human beings are able to exert this higher level of abstraction during the thought process.

15. For this reason, Fuzzy Logic has been compared to the human decision making process. Conventional Logic (and computing systems for that matter) are by nature related to the Boolean Conditions (true/false). What Fuzzy Logic attempts to encompass is that area where a partial truth can be established. In fuzzy set theory, although it is still possible to have an exact yes/no answer as to set membership, elements can now be partial members in a set.

16. Fuzzy Sets Fuzzy logic is a superset of Boolean (conventional) logic that handles the concept of partial truth, which is truth values between "completely true" and "completely false". This section of the fuzzy logic describes: –Basic Definition of Fuzzy Set –Similarities and Differences of Fuzzy Sets with Traditional Set Theory –Examples Illustrating the Concepts of Fuzzy Sets –Logical Operation on Fuzzy Sets –Hedging

17. Fuzzy Set A fuzzy set is a set whose elements have degrees of membership. That is, a member of a set can be full member (100% membership status) or a partial member (eg. less than 100% membership and greater than 0% membership). To fully understand fuzzy sets, one must first understand traditional sets. A traditional or crisp set can formally be defined as the following: A subset U of a set S is a mapping from the elements of S to the elements of the set {0,1}. This is represented by the notation: U: S-> {0,1} The mapping is represented by one ordered pair for each element S where the first element is from the set S and the second element is from the set {0,1}. The value zero represents non-membership, while the value one represents membership. Essentially this says that an element of the set S is either a member or a non-member of the subset U. There are no partial members in traditional sets.

18. A fuzzy set is a set whose elements have degrees of membership. These can formally be defined as the following: A fuzzy subset F of a set S can be defined as a set of ordered pairs. The first element of the ordered pair is from the set S, and the second element from the ordered pair is from the interval [0,1]. The value zero is used to represent non- membership; the value one is used to represent complete membership, and the values in between are used to represent degrees of membership.

19. Example Consider a set of young people using fuzzy sets. In general, young people range from the age of 0 to 20. But, if we use this strict interval to define young people, then a person on his 20th birthday is still young (still a member of the set). But on the day after his 20th birthday, this person is now old (not a member of the young set). How can one remedy this? By RELAXING the boundary between the strict separation of young and old. This separation can easily be relaxed by considering the boundary between young and old as "fuzzy". The figure below graphically illustrates a fuzzy set of young and old people.

20. Notice in the figure that people whose ages are >= zero and <= 20 are complete members of the young set (that is, they have a membership value of one). Also note that people whose ages are > 20 and < 30 are partial members of the young set. For example, a person who is 25 would be young to the degree of 0.5. Finally people whose ages are >= 30 are non-members of the young set.

21. Logical Operations on Fuzzy Sets Negation Intersection Union

22. Negation the red line is a fuzzy set. To negate this fuzzy set, subtract the membership value in the fuzzy set from 1. For example, the membership value at 5 is one. In the negation, the membership value at 5 would be (1- 1=0) and if the membership value is 0.4 the membership value would be (1-0.4=0.6).

23. Intersection In this figure, the red and green lines are fuzzy sets. To find the intersection of these sets take the minimum of the two membership values at each point on the x-axis. For example, in the figure the red fuzzy set has a membership of ZERO when x = 4 and the green fuzzy set has a membership of ONE when x = 4. The intersection would have a membership value of ZERO when x = 4 because the minimum of zero and one is zero.

25. Union To find the union of these sets take the maximum of the two membership values at each point on the x-axis. For example, in the figure the red fuzzy set has a membership of ZERO when x = 4 and the green fuzzy set has a membership of ONE when x = 4. The union would have a membership value of ONE when x = 4 because the maximum of zero and one is One.

27. Limitation Fuzzy logic cannot be used for unsolvable problems. An obvious drawback to fuzzy logic is that it's not always accurate. The results are perceived as a guess, so it may not be as widely trusted as an answer from classical logic. Certainly, though, some chances need to be taken. How else can dressmakers succeed in business by assuming the average height for women is 5'6"? Fuzzy logic can be easily confused with probability theory, and the terms used interchangeably. While they are similar concepts, they do not say the same things. Probability is the likelihood that something is true. Fuzzy logic is the degree to which something is true (or within a membership set).

28. Classical logicians argue that fuzzy logic is unnecessary. Anything that fuzzy logic is used for can be easily explained using classic logic. For example, True and False are discrete. Fuzzy logic claims that there can be a gray area between true and false. Fuzzy logic has traditionally low respectability. That is probably its biggest problem. While fuzzy logic may be the superset of all logic, people don't believe it. Classical logic is much easier to agree with because it delivers precision.

29. References http://www.dementia.org/~julied/logic /index.html