1. Fuzzy Logic 1

2. Introduction Form of multivalued logic Deals reasoning that is approximate rather than precise The fuzzy logic variables may have a membership value of not only 0 or 1 – that is, the degree of truth of a statement can range between 0 and 1 and is not constrained to the two truth values of classic propositional logic. 2

3. Introduction Fuzzy logic has been applied to many fields, from control theory to artificial intelligence it still remains controversial among most statisticians, who prefer Bayesian logic, and some control engineers, who prefer traditional two-valued logic. 3

4. Fuzzy Logic A logic based on the two truth values True and False is sometimes inadequate when describing human reasoning. Fuzzy logic uses the whole interval between 0 (False) and 1 (True) to describe human reasoning. A system of logic developed for representing conditions that deals with degrees of membership and degrees of truth. The concept was introduced by Lotfi Zadeh in 1965.

5. Degrees of truth let a 100 ml glass contain 30 ml of water. Then we may consider two concepts: Empty and Full. The meaning of each of them can be represented by a certain fuzzy set. Then one might define the glass as being 0.7 empty and 0.3 full. The concept of emptiness would be subjective and thus would depend on the observer or designer. 5

6. An image that describe fuzzy logic 6

7. A point on that scale has three "truth values" — one for each of the three functions. Since the red arrow points to zero, this temperature may be interpreted as "not hot". The orange arrow (pointing at 0.2) may describe it as "slightly warm" and the blue arrow (pointing at 0.8) "fairly cold". 7

8. Fuzzy Rules fuzzy logic usually uses IF-THEN rules Rules are usually expressed in the form: IF variable IS property THEN action For example, a simple temperature regulator that uses a fan might look like this: – IF temperature IS very cold THEN stop fan – IF temperature IS cold THEN turn down fan – IF temperature IS normal THEN maintain level – IF temperature IS hot THEN speed up fan 8

9. Fuzzy Rules There is no "ELSE" – all of the rules are evaluated, because the temperature might be "cold" and "normal" at the same time to different degrees. The AND, OR, and NOT operators of boolean logic exist in fuzzy logic, usually defined as the minimum, maximum, and complement when they are defined this way, they are called the Zadeh operators 9

10. Zadeh Operators NOT x = (1 - truth(x)) x AND y = minimum(truth(x), truth(y)) x OR y = maximum(truth(x), truth(y)) 10

11. Hedges There are also other operators, more linguistic in nature, called hedges that can be applied. These are generally adverbs such as "very", or "somewhat" 11

12. Fuzzy Logic Applications Air conditioning Washing Machines (LG is the pioneer) Mono-rails (first used in Tokyo) Digital image processing (specially in medical imaging) Elevators (in case of power failure) Rice cookers Video game engines (disperse intelligence in prince of Persia) Special effects (swarm intelligence in Batman Begins, Terminator Salvation, The Lord of the Rings) 12

13. Objections against Fuzzy Logic The concept of "coldness" cannot be expressed in an equation, because although temperature is a quantity, "coldness" is not people have an idea of what "cold" is, and agree that there is no sharp cutoff between "cold" and "not cold" where something is "cold" at N degrees but "not cold" at N+1 degrees — a concept classical logic cannot easily handle 13

14. Objections against Fuzzy Logic The result has no set answer so it is believed to be a 'fuzzy' answer. Fuzzy logic simply provides a mathematical model of the vagueness which is manifested in the above example. 14

15. A new way to represent probabilistic logic? fuzzy set theory uses the concept of fuzzy set membership (i.e., how much a variable is in a set) probability theory uses the concept of subjective probability (i.e., how probable do I think that a variable is in a set). 15

16. Example 1: Classifying Houses Problem. A realtor wants to classify the houses he offers to his clients. One indicator of comfort of these houses is the number of bedrooms in them. Let the available types of houses be represented by the following set. U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} 16

17. Example 1: Classifying Houses 17 The houses in this set are described by u number of bedrooms in a house. The realtor wants to describe a "comfortable house for a 4- person family," using a fuzzy set. Solution. The fuzzy set "comfortable type of house for a 4-person family" may be described using a fuzzy set in the following manner.

18. Example 1: Classifying Houses 18