Fuzzy modeling is a powerful approach found by Zadeh for the modeling of complex and uncertain systems [2]. Fuzzy logic has a distinctive advantage where the precise definition of a control process is unachievable. Fuzzy models have the ability to establish a relationship between input and output variables by employing predefined rules. The technique provides simple solutions which are based on natural language statements. Fuzzy logic takes the inputs and outputs in the form of fuzzy sets where each set contains elements that have varying degrees of membership. A fuzzy set then enables transforming real numbers to the membership degrees changing from 0 to 1. Fuzzy rules relate input variables to output variables. These rules represent the expert knowledge in the system. Indeed, the intuition behind fuzzy logic is, it works with perception-based data instead of measurement-based which are crisp and numeric. Hence, it tries to capture how human use perceptions of time, direction, speed, shape, possibility, likelihood, truth, and other attributes of physical and mental objects. Perceptions in this manner are inherently imprecise when compared to crisp values, for example, a human might express his intuition about the weather as being not very hot while a sensor would read the heat in degrees and give us a crisp value. Therefore, perceptions are very subjective and reflect the partiality of human concepts. In 2001, Prof. Zadeh proposed his computational theory of perceptions (CTP) where the objects of computations are words and propositions drawn from natural language rather than crisp numeric values. The idea of the theory came due to the unavailability of a methodology for reasoning and computing with perceptions rather than measurements. Hence, the CPT was the ground for allowing a computer to make subjective judgments which often refered as perceptual computing. E.H. Mamdani, Application of fuzzy algorithms for control of simple dynamic plant, in: Proceedings of the Institution of Electrical Engineers, IET, 1974, pp. 1585-1588. Zadeh, Lotfi A. "Fuzzy sets." Information and control 8, no. 3 (1965): 338-353. Zadeh, Lotfi A. "A new direction in AI: Toward a computational theory of perceptions." AI magazine 22, no. 1 (2001): 73.

1. Fuzzy Logic
Amir, Hussein

2. Introduction
Uncertainty
● When A is a fuzzy set and x is a relevant object, the proposition “x is a member of A”
is not necessarily either true or false. It may be true only to some degree, the degree
to which x is actually a member of A.
For example: the weather today
● Sunny: If we define any cloud cover of 25% or less is sunny.
● This means that a cloud cover of 26% is not sunny?
● “Vagueness” should be introduced.

3. Introduction
The crisp set v.s. the fuzzy set
● The crisp set is defined in such a way as to partition the individuals in some given universe of
discourse into two groups: members and nonmembers.
● However, many classification concepts do not exhibit this characteristic.
● For example, the set of tall people, expensive cars, or sunny days.
A fuzzy set can be defined mathematically by assigning to each possible individual in the universe of
discourse a value representing its degree of membership in the fuzzy set.

4. Fuzzy sets
A membership function:
● The values assigned to the elements of the universal set fall within a specified range and
indicate the membership degree of these elements in the set.
● Larger values denote higher degrees of set membership.
● A set defined by membership functions is a fuzzy set.

5. Example of Membership Functions
● Consider three fuzzy sets that represent the concepts of a young, middle-aged, and old person.
The membership functions are defined as follows:

6. Example of Membership Functions...

7. Fuzzy Logic
● Many decision-making and problem-solving tasks are too complex to be
defined precisely
● However, people succeed by using imprecise knowledge
● Fuzzy logic resembles human reasoning in its use of approximate
information and uncertainty to generate decisions.

8. Natural Language
● Consider:
○ Joe is tall -- what is tall?
○ Joe is very tall -- what does this differ from tall?
● Natural language (like most other activities in life and indeed the universe) is
not easily translated into the absolute terms of 0 and 1.
● Unlike statistics and probabilities, the degree is not describing
probabilities that the item is in the set, but instead describes to what
extent the item is the set.

9. ● Fuzzy expert system is a collection of membership functions and rules that
are used to reason about data.
Operation of Fuzzy System

10. Tipping Problem -The Fuzzy Approach
● Given two sets of numbers between 0 and 10 (where 10 is excellent) that
respectively represent the quality of the service and the quality of the food
at a restaurant,
● If service is poor, then tip is cheap
● If service is good, then tip is average
● If service is excellent, then tip is generous

11. Fuzzy Logic System

12. Implementation (Mamdani’s approach)

13. Fuzzy Logic Applications
● Fuzzy logic is a convenient way to map an input space to an output space.
● With information about how good your service was at a restaurant, a fuzzy
logic system can tell you what the tip should be.
● With information about how fast the car is going and how hard the motor is
working, a fuzzy logic system can shift gears for you.
● With information about how far away the subject of your photograph is, a
fuzzy logic system can focus the lens for you.

14. Perceptual Computing From The Fuzzy
Logic Perspective
Lotfi Zadeh Work published between 1999 - 2002

15. Perceptions
● Measurements are crisp | perceptions are fuzzy.
● Perception-based information is drawn from
natural language.
● Humans use perceptions of time, direction,
speed, shape, possibility, likelihood, truth, and
other attributes of physical and mental objects.
● perceptions are imprecise.
○ Example: not very high, about 0.8

16. Zadeh’s Computational theory of perceptions
● Motivation:
○ Humans are remarkable in manipulating perceptions-- perceptions of direction, likelihood, truth and intent.
○ Partiality of human concepts. The validity of a human concept is a matter of degree.
○ The unavailability of a methodology for reasoning and computing with perceptions rather than measurements.
● Grounded on a methodology called computing with words (CWW)
○ objects of computation are words and propositions drawn from a natural language, e.g., small, large, not very
likely, the price of gas is low and declining.
● CWW is for making subjective judgments, which they call perceptual computing

17. Subjective judgements
● A subjective judgment is
○ a personal opinion that has been influenced by one’s personal views, experience, or background.
○ a variable personal assessment made using a mixture of qualitative and quantitative information.
● Example of CWW for making subjective judgements:
○ Social Judgment Making
■ The meaning of another’s behavior to judge the level of the variable interest.
■ Example:
● examining the level of kindness depending on how kind a person is to the other individual.
● What is considered to be kind?
● How to measure kindness (number of times you brought me cookies or smiling in my face)
● Final result:
○ IF you brought me cookies every day THEN you are so kind

18. Reasoning with perceptions vs. traditional methods
Perceptual reasoning
involve a goal-directed propagation of constraints from
premises to conclusions.
Faulty conclusions

19. References
● Zadeh, Lotfi A. "Toward a perception-based theory of probabilistic
reasoning with imprecise probabilities." Journal of statistical planning and
inference 105, no. 1 (2002): 233-264.
● Zadeh, Lofti. "From computing with numbers to computing with
words—from manipulation of measurements to manipulation of
perceptions." In Logic, Thought and Action, pp. 507-544. Springer
Netherlands, 1999.
● Zadeh, Lotfi A. "A new direction in AI: Toward a computational theory of
perceptions." AI magazine 22, no. 1 (2001): 73.
● Mendel, Jerry, and Dongrui Wu. Perceptual computing: aiding people in
making subjective judgments. Vol. 13. John Wiley & Sons, 2010.