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Fuzzy Logic

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.

Fuzziness Vs. Vagueness

3

TRADITIONAL REPRESENTATION OF LOGIC

Slow

Fast

Speed 0

Speed 1

bool speed get the speed if ( speed 0)

// speed is slow else // speed is

fast

FUZZY LOGIC REPRESENTATION

Slowest

- Every problem must represent in terms of fuzzy

sets.

0.0 0.25

Slow

0.25 0.50

Fast

0.50 0.75

Fastest

0.75 1.00

FUZZY LOGIC REPRESENTATION CONT.

Slowest

Fastest

Slow

Fast

float speed get the speed if ((speed gt

0.0)(speed lt 0.25)) // speed is slowest

else if ((speed gt 0.25)(speed lt 0.5)) //

speed is slow else if ((speed gt 0.5)(speed lt

0.75)) // speed is fast else // speed gt

0.75 speed lt 1.0 // speed is fastest

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.

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.

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.

- 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.

Fuzzy Logic

Boolean Logic

- 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.

- 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.

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.

- 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.

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

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-gt

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.

- 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.

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.

- Notice in the figure that people whose ages are

gt zero and lt 20 are - complete members of the young set (that is, they

have a membership value of one). - Also note that people whose ages are gt 20 and lt

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 gt 30 are

non-members of the young set.

Logical Operations on Fuzzy Sets

- Negation
- Intersection
- Union

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

(1-10) and if the membership value is 0.4 the

membership value would be (1-0.40.6).

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.

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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.

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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).

- 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.

References

- http//www.dementia.org/julied/logic/index.html