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Fuzzy Logic and its Applications

Fuzzy Logic and its Applications

- When the boundaries of concepts are continuous

Fuzzy Logic and its Applications

- Outline
- Introduction to Fuzzy Logic
- Fuzzy Numbers
- Fuzzy Knowledge Data Engineering
- Fuzzy Control
- Other Applications

Introduction To Fuzzy Logic

- Human knowledge is extracted and transferred to

knowledge base. - Imprecision and fuzziness are common features in

human knowledge. - Reasons
- Human thinking always manipulate data which are

imprecisely defined properties or quantities - e.g. the price is low, a long story, a tall man
- low, long, tall are fuzzy concepts.
- Knowledge is often derived in linguistic form

from expert - e.g. If the price is low, it implies that the

profit will be below normal. - Fuzzy logic offers a way (amongst other methods)

to manipulate this kind of inexact knowledge.

Introduction To Fuzzy Logic

- For a logic system

Axioms

built on axioms and linked together rigorously

according to stated rules, without contradiction

based on

?

Propositions

Fuzzy logic based on

fuzzy set theory

?

Boolean logic based on

ordinary set theory

?

Various applications (fuzzy logic)

- Approximate reasoning
- Expert system
- Computational linguistic
- Database interface
- Pattern recognition
- Image processing
- Decision making
- Industrial process control

Basic Theory of Fuzzy Set

- 2.1 Definition (originated by Lotfi Zadeh)
- Ordinary Set
- X set of objects, called universe
- A ordinary set in X
- For an element x in X, define membership function

as - Fuzzy Set
- If the value (degree of membership) is allowed to

vary in between the real interval 0, 1, then - A fuzzy set
- e.g. Fuzzy set A of integers approximately equal

to 10

(The universe, X is the set of all integers)

Basic Theory of Fuzzy Set (Cont)

- 2.2 Notation
- Let X, the universe be a finite set x1, x2,

xn - Fuzzy set A on X is expressed as
- A ?A (x1) / x1 ?A (x2) / x2 ?A

(xn) / xn - (where is union)
- e.g. the above example
- A 0.1/7 0.5/8 0.8/9 1/10 0.8/11

0.5/12 0.1/13 - When the universe X is not finite set, we write
- Two fuzzy sets are equal iff

Fuzzy Set Operators

- 2.3 Basic Fuzzy Set Operators (3 operators)
- Intersection / Union ?x ? X
- Intersection ?A?B(x) min(?A(x), ?B(x))
- Union ?A?B(x) max(?A(x), ?B(x))
- Example Let A fuzzy set of integers around

7 - B fuzzy set of

integers slightly less than 10 - A 0.1/4 0.5/5 0.8/6 1/7 0.8/8 0.5/9

0.1/10 - B 0.1/6 0.5/7 0.8/8 1/9
- Intersection
- A?B fuzzy set of integers around 7 and

slightly less than 10 - A?B 0.1/6 0.5/7 0.8/8 0.5/9
- Union
- A?B fuzzy set of integers around 7 or slightly

less than 10 - A?B 0.1/4 0.5/5 0.8/6 1/7 0.8/8 1/9

0.1/10

Fuzzy Set Operators (Cont)

A?B

A?B

Note The choice of MAX and MIN has been

justified by Bellman and Giertz. The conclusions

in these definitions are natural and reasonable.

Fuzzy Set Operators (Cont)

- Complement
- The complement of A is defined by

e.g. We define fuzzy set NOT TALL from TALL

complement ?

Fuzzy Logic

- 2.4 Fuzzy Logic (deal with fuzziness)
- Based on fuzzy set theory
- Fuzzy terms represented by fuzzy set

E.g. tall

A psychological continuum of human

perception e.g. good, bad, red, beautiful, smart

Fuzzy Logic (Cont)

- Modifiers like very, quite can be treated as

applying arithmetic operators on that of the base

fuzzy set

x

x2

Fuzzy Logic (Cont)

- Rules relating fuzzy terms can be modeled by

fuzzy relations. - e.g. ANT1 if the price is high then the profit

should be -

good (R) - ANT2 the price is very high

(S1) - Conclusion the profit should be very good (S2)

In Approximate Reasoning

- Examples
- Modeling by Fuzzy Logic

(antecedent) (consequence)

If x is P1 then y is Q1 x is P1 y is

Q1

A1

A2

C

P1, P1, Q1, Q1 are fuzzy concepts

In Approximate Reasoning (Cont)

- To find Q, apply fuzzy composition

?

Q P R where ? is composition, R (which is

a matrix) is obtained by some fuzzy operations

between P and Q

?

composition if C D ? R, then ?C(x)

MAXW(MINW(?D(w), ?R(w, x))) where C and

D are fuzzy sets (vectors) R is a fuzzy

relation matrix ?(x) is a membership function

In Approximate Reasoning (Cont)

- Methods of Obtaining Fuzzy Relation R
- (i) (proposed by

Mamdani) - where is Cartesian product

(minimum) - (ii)

(proposed by Zadeh) - (iii)

(proposed by Mitumoto) - where

- (iv) (proposed by Mitumoto)
- where
- (v)

In Approximate Reasoning (Cont)

- Many other methods had been proposed!
- Different methods will have slightly different

result after making the inference. - Exact reasoning (predicate logic) is only a

special case of approximate reasoning. - Note

The End