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## Fuzzy Logic and its Applications

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Title: Fuzzy Logic and its Applications

1
Fuzzy Logic and its Applications
2
Fuzzy Logic and its Applications
• When the boundaries of concepts are continuous

3
Fuzzy Logic and its Applications
• Outline
• Introduction to Fuzzy Logic
• Fuzzy Numbers
• Fuzzy Knowledge Data Engineering
• Fuzzy Control
• Other Applications

4
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.

5
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
?
6
Various applications (fuzzy logic)
• Approximate reasoning
• Expert system
• Computational linguistic
• Database interface
• Pattern recognition
• Image processing
• Decision making
• Industrial process control

7
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)
8
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

9
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

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.
11
Fuzzy Set Operators (Cont)
• Complement
• The complement of A is defined by

e.g. We define fuzzy set NOT TALL from TALL
complement ?
12
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
13
Fuzzy Logic (Cont)
• Modifiers like very, quite can be treated as
applying arithmetic operators on that of the base
fuzzy set

x
x2
14
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)

15
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
16
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
17
In Approximate Reasoning (Cont)
• Methods of Obtaining Fuzzy Relation R
• (i) (proposed by
Mamdani)
• where is Cartesian product
(minimum)
• (ii)
• (iii)
(proposed by Mitumoto)
• where

18
• (iv) (proposed by Mitumoto)
• where
• (v)

19
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

20
The End