Fuzzy Logic and its Applications

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Fuzzy Logic and its Applications
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Fuzzy Logic and its Applications

• When the boundaries of concepts are continuous

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Fuzzy Logic and its Applications

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

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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.

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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
?
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Various applications (fuzzy logic)

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

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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)
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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

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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

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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.
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Fuzzy Set Operators (Cont)

• Complement
• The complement of A is defined by

e.g. We define fuzzy set NOT TALL from TALL
complement ?
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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
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Fuzzy Logic (Cont)

• Modifiers like very, quite can be treated as
  applying arithmetic operators on that of the base
  fuzzy set

x
x2
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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)

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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
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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
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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

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

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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

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The End