FLEA

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FLEA

• Prof. dr. ir. J. Hellendoorn

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1. Overview of fuzzy logic

• Fuzzy concepts
• Fuzzy vs. probability
• Sources of non-probabilistic uncertainty
• Fuzziness vs. possibility
• The goal of fuzzy logic
• Branches of fuzzy logic

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Crisp, semi-crisp, and fuzzy
Consider the integer numbers between 0 and
25. Which numbers have the property of being
large? The crisp answer
large
1
0.5
not large
0
20
25
10
15
5
4
The semi-crisp answer
large
1
dont know
0.5
not large
0
20
25
10
15
5
Is 9 a large number? No! Is 11 a large
number? Dont know! Is 21 a large number? Yes
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The fuzzy answer
The closer to 25 a number is, the bigger the
possibility to classify it as a large number. The
closer to 10 a number is, the smaller the
possibility to classify it as a large number.
1
0.5
0
20
25
10
15
5
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Reasoning

• Reasoning starts with atomic expressions
• It uses operators for and,or and not to
  build compound expressions
• Causality is expressed by if-then rules, e.g.,
  if x is large, then y is small
• The modus ponens combines if-then rules with
  expressions

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Crisp reasoning
If x is large, then y is small
1
large
0.5
not large
0
X
20
25
10
15
5
1
small
0.5
not small
0
Y
20
25
10
15
5
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Questions to crisp reasoning

• If x is 10, what is y ?
• If x is 20, what is y ?
• If x is 25, what is y ?

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Fuzzy reasoning
If x is large, then y is small
1
0.7
0.5
large number
0
X
20
25
10
15
5
22
1
0.7
0.5
small number
0
Y
4
20
25
10
15
5
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2. Fuzziness vs. probability

• What is the chance of throwing 6?
• Answer Pr(6) 1/6
• What is the chance of throwing 4, 5, or 6?
• Answer Pr(4, 5, or 6) 1/2
• What is the chance of throwing a large number?

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Throwing a large number

• Answer 1 If the meaning of large number is the
  crisp set 4,5,6 thenPr(large number) 1/2
• Answer 2 If the meaning of large number is the
  fuzzy set 0.3/4,0.7/5,1/6 thenPr(large number)
  1/60.31/60.71/61 ? 0.37

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Probability and fuzziness

• Probability the frequency with which an event
  occurs
• Fuzziness how to define the meaning of an
  event
• Fuzziness and probability the frequency of a
  fuzzy event

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Sources of non-probabilistic uncertainty

• Consider the following data categories
• Crisp data e.g., the rational numbers. Age(John)
  25 years
• Crisp-imprecise data e.g., intervals defined on
  rational numbers. Age(John) 19,30
• Fuzzy data e.g., a fuzzy set defined on the
  rational numbers. Age(John) young

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Knowledge and retrieval

• Knowledge can be stored in a data base, and can
  be
• crisp,
• imprecise, or
• fuzzy
• We can ask our data base questions, these
  questions can also be
• crisp,
• imprecise, or
• fuzzy

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Characteristic functions

• Let P be the domain of all possible values of
  pressure, e.g.,
• The notion high pressure is defined via a
  characteristic function, e.g.,
• For example and

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Universe of discourse approach

• Let U be the universe of discourse, and A a set
  defined on U
• Then
• A fuzzy set is defined via the notion of a
  characteristic (membership) function

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A fuzzy set

• Let U be the universe of discourse,
• A fuzzy set F of U is defined by the membership
  function
• Note is a classical
  set

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Alternatively

• F is defined by the set of tuples
• where is the degree of membership of
  u in F and

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Notation

• For practical reasons the notation
• is used instead of
• and a is used instead of ,
• e.g.,

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Discrete fuzzy sets

• Around 10ºC

m
1
0.5
0
Temperature
13
11
12
7
8
9
10
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Continuous fuzzy sets

• Around 10ºC
• or, alternatively,

m
1
0
Temperature
10
13
7
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Concepts

• Core(A)
• Support(A)
• A is normal iff

m
1
Core
0
Support
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a-level sets

• The a-level set of F is a crisp set defined by
• If then

1
a
0
a-level
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Convex fuzzy set

• A is convex iff

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convex
non-convex
0
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Cardinality of a fuzzy set

• From a discrete fuzzy set
• Relative cardinality
• From a continuous fuzzy set

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Entropy of a fuzzy set

• How fuzzy is a fuzzy set?
• Let d be a measure of fuzziness
• if A is a crisp set then d(A) 0
• d(A) has a maximum at mA(x) 0.5
• if mA(x) ? mA'(x) ? 0.5or mA(x) ? mA' (x) ?
  0.5then d(A) ? d'(A)
• d(A) d(A)

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3. Operations on fuzzy sets

• Intersection (AND-function/conjunction)
• Example temp is around 5ºC
• temp is around 8ºC
• Intersection

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Intersection example
Hence
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Union (OR or Disjunction)
F1 Temperature is around 5C
F2 Temperature is around 8C
m
m
1
1
0
0
5
8
5
8
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Negation (complement)
F1 Temperature is smaller than 10 C
F2 Temperature is not smaller than 10 C
F3 Temperature is higher than 10 C
F1
F2
F3
1
0
10
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Properties
(similarly for ?)
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Other alternatives
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Other alternatives II
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4. Fuzzy relations
REL(T1,T2) T2 is much lower than T1
Examples m(1,1)0, m(2.5,1)0.15, m(3.6,1)0.26,
m(10,1)0.9, m(60,1)1
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A discrete fuzzy relation
Consider the relation x is roughly equal to
y, where x,y ? 1,2,3,4
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Union
F1 x is roughly equal to y
F2 x is much larger than y
F1 or F2
F1 and F2
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5. Composition

• The domain is 1,2,3,4,5
• R1 x is much larger than y
• R2 x is large
• What is y?
• Answer

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Example 1
Compare
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Example 2
R1 x is taller than y
R2 y is taller than z
What is the relation R3 between (John, Pam, Jim)
and (Kay, Ron, Clara)?
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Composition properties

• Associativity (R1 R2) R3 R1 (R2 R3)
• Reflexivity
• R is reflexive if mR(x,x) 1
• if R1, R2 are reflexive then R1 R2 is reflexive
  
• Symmetricity
• R is symmetric if mR(x,y) mR(y,x)
• if R1, R2 are symmetric then R1 R2 is symmetric