Page 1
Mohammad Shaker Seminar of Fuzzy Logic. IT Engineering of Damascus, Syria
ZGTR, April, 15 2013
Social Relationship
and Decision Making
Explained by Fuzzy
Logic
Page 2
Mohammad Shaker Student at Information Technology Engineering, Damascus - Syria,
Dept. of Artificial Intelligence
@ZGTRShaker
Dr. Maisa Abu-Alkassem Supervisor and Dr. at Information Technology Engineering, Damascus - Syria,
Dept. of Artificial Intelligence
Page 4
“
Social
Relationship
”
Page 10
What we really see
Page 11
What we think about others
Page 15
Thinking and acting
Page 17
Social Relationship
Page 18
Modeling “The Basics”
Page 20
Logic Mapping Validity of statement;
not if the premises and conclusion are true or false.
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Statement P1, P2,..., Pn ⇒ Q
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P1, P2,..., Pn ⇒ Q
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Antecedent
P1, P2,..., Pn ⇒ Q
Page 24
Consequent
P1, P2,..., Pn ⇒ Q
Page 25
Proposition
P1, P2,..., Pn ⇒ Q
Page 26
Premises
P1, P2,..., Pn ⇒ Q
Page 27
Conclusion
P1, P2,..., Pn ⇒ Q
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Statement P1, P2,..., Pn ⇒ Q
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Valid Statement P1 ∧ P2 ∧ ... ∧ Pn → Q
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Valid Statement P1 ∧ P2 ∧ ... ∧ Pn → Q
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Conditional Proposition P1 ∧ P2 ∧ ... ∧ Pn → Q
Page 32
Conditional Proposition P1 ∧ P2 ∧ ... ∧ Pn → Q
Valid Statement P1 ∧ P2 ∧ ... ∧ Pn → Q
Statement P1, P2,..., Pn ⇒ Q
Page 33
Perfect knowledge and
human reasoning
Page 34
P1 ∧ P2 ∧ ... ∧ Pn → Q
Pi, only one input universe of discourse
Page 35
Set Theory μ(x) → {0, 1}
Page 36
Aristotelic logic
Page 37
𝑥1
𝑥2
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
Aristotelic logic
P1 ∧ P2 ∧ ... ∧ Pn → Q
Page 38
𝑥1
𝑥2
𝑃2
𝑃1
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
Aristotelic logic
P1 ∧ P2 ∧ ... ∧ Pn → Q
Page 39
𝑥1
𝑥2
𝑃2
𝑃1
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
(𝑥0, 𝑦0)
Aristotelic logic
P1 ∧ P2 ∧ ... ∧ Pn → Q
Page 40
Vectorial human mind description
Page 41
𝑥1
𝑥2
𝑃2
𝑃1
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
(𝑥0, 𝑦0)
Vectorial human mind description
Page 42
𝑥1
𝑥2
𝑃2
𝑃1
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
(𝑥0, 𝑦0)
Human mental description
singleton values, inference and decision-making
Page 43
𝑥1
𝑥2
𝑃2
𝑃1
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
(𝑥0, 𝑦0)
Not a coincidence that point-of-view,
mathematical conditional proposition, and
premises when using the logic operator ∧
are immediate and close related
”
“
Page 44
𝑌
ߤ 𝑄 (𝑦)
Individual is associated to the point
𝑥1
𝑥2
𝑃2
𝑃1
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
(𝑥0, 𝑦0)
P1 ∧ P2 ∧ ... ∧ Pn → Q
Page 45
𝑌
𝑄
ߤ 𝑄 (𝑦)
Individual is associated to the point
𝑥1
𝑥2
𝑃2
𝑃1
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
(𝑥0, 𝑦0)
P1 ∧ P2 ∧ ... ∧ Pn → Q
Page 46
SOCIAL RELATIONSHIP FOR
HUMAN MENTAL DESCRIPTION
Page 47
Social (Aristotelic) Relationship
Page 48
𝑥1
𝑥2
𝑃2
𝑃1
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
(𝑥0, 𝑦0)
Social (Aristotelic) Relationship
Page 49
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃21
Social (Aristotelic) Relationship
(𝑥0, 𝑦0)
Page 50
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃21
Social (Aristotelic) Relationship
Page 51
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃21
𝑃12
𝑃22
Social (Aristotelic) Relationship
Page 52
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃21
𝑃12
𝑃22
𝑃13
𝑃23
Social (Aristotelic) Relationship
Page 53
Multi-vectorial,
Social human description
Page 54
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃21
𝑃12
𝑃22
𝑃13
𝑃23
Social (Aristotelic) Relationship
Page 55
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃21
𝑃12
𝑃22
𝑃13
𝑃23
Social (Aristotelic) Relationship
Page 56
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃21
𝑃12
𝑃22
𝑃13
𝑃23
Social (Aristotelic) Relationship
Page 57
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃21
𝑃12
𝑃22
𝑃13
𝑃23
𝑑12𝑥1
Social (Aristotelic) Relationship
Page 58
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃21
𝑃12
𝑃22
𝑃13
𝑃23
𝑑12𝑥1 𝑑23𝑥1
𝑑23𝑥2 𝑑13𝑥2
Social (Aristotelic) Relationship
Page 59
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
There is a closer thinking between 1 and 2, than between 1 and 3.
𝑃21
𝑃12
𝑃22
𝑃13
𝑃23
𝑑12𝑥1 𝑑23𝑥1
𝑑23𝑥2 𝑑13𝑥2
Social (Aristotelic) Relationship
Page 60
𝑥1
Do not seem to be an ideal representative of real world relationship. Even presenting a
barely small distance, individuals represented by P1(·) and
P2(·) does not share the same space in the n-dimensional
plane, either in the i-th universe of discourse.
”
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃21
𝑃12
𝑃22
𝑃13
𝑃23
𝑑12𝑥1 𝑑23𝑥1
𝑑23𝑥2 𝑑13𝑥2
“
Page 62
Fuzzy Sets flexible; in order to accommodate small social differences
Page 64
Singleton Representation is enlarged, but keeping the core unaltered
Page 65
𝑌
𝑄
ߤ 𝑄 (𝑦)
Mapping, Fuzziness
𝑥1
𝑥2
𝑃2
𝑃1
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
(𝑥0, 𝑦0)
Page 66
𝑥1
𝑥2
𝑃2
𝑃1
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
(𝑥0, 𝑦0)
Mapping, Fuzziness
Page 67
𝑥1
𝑥2
𝑃2
𝑃1
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
Mapping, Fuzziness
2 universes of discourse
Page 68
𝑥1
𝑥2
𝑃2
𝑃1
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
Mapping, Fuzziness
Page 69
𝑥1
𝑥2
𝑃2
𝑃1
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
(𝑥0, 𝑦0)
𝑌
𝑄
ߤ 𝑄 (𝑦)
Mapping, Fuzziness
Page 70
Most of time the human
reasoning is approximate
𝑥1
𝑥2
𝑃2
𝑃1
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
(𝑥0, 𝑦0)
𝑌
𝑄
ߤ 𝑄 (𝑦)
Page 71
Individuals Ideas Representation
Page 72
𝑥1
𝑥2
𝑃2
𝑃1
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
(𝑥0, 𝑦0)
Individuals Ideas Representation
Page 73
𝑥1
𝑥2
𝑃2
𝑃1
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
(𝑥0, 𝑦0)
Individuals Ideas Representation
Page 74
𝑥1
𝑥2
𝑃2
𝑃1
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
Individuals Ideas Representation
Page 75
𝑥1
𝑥2
𝑃2
𝑃1
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
Source of uncertainty
and imprecision
Individuals Ideas Representation
Page 76
𝑥1
𝑥2
𝑃2
𝑃1
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
Source of uncertainty
and imprecision
The vector that represents the idea can be related to the situation where the individual is in doubt or even
is flexible in making decisions, judging or analyzing a situation.
Page 77
APPROXIMATE SOCIAL RELATIONSHIP FOR
HUMAN MENTAL DESCRIPTION
Page 78
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
Social (Aristotelic) Relationship
𝑃21
𝑃12
𝑃22
𝑃13
𝑃23
Page 79
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃21
𝑃12
𝑃22
𝑃13
𝑃23
Social (Aristotelic) Relationship
Page 80
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
Fuzzy Social Relationship
𝑃21
𝑃12
𝑃22
𝑃13
𝑃23
Page 81
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃21
𝑃12
𝑃22
𝑃13
𝑃23
Fuzzy Social Relationship
Page 82
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃21
𝑃12
𝑃22
𝑃13
𝑃23
Fuzzy Social Relationship
Page 83
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃21
𝑃12
𝑃22
𝑃13
𝑃23
Fuzzy Social Relationship
Page 84
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃21
𝑃12
𝑃22
𝑃13
𝑃23
Fuzzy Social Relationship
Page 85
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃21
𝑃12
𝑃22
𝑃13
𝑃23
𝑃𝑖: individual 𝑖
Fuzzy Social Relationship
Page 86
Social Relationship 𝑃𝑟𝑜𝑗𝑋𝑗 = 𝑠𝑢𝑝𝑋𝑗(𝑃
𝑖∧ 𝑃≠𝑖)
Page 87
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃21
𝑃12
𝑃22
𝑃13
𝑃23
𝑃𝑟𝑜𝑗𝑋𝑗 = 𝑠𝑢𝑝𝑋𝑗(𝑃𝑖∧ 𝑃≠𝑖)
The social relationship of two individuals, 𝑃𝑖 and 𝑃≠𝑖, each
represented by a n-th relation, 𝑅𝑛, projected in a j-th universe
of discourse is:
Page 88
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃21
𝑃12
𝑃22
𝑃13
𝑃23
𝑃𝑟𝑜𝑗𝑋𝑗 = 𝑠𝑢𝑝𝑋𝑗(𝑃𝑖∧ 𝑃≠𝑖)
Fuzzy Social Relationship
Page 89
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃21
𝑃12
𝑃22
𝑃13
𝑃23
𝑃𝑟𝑜𝑗𝑋𝑗 = 𝑠𝑢𝑝𝑋𝑗(𝑃𝑖∧ 𝑃≠𝑖)
Fuzzy Social Relationship
Page 90
Decision Making in fuzzy environment
Page 92
D = G ∩ C D
eci
sion
Page 93
D = G ∩ C Fuzzy G
oal
Page 94
D = G ∩ C Fuzzy Constraint
Page 95
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃21
𝑃12
𝑃22
𝑃13
𝑃23
Decision Making in fuzzy environment
Page 96
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃21
𝑃12
𝑃22
𝑃13
𝑃23
Decision Making in fuzzy environment
Page 97
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃21
𝑃12
𝑃22
𝑃13
𝑃23
Decision Making in fuzzy environment
Page 98
Traditional Decision-making in Fuzzy environment
0 120 Input universe of discourse x1
Page 99
Mem
bers
hip
funct
ion, G
, C, a
nd G
and C
Traditional Decision-making in Fuzzy environment
0
1
120 Input universe of discourse x1
Page 100
Input universe of discourse x1
Mem
bers
hip
funct
ion, G
, C, a
nd G
and C
Traditional Decision-making in Fuzzy environment
0
1
120
Constraint, 𝑃12
Constraint, 𝑃12
Page 101
Input universe of discourse x1
Mem
bers
hip
funct
ion, G
, C, a
nd G
and C
Traditional Decision-making in Fuzzy environment
0
1
120
Constraint, 𝑃12
Goal, 𝑃11
Goal, 𝑃11 Constraint, 𝑃1
2
Page 102
Input universe of discourse x1
Mem
bers
hip
funct
ion, G
, C, a
nd G
and C
Traditional Decision-making in Fuzzy environment
0
1
120
Constraint, 𝑃12
Goal, 𝑃11
Goal, 𝑃11 Constraint, 𝑃1
2
Page 103
Input universe of discourse x1
Mem
bers
hip
funct
ion, G
, C, a
nd G
and C
Traditional Decision-making in Fuzzy environment
0
1
120
Constraint, 𝑃12
Goal, 𝑃11
Decision by Defuzzification
Goal, 𝑃11 Constraint, 𝑃1
2
Decision by
Defuzzification
Page 104
Input universe of discourse x1
Mem
bers
hip
funct
ion, G
, C, a
nd G
and C
Traditional Decision-making in Fuzzy environment
0
1
120
Constraint, 𝑃12
Goal, 𝑃11
Decision by Defuzzification
Goal, 𝑃11 Constraint, 𝑃1
2
Decision by
Defuzzification
D = G ∩ C
Page 106
𝑃𝑗𝑖 = 𝐺𝑗
𝑖
j-th propositions, 𝑃𝑗
Page 107
𝑃𝑗𝑖 = 𝐺𝑗
𝑖
i-th the individual, 𝑃𝑖
Page 108
𝑃𝑗𝑖 = 𝐺𝑗
𝑖
is related to individual goal, 𝐺𝑖
Page 109
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃21
𝑃12
𝑃22
𝑃13
𝑃23
Decision Making in fuzzy environment
Page 110
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃21
𝑃12
𝑃22
𝑃13
𝑃23
Decision Making in fuzzy environment
Page 111
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃21
𝑃12
𝑃22
𝑃13
𝑃23
Decision Making in fuzzy environment
Page 112
𝑃𝑗𝑖 = 𝐺𝑗
𝑖 = 𝐶𝑗≠𝑖
the goal for one individual is, actually, the constraint for the other one
Page 113
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃21
𝑃12
𝑃22
𝑃13
𝑃23
Decision Making in fuzzy environment
Page 114
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃21
𝑃12
𝑃22
𝑃13
𝑃23
𝑃𝑟𝑜𝑗𝑋𝑗 = 𝑠𝑢𝑝𝑋𝑗(𝑃𝑖∧ 𝑃≠𝑖)
Decision Making in fuzzy environment
Page 115
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃21
𝑃12
𝑃22
𝑃13
𝑃23
𝑃𝑟𝑜𝑗𝑋𝑗 = 𝑠𝑢𝑝𝑋𝑗(𝑃𝑖∧ 𝑃≠𝑖)
𝑃11 = 𝐺1
1 = 𝐶12 = 𝐶1
3
Decision Making in fuzzy environment
Page 116
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃21
𝑃12
𝑃22
𝑃13
𝑃23
𝐷𝑃1,𝑃2,𝑥1 = 𝐺1
1 ∩ 𝐶12= 𝑃1
1 ∧ 𝑃12 ,
Decision Making in fuzzy environment
Page 117
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃21
𝑃12
𝑃22
𝑃13
𝑃23
𝐷𝑃1,𝑃2,𝑥1 = 𝐺1
1 ∩ 𝐶12= 𝑃1
1 ∧ 𝑃12 ,
𝐷𝑃1,𝑃2,𝑥1 = 𝐺1
2 ∩ 𝐶11= 𝑃1
2 ∧ 𝑃11
Decision Making in fuzzy environment
Page 118
if there is a non-coincidental
goal for people that want interact
to, their feasible area of decision
is related to what they have in
common
“
”
Page 119
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃21
𝑃12
𝑃22
𝑃13
𝑃23
Decision Making in fuzzy environment
Page 120
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃12 𝑃1
3
Decision Making in fuzzy environment
𝑃21
𝑃22
𝑃23
Page 121
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃12 𝑃1
3
Decision Making in fuzzy environment
𝑃21
𝑃22
𝑃23
Page 122
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃12 𝑃1
3
Decision Making in fuzzy environment
𝑃21
𝑃22
𝑃23
Page 123
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃12 𝑃1
3
Decision Making in fuzzy environment
𝑃21
𝑃22
𝑃23
Page 124
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃12 𝑃1
3
Decision Making in fuzzy environment
𝑃21
𝑃22
𝑃23
𝑃21 𝑃2
2 = G ∩ C
𝑃22 𝑃2
1 = G ∩ C
Page 125
it can be noticed both
mathematically and visually that
when there is overlapping in the
fuzzy sets in the n-dimensional
mathematical relation, a true
relationship exists.
“
”
Page 126
when there is flexibility and a
sort of representative behavior
that worth sharing space in
debate, the chances of achieving
a common decision-making is
greater.
“
”
Page 127
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃12 𝑃1
3
Affinity and Friendship
𝑃21
𝑃22
𝑃23
Page 128
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃12 𝑃1
3
Affinity and Friendship
𝑃21
𝑃22
𝑃23
Page 129
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃12 𝑃1
3
Affinity and Friendship
𝑃21
𝑃22
𝑃23
Page 130
So, does it all works everytime?
Page 132
What will
others think?
Page 133
Can I
do it?!
Should I
do it?
What will
others think?
Am I
prepared?
Page 134
Can I
do it?!
Should I
do it?
What will
others think?
She’s not
gonna like
it
Am I
prepared?
Page 135
Can I
do it?!
Should I
do it?
What will
others think?
She’s not
gonna like
it
This’s
definitely
wrong!
This will turn
out good I’m good at it, I
should do it
Am I
prepared?
Page 136
Can I
do it?!
Should I
do it?
What will
others think?
She’s not
gonna like
it
This’s
definitely
wrong!
This will turn
out good I’m good at it, I
should do it
Am I
prepared?
Page 138
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