Mohammad Shaker Seminar of Fuzzy Logic. IT Engineering of Damascus, Syria
ZGTR, April, 15 2013
Social Relationship
and Decision Making
Explained by Fuzzy
Logic
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
“
Social
Relationship
”
What we really see
What we think about others
Thinking and acting
Social Relationship
Modeling “The Basics”
Logic Mapping Validity of statement;
not if the premises and conclusion are true or false.
Statement P1, P2,..., Pn ⇒ Q
P1, P2,..., Pn ⇒ Q
Antecedent
P1, P2,..., Pn ⇒ Q
Consequent
P1, P2,..., Pn ⇒ Q
Proposition
P1, P2,..., Pn ⇒ Q
Premises
P1, P2,..., Pn ⇒ Q
Conclusion
P1, P2,..., Pn ⇒ Q
Statement P1, P2,..., Pn ⇒ Q
Valid Statement P1 ∧ P2 ∧ ... ∧ Pn → Q
Valid Statement P1 ∧ P2 ∧ ... ∧ Pn → Q
Conditional Proposition P1 ∧ P2 ∧ ... ∧ Pn → Q
Conditional Proposition P1 ∧ P2 ∧ ... ∧ Pn → Q
Valid Statement P1 ∧ P2 ∧ ... ∧ Pn → Q
Statement P1, P2,..., Pn ⇒ Q
Perfect knowledge and
human reasoning
P1 ∧ P2 ∧ ... ∧ Pn → Q
Pi, only one input universe of discourse
Set Theory μ(x) → {0, 1}
Aristotelic logic
𝑥1
𝑥2
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
Aristotelic logic
P1 ∧ P2 ∧ ... ∧ Pn → Q
𝑥1
𝑥2
𝑃2
𝑃1
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
Aristotelic logic
P1 ∧ P2 ∧ ... ∧ Pn → Q
𝑥1
𝑥2
𝑃2
𝑃1
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
(𝑥0, 𝑦0)
Aristotelic logic
P1 ∧ P2 ∧ ... ∧ Pn → Q
Vectorial human mind description
𝑥1
𝑥2
𝑃2
𝑃1
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
(𝑥0, 𝑦0)
Vectorial human mind description
𝑥1
𝑥2
𝑃2
𝑃1
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
(𝑥0, 𝑦0)
Human mental description
singleton values, inference and decision-making
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
”
“
𝑌
ߤ 𝑄 (𝑦)
Individual is associated to the point
𝑥1
𝑥2
𝑃2
𝑃1
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
(𝑥0, 𝑦0)
P1 ∧ P2 ∧ ... ∧ Pn → Q
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
SOCIAL RELATIONSHIP FOR
HUMAN MENTAL DESCRIPTION
Social (Aristotelic) Relationship
𝑥1
𝑥2
𝑃2
𝑃1
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
(𝑥0, 𝑦0)
Social (Aristotelic) Relationship
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃21
Social (Aristotelic) Relationship
(𝑥0, 𝑦0)
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃21
Social (Aristotelic) Relationship
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃21
𝑃12
𝑃22
Social (Aristotelic) Relationship
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃21
𝑃12
𝑃22
𝑃13
𝑃23
Social (Aristotelic) Relationship
Multi-vectorial,
Social human description
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃21
𝑃12
𝑃22
𝑃13
𝑃23
Social (Aristotelic) Relationship
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃21
𝑃12
𝑃22
𝑃13
𝑃23
Social (Aristotelic) Relationship
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃21
𝑃12
𝑃22
𝑃13
𝑃23
Social (Aristotelic) Relationship
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃21
𝑃12
𝑃22
𝑃13
𝑃23
𝑑12𝑥1
Social (Aristotelic) Relationship
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃21
𝑃12
𝑃22
𝑃13
𝑃23
𝑑12𝑥1 𝑑23𝑥1
𝑑23𝑥2 𝑑13𝑥2
Social (Aristotelic) Relationship
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
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
“
Fuzzy Sets flexible; in order to accommodate small social differences
Singleton Representation is enlarged, but keeping the core unaltered
𝑌
𝑄
ߤ 𝑄 (𝑦)
Mapping, Fuzziness
𝑥1
𝑥2
𝑃2
𝑃1
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
(𝑥0, 𝑦0)
𝑥1
𝑥2
𝑃2
𝑃1
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
(𝑥0, 𝑦0)
Mapping, Fuzziness
𝑥1
𝑥2
𝑃2
𝑃1
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
Mapping, Fuzziness
2 universes of discourse
𝑥1
𝑥2
𝑃2
𝑃1
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
Mapping, Fuzziness
𝑥1
𝑥2
𝑃2
𝑃1
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
(𝑥0, 𝑦0)
𝑌
𝑄
ߤ 𝑄 (𝑦)
Mapping, Fuzziness
Most of time the human
reasoning is approximate
𝑥1
𝑥2
𝑃2
𝑃1
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
(𝑥0, 𝑦0)
𝑌
𝑄
ߤ 𝑄 (𝑦)
Individuals Ideas Representation
𝑥1
𝑥2
𝑃2
𝑃1
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
(𝑥0, 𝑦0)
Individuals Ideas Representation
𝑥1
𝑥2
𝑃2
𝑃1
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
(𝑥0, 𝑦0)
Individuals Ideas Representation
𝑥1
𝑥2
𝑃2
𝑃1
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
Individuals Ideas Representation
𝑥1
𝑥2
𝑃2
𝑃1
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
Source of uncertainty
and imprecision
Individuals Ideas Representation
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.
APPROXIMATE SOCIAL RELATIONSHIP FOR
HUMAN MENTAL DESCRIPTION
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
Social (Aristotelic) Relationship
𝑃21
𝑃12
𝑃22
𝑃13
𝑃23
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃21
𝑃12
𝑃22
𝑃13
𝑃23
Social (Aristotelic) Relationship
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
Fuzzy Social Relationship
𝑃21
𝑃12
𝑃22
𝑃13
𝑃23
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃21
𝑃12
𝑃22
𝑃13
𝑃23
Fuzzy Social Relationship
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃21
𝑃12
𝑃22
𝑃13
𝑃23
Fuzzy Social Relationship
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃21
𝑃12
𝑃22
𝑃13
𝑃23
Fuzzy Social Relationship
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃21
𝑃12
𝑃22
𝑃13
𝑃23
Fuzzy Social Relationship
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃21
𝑃12
𝑃22
𝑃13
𝑃23
𝑃𝑖: individual 𝑖
Fuzzy Social Relationship
Social Relationship 𝑃𝑟𝑜𝑗𝑋𝑗 = 𝑠𝑢𝑝𝑋𝑗(𝑃
𝑖∧ 𝑃≠𝑖)
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:
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃21
𝑃12
𝑃22
𝑃13
𝑃23
𝑃𝑟𝑜𝑗𝑋𝑗 = 𝑠𝑢𝑝𝑋𝑗(𝑃𝑖∧ 𝑃≠𝑖)
Fuzzy Social Relationship
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃21
𝑃12
𝑃22
𝑃13
𝑃23
𝑃𝑟𝑜𝑗𝑋𝑗 = 𝑠𝑢𝑝𝑋𝑗(𝑃𝑖∧ 𝑃≠𝑖)
Fuzzy Social Relationship
Decision Making in fuzzy environment
D = G ∩ C D
eci
sion
D = G ∩ C Fuzzy G
oal
D = G ∩ C Fuzzy Constraint
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃21
𝑃12
𝑃22
𝑃13
𝑃23
Decision Making in fuzzy environment
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃21
𝑃12
𝑃22
𝑃13
𝑃23
Decision Making in fuzzy environment
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃21
𝑃12
𝑃22
𝑃13
𝑃23
Decision Making in fuzzy environment
Traditional Decision-making in Fuzzy environment
0 120 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 Input universe of discourse x1
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
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
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
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
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
𝑃𝑗𝑖 = 𝐺𝑗
𝑖
j-th propositions, 𝑃𝑗
𝑃𝑗𝑖 = 𝐺𝑗
𝑖
i-th the individual, 𝑃𝑖
𝑃𝑗𝑖 = 𝐺𝑗
𝑖
is related to individual goal, 𝐺𝑖
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃21
𝑃12
𝑃22
𝑃13
𝑃23
Decision Making in fuzzy environment
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃21
𝑃12
𝑃22
𝑃13
𝑃23
Decision Making in fuzzy environment
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃21
𝑃12
𝑃22
𝑃13
𝑃23
Decision Making in fuzzy environment
𝑃𝑗𝑖 = 𝐺𝑗
𝑖 = 𝐶𝑗≠𝑖
the goal for one individual is, actually, the constraint for the other one
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃21
𝑃12
𝑃22
𝑃13
𝑃23
Decision Making in fuzzy environment
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃21
𝑃12
𝑃22
𝑃13
𝑃23
𝑃𝑟𝑜𝑗𝑋𝑗 = 𝑠𝑢𝑝𝑋𝑗(𝑃𝑖∧ 𝑃≠𝑖)
Decision Making in fuzzy environment
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
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
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
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
“
”
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃21
𝑃12
𝑃22
𝑃13
𝑃23
Decision Making in fuzzy environment
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃12 𝑃1
3
Decision Making in fuzzy environment
𝑃21
𝑃22
𝑃23
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃12 𝑃1
3
Decision Making in fuzzy environment
𝑃21
𝑃22
𝑃23
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃12 𝑃1
3
Decision Making in fuzzy environment
𝑃21
𝑃22
𝑃23
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃12 𝑃1
3
Decision Making in fuzzy environment
𝑃21
𝑃22
𝑃23
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
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.
“
”
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.
“
”
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃12 𝑃1
3
Affinity and Friendship
𝑃21
𝑃22
𝑃23
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃12 𝑃1
3
Affinity and Friendship
𝑃21
𝑃22
𝑃23
𝑥1
𝑥2
𝑃11
ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)
𝑃12 𝑃1
3
Affinity and Friendship
𝑃21
𝑃22
𝑃23
So, does it all works everytime?
What will
others think?
Can I
do it?!
Should I
do it?
What will
others think?
Am I
prepared?
Can I
do it?!
Should I
do it?
What will
others think?
She’s not
gonna like
it
Am I
prepared?
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?
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?
http://picol.org/icon_library.php