Social Relationship and Decision-Making explained by Fuzzy Logic

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

“

Relationship

”

“

Social

Relationship

”

Relationship

What we see

What we really see

What we think about others

Thinking and acting

Profiling

Social Relationship

Modeling “The Basics”

Logic Mapping

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

Valid Statement P1 ∧ P2 ∧ ... ∧ Pn → Q

Conditional Proposition P1 ∧ P2 ∧ ... ∧ Pn → Q

Conditional Proposition P1 ∧ P2 ∧ ... ∧ Pn → Q

Valid 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)


𝑥1

𝑥2

𝑃11

ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)

𝑃21


(𝑥0, 𝑦0)

𝑥1

𝑥2

𝑃11

ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)

𝑃21


𝑥1

𝑥2

𝑃11

ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)

𝑃21

𝑃12

𝑃22


𝑥1

𝑥2

𝑃11

ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)

𝑃21

𝑃12

𝑃22

𝑃13

𝑃23


Multi-vectorial,

Social human description

𝑥1

𝑥2

𝑃11

ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)

𝑃21

𝑃12

𝑃22

𝑃13

𝑃23


𝑥1

𝑥2

𝑃11

ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)

𝑃21

𝑃12

𝑃22

𝑃13

𝑃23

𝑑12𝑥1


𝑥1

𝑥2

𝑃11

ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)

𝑃21

𝑃12

𝑃22

𝑃13

𝑃23

𝑑12𝑥1 𝑑23𝑥1

𝑑23𝑥2 𝑑13𝑥2


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


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

Fuzzy Sets flexible; in order to accommodate small social differences

Singleton

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)


𝑥1

𝑥2

𝑃2

𝑃1

ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)


𝑥1

𝑥2

𝑃2

𝑃1

ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)

Source of uncertainty

and imprecision


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)


𝑃21

𝑃12

𝑃22

𝑃13

𝑃23

𝑥1

𝑥2

𝑃11

ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)

𝑃21

𝑃12

𝑃22

𝑃13

𝑃23


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


𝑥1

𝑥2

𝑃11

ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)

𝑃21

𝑃12

𝑃22

𝑃13

𝑃23

𝑃𝑖: individual 𝑖


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



Decision Making in fuzzy environment

D = G ∩ C

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


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


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


0

1

120

Constraint, 𝑃12

Constraint, 𝑃12


Mem

bers

hip

funct

ion, G

, C, a

nd G

and C


0

1

120

Constraint, 𝑃12

Goal, 𝑃11

Goal, 𝑃11 Constraint, 𝑃1

2


Mem

bers

hip

funct

ion, G

, C, a

nd G

and C


0

1

120

Constraint, 𝑃12

Goal, 𝑃11

Decision by Defuzzification


2

Decision by

Defuzzification


Mem

bers

hip

funct

ion, G

, C, a

nd G

and C


0

1

120

Constraint, 𝑃12

Goal, 𝑃11

Decision by Defuzzification


2

Decision by

Defuzzification

D = G ∩ C

𝑃𝑗𝑖 = 𝐺𝑗

𝑖


𝑖

j-th propositions, 𝑃𝑗


𝑖

i-th the individual, 𝑃𝑖


𝑖

is related to individual goal, 𝐺𝑖

𝑥1

𝑥2

𝑃11

ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)

𝑃21

𝑃12

𝑃22

𝑃13

𝑃23



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


𝑥1

𝑥2

𝑃11

ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)

𝑃21

𝑃12

𝑃22

𝑃13

𝑃23


𝑃11 = 𝐺1

1 = 𝐶12 = 𝐶1

3


𝑥1

𝑥2

𝑃11

ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)

𝑃21

𝑃12

𝑃22

𝑃13

𝑃23

𝐷𝑃1,𝑃2,𝑥1 = 𝐺1

1 ∩ 𝐶12= 𝑃1

1 ∧ 𝑃12 ,


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


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


𝑥1

𝑥2

𝑃11

ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)

𝑃12 𝑃1

3


𝑃21

𝑃22

𝑃23

𝑥1

𝑥2

𝑃11

ߤ 𝑃1𝑥𝑃2 (𝑥1, 𝑥2)

𝑃12 𝑃1

3


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

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?

http://picol.org/icon_library.php

@ZGTRShaker

Thx

Social Relationship and Decision-Making explained by Fuzzy Logic

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