preference relation in pliant system dombi/dr university of szeged department of informatics...

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Preference relation in pliant system

http://www.inf.u-szeged.hu/~dombi/dr

University of SzegedDepartment of Informatics

Pamplona 2009

2

17-Sept-2009

Elements of pliant system

1. Conjunction, disjunction, negation

2. Aggregation

3. Preference relation

4. Distending function

5. Distending function as preference

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17-Sept-2009

Conjunction, disjunction, negation

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17-Sept-2009

Conjunctive and disjunctive operator

We shall be looking for the general form of c(x,y) and

d(x,y) :

1. is continuous

2. Strict monotonous increasing

3. Compatible with the two valued logic

4. Associative

5. Archimedian

]1,0[]1,0[]1,0[: c

0and'if)',(),( xyyyxcyxc

1)0,1(0)1,0(

1)1,1(0)0,0(

cc

cc

zyxcczycxc ,,,,

.),( xxxc

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17-Sept-2009

Conjunctive and disjunctive operator

Theorem: (Aczél)

If with u and v, h(u,v) also always lies in a given

(possibly infinite) interval and h(u,v) is reducible on

both sides, then

.)()(),( 1 yfxffyxh

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17-Sept-2009

Operators and DeMorgan law

Let’s generalize tha conjunctive and disjunctive

operators and let:

where

,)(),;...;,;,(),(1

12211

i

n

icicnn xfwfxwxwxwcxwc

,)(),;...;,;,(),(1

12211

i

n

ididnn xfwfxwxwxwdxwd

.0iw

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17-Sept-2009

Negation

Definition:

(x) is a negation iff satisfies the following

conditions:

1. (x) is continuous

2. Boundary conditions are and

3. Monotonicity: for

4. Involutivness:

1,01,0:

1)0( 0)1(

)()( yx yx

xx ))((

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17-Sept-2009

Negation

Other properties:

* fix point of the negation, where

- The decision value:

)(

0

0

)(

)(

xthenx

xthenx

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Negation

- On Figure there are some negation functions with

different * and values:

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17-Sept-2009


Definition:

The DeMorgan law for general conjunctive and

disjunctive operator is:

where (x) is the negation function.

)),,;...;,;,(())(,);...;(,);(,( 22112211 nnnn xwxwxwdxwxwxwc

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Theorem: (DeMorgan law)

The generalized DeMorgan law is valid iff

where

,)(1

)( 1

xfa

fx dc

.0a

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17-Sept-2009

Negation and DeMorgan law

Parametrical form of the negation is:

.)()(

)()( 1

xff

ffx d

d

cc

,)()(

)()( 1

xff

ffx c

c

dd

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17-Sept-2009

Representation theorem of negation

For all given (x) there exist an f(x) such that

where k(x) is a strictly decreasing function with the

property

and f is the generator function of a conjunctive, or

disjunctive operator.----------------------------------------------------------------------Trillas’ result:

,))(()( 1 xfkfx

)()( 1 xkxk

)(1)( 1 xffx

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17-Sept-2009

Operator with various negations

Theorem:

c(x,y) and d(x,y) build DeMorgan system for

where if and only if

)(x

)1,0(

.1)()( xfxf dc

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17-Sept-2009

Multiplicative pliant system

Definition:

If k(x) = 1/x, i.e.

and then we call the generated

connectives multiplicative pliant system.

,1)()( xfxf dc

)()( xfxf

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Theorem:

The general form of the multiplicative pliant system is

where f(x) is the generator function of either the

conjunctive or the disjunctive operator.

11 )()(),( yfxffyx

)(

)()()( 0

1, 0 xf

fffx

,)(

)()(

21

xf

ffx

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If f = fc , then depending on thevalue of the

operator is

),(),( yxcyx

),(),( yxdyx 0

0

),min(),( yxyx

),max(),( yxyx

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17-Sept-2009

Dombi operator system

Let choose then we get

x

xxf

1)(

1

1

11

1)(

n

i i

i

xx

xc

1

1

11

1)(

n

i i

i

xx

xd

xx

x

111

1

1)(

0

0, 0

xx

x

11

1

1)( 2

0

0

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17-Sept-2009

Aggregation

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17-Sept-2009

Aggregation

Let us consider a set of objects .Let us

characterize every object with a number m of its

properties ,where and i = 1,…,n.

Thus, if the aggregative operator as denoted as

, for a decision level we have

),...,,( 21 nOOO

),...,,(21 miii xxx )1,0(ix

),...,( 1 nxxa

.)(),...,,(|

,),...,,(|

21

21

2,

1,

m

m

iiii

iiii

xxxaOC

xxxaOC

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17-Sept-2009

Aggregation

Let us next substitute every property by its antithetic

one (in the following its negative form and carry

out division into classes at the level:

)(jix

.)())(),...,((|

,))(),...,((|

1

1

2,

1,

m

m

iii

iii

xxaOC

xxaOC

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17-Sept-2009

Aggregative operators and representable uninorms

Definition: (of correct decision formation)

The condition of correct formation is thus

Theorem:

It is necessary and sufficient condition of the aggre-

gative operator satisfying correct decision formation

that

should hold.

., 1,2,2,1, CCCC

)(),(),( yxayxa

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17-Sept-2009


Definition:

An aggregative operator is a strictly increasing

function with the properties:

1. Continuous on

2. Boundary conditions are and

3. Associativity:

4. There exists a strong negation such that

(self DeMorgan identity)

1,01,0: 2 a

0,1,1,0\1,0 2

00,0 a 11,1 a

)),,(()),(,( zyxaazyaxa

))(),((),( yxayxa

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17-Sept-2009


Definition:

A uninorm U is a mapping having the

following properties :

1. Commutativity:

2. Monotonicity: if and

3. Associativity:

4. Neutral element:

1,01,0: 2 U

),(),( xyUyxU

),(),( 2211 yxUyxU 21 xx 21 yy

zyxUUzyUxU ),,(),(,

1,0 1,0x xxU ),(

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17-Sept-2009


Theorem:

Let be a function. It is an aggregative

operator if and only if there exists a continuous and

strictly monotone function with

such that for all

1,01,01,0: a

,1,0:g ,0)( g

1,0 21,0),( yx

.)()(),( 1 ygxggyxa

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17-Sept-2009

Aggregation

Theorem:

It holds that:

Theorem:

It holds for the aggregative operator that

1.

2.

3.

4.

.)(, xxa

1,0,)(, xxxxa

xxa ,

0if11, xxa

1if00, xxa

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17-Sept-2009

Aggregation

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17-Sept-2009

The neutral value

Theorem: (Additive form of negations)

Let be a continuous function, then the

following are equivalent:

1. is a negation with neutral value *.

2. There exists a continuous and strictly monotone

function and such that for

all

1,01,0:

,1,0:g 1,0

1,0x

.)()(2)( 1 xgggx

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17-Sept-2009

Conjunctive, disjunctive and aggregative operators

Definition:

We will use the term conjunctive operator for strict,

continuous t-norms, and disjunctive operator for

strict, continuous t-conorms. The expression logical

operators will refer to both of them.

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17-Sept-2009


Theorem:

The following are equivalent:

1. is a logical operator.

2. is an aggregative aoperator.

)()(),( 1 yfxffyx

)()(),( 1 yfxffyxa

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17-Sept-2009

Aggregation and Pan operators

Pan operator:

Theorem:

Let c and d be a conjunctive and a disjunctive opera-

tor with additive generator functions fc and fd .

Suppose their corresponding negations are equivalent

(i.e. ), denoted by ((*) = * ). The

three connectives c, d and form a De Morgan triplet

if and only if fc(x)fd(x) = 1 .

)()(),(

)()(),(1

1

yfxffyxa

yfxffyxa

dddd

cccc

0),()( kxfxf kcd

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17-Sept-2009


Definition:

Let f be the additive generator of a logical operator.

The aggregative operator is called

the corresponding aggregative operator of the

conjunctive or disjunctive operator, and vice versa.

)()(),( 1 yfxffyxa

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17-Sept-2009


Multiplicative form of negations:

The function is a negation with neutral

value if and only if

where f is a generator function of a logical operator.

1,01,0:

,)(

)()(

21

xf

ffx

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17-Sept-2009

Pliant operators

Theorem:

Let c and d be a conjunctive and disjunctive operator

with additive generator functions fc and fd . Suppose

their corresponding negations are equivalent ( i.e.

), denoted by . The three

connectives c, d and n form a DeMorgan triplet if and

only if

0),()( kxfxf kcd

)(n

.1k

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17-Sept-2009

Unary operators

The general form of the unary operator:

Special case of the function:

if =1 and > 0 then concentration operator

if =1 and < 0 then dilutor operator

if =-1 then negation operator

if f(0)= f() = 1 then sharpness operator

)(

)()()( 0

1)(

f

xfxffx

)()( x

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17-Sept-2009

Pliant operators

The Dombi operator case:

1

1

11

1)(

n

i i

i

xx

xc

1

1

11

1)(

n

i i

i

xx

xd

n

i i

i

xx

xa

1

11

1)(

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17-Sept-2009

Pliant operators

Modifier:

if =1 and > 0 then concentration operator

if =1 and < 0 then dilutor operator

if =-1 then negation operator

Negation:

xx

x

111

1

1)(

0

0

xx

x1

11

1

1)(

0

0

)(

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17-Sept-2009

Preference relation

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17-Sept-2009

Preference operator on the [0,1] interval

We define the preference function in the following

way:

yxayxp

yxayxp

),(),(

),(),(

,,

,,

00

00

)(truth),( yxyxp

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17-Sept-2009

Properties of preference operator

Theorem:

Let the pliant operations:

and the preference operator

)(

1)(

)()(

)()(

)()(

1

1

1

1

11

1

1

xffx

xffxa

xffxa

xffxa

n

ii

ww

n

i

ni

n

ii

i

.)(

)(),( 1

xf

yffyxp

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17-Sept-2009


The following properties hold for the preference rela-tions:

I. Preference properties

1. Continuity:

2. Monotonicity:

3.Compatibility conditions:

continuous)1,0()1,0()1,0(: p

),(),(thenif)

),(),(thenif)

2121

2121

yxpyxpxxb

yxpyxpyya

0)0,1(1)1,0( pp

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17-Sept-2009


4. Boundary conditions: if then

5. Neutrality:

6. Preference property:

0)0,(

1)1,(

1),0(

0),1(

xp

xp

xp

xp

0),( xxp

),(thenif)

),(thenif)

yxpyxb

yxpyxa

)1,0(x

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17-Sept-2009


7. Bisymmetric property:

8. Common basis property: for all z

II. Preference and negation operator

1.

2.

3.

),(),,(),(),,( 21212211 yypxxppyxpyxpp

),(),,(),( yzpxzppyxp

),(),( xypyxp

)(),(),( xypyxp

)(),(),( yxpyxp

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17-Sept-2009


III. Preference and aggregation

1. Transitivity with aggregation:

2. Common basis principles

3. Inverse property:

4. Neutrality:

),(),(),,( zxpzypyxpa

),(),,(),( xzpzypayxp

),(, zypxay

),(),,(0 xypyxpa

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17-Sept-2009


5. Exchangeability:

6. Preference of aggregation:

yzxpazyaxp ),,(),(,

),(),,(),(),,(

:casevariable2

),,(),...,,(),,()(),()

22112211

2211

yxpyxpayxayxap

yxpyxpyxpayaxapa nn

),,(),...,,(),,()(),() 2211 nnwww yxpyxpyxpayaxapb

),(),,()),,(

:casevariable2

),(),...,,(),,(),() 2211

zypzxpazyxap

yxpyxpyxpaayxapc nnww

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17-Sept-2009


IV. Threshold property

1. Threshold transitivity:

p(x,y) is threshold transitiv if:

2. Strong completeness:

3. Antisymmetricity:

000 ),(then),(and),( zxpzypyxp

000 ),(or),(or),( yxpyxpyxp

00

00

),(),,(),(),,(

),(),,(),(),,(

xypyxpdxypyxpd

xypyxpcxypyxpc

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17-Sept-2009

Preference and multicriteria decision making

We can express the preference relation in additive

form:

where g(x)=ln(f(x)) .

In multicriteria decision the preference is

)(

)(),(),( 1

xf

yffyxayxp

)1())(ln())(ln(1 xfyfef

)()(1 xgygg

)3(),(

)2()()(),(

xyyxpor

xgygyxp

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17-Sept-2009

Preference and multicriteria decision making

In pliant concept and so (1) and (2) are

the same. Most cases in the framework of multicriteria

decision (3) are used. We can approximate (3) using

Rolle theorem: i.e.

Substituting it into (1)

where

i.e. the preference depends on y and x .

)()( 1 xgx

],[)()(

)(' yxxy

xgygg

)())(('),( 1 xyxyggyxp

)(')( 1 xggx

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17-Sept-2009

Distending function

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17-Sept-2009

Distending function instead of membership function

Let choose an often used one the term “old”. The same

example exist in Zadeh’s seminal paper . We suppose

now that the term “old” depends only on age, and we

do not care that most polar terms are always context

dependant i.e. old professor is defined in an other

domain than old student. In classical logic we have to

fix a dividing line, in our case let it be 63 years (a=63).

If somebody is older than 63 years then he/she

belongs to the class (set) of old people, otherwise does

not.

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17-Sept-2009


We can write this in an inequality form, using a

characteristic function:

The expression a<x is equivalent with the expression

0 < x-a , so the above form could be written as:

xaif

xaifxa

0

1)(

axif

axifax

00

01)(

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17-Sept-2009


Generaly, on the left side of the inequality could beany g(x) function.

In the pliant concept we introduce the distending

function. We will use the notation

We can generalize this in the following way:

)(00

)(01))((

xgif

xgifxg

.)0()( Rxxtruthx

.)(0))(( nRxxgtruthxg

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17-Sept-2009

General form of the distending function

Let start with the aggregation concept. The weighted

aggregation operator is:

where xi are the distending values and f is

the generator function of the logical operator.

Intuitively aggregation is a weighted average of the

values,

The following theorem gives the exact description of

n

ii

wnw xffxxxa i

1

121 )(),...,,(

)( ii tx

.1

n

iiitwt

).( it

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17-Sept-2009

Distending function

Theorem:

Using the aggregation

if and only if

----------------------------------------------------------------------

Dombi operator case:

.)(and)()(1

1ii

n

ii

ww txxffxa i

n

iiinw twttta

121 )(),...,(),(

.)()( 1 teft

)(0

1)( )()(truth)( axa effxax

)()(

1

1)(

axa ex

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17-Sept-2009

Sigmoid function and logistic regression

The sigmoid function has the following properties.

The sigmoid function is able to modelize inequality.

0if2

1)(

0if2

1)(

0if2

1)(

,1

1)(

xx

xx

xx

ex

x

0)(if2

1))((

0)(if2

1))((

0)(if2

1))((

,1

1))((

)(

xgxg

xgxg

xgxg

exg

xg

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17-Sept-2009

Distending function as preference

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17-Sept-2009

Distending function as preferences on the real line

The distending function has the following form:

We can define a preference function:

)(1)( )( axa efx

RefyxP yx )(1)( ),(

)(truth xy

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17-Sept-2009


The following properties hold for .

I. Preference properties

1. Continuity:

2. Monotonicity:

3. Limes property:

continuous)1,0(),(),(: P

),(),(thenif)

),(),(thenif)

2)(

1)(

21

2)(

1)(

21

yxPyxPxxb

yxPyxPyya

0),(1),( )()( PP

),()( yxP

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17-Sept-2009


4. Boundary conditions:

5. Neutrality:

6. Preference property:

0),(

1),(

1),(

0),(

)(

)(

)(

)(

xP

xP

xP

xP

2

1),( 0

)( xxP

0)(

0)(

),(thenif)

),(thenif)

yxPyxb

yxPyxa

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17-Sept-2009


7. Translation property:

II. Preference and negation operator

III. Preference and aggregation

1. Transitivity with aggregation:

2. Common basis principles

),(),( )()( yxPzyzxP

),(),( )()( xyPyxP

),(),(),,( )()()( zxPzyPyxPa

),(),,(),( )()()( yzPzxPayxP

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17-Sept-2009


4. Neutrality:

IV. Threshold property

1. Threshold transitivity:

P(λ) is threshold transitiv if:

2. Strongly complete:

),(),,( )()(0 xyPyxPa

0)(

0)(

0)( ),(then),(and),( zxPzyPyxP

0)(

0)(

0)( ),(or),(or),( yxPyxPyxP

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17-Sept-2009


3. Antisymmetric:

0

)()(0

)()(

0)()(

0)()(

),(),,(),(),,(

),(),,(),(),,(

xyPyxPdxyPyxPd

xyPyxPcxyPyxPc

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Animation

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Animation

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Animation

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Animation

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Animation

Animation17-Sept-2009

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Thank you for your attention!

17-Sept-2009

preference relation in pliant system dombi/dr university of szeged department of informatics...

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