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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
3
17-Sept-2009
Conjunction, disjunction, negation
4
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
5
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
6
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
7
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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17-Sept-2009
Negation
- On Figure there are some negation functions with
different * and values:
10
17-Sept-2009
Operators and DeMorgan law
Definition:
The DeMorgan law for general conjunctive and
disjunctive operator is:
where (x) is the negation function.
)),,;...;,;,(())(,);...;(,);(,( 22112211 nnnn xwxwxwdxwxwxwc
11
17-Sept-2009
Operators and DeMorgan law
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
13
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
15
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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17-Sept-2009
Multiplicative pliant system
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
17
17-Sept-2009
Multiplicative pliant system
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
19
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Aggregation
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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
21
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
22
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
23
17-Sept-2009
Aggregative operators and representable uninorms
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
24
17-Sept-2009
Aggregative operators and representable uninorms
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 ),(
25
17-Sept-2009
Aggregative operators and representable uninorms
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
26
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
27
17-Sept-2009
Aggregation
28
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
29
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.
30
17-Sept-2009
Conjunctive, disjunctive and aggregative operators
Theorem:
The following are equivalent:
1. is a logical operator.
2. is an aggregative aoperator.
)()(),( 1 yfxffyx
)()(),( 1 yfxffyxa
31
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
32
17-Sept-2009
Conjunctive, disjunctive and aggregative operators
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
33
17-Sept-2009
Conjunctive, disjunctive and aggregative operators
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
34
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
35
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
36
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)(
37
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
)(
38
17-Sept-2009
Preference relation
39
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
40
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
41
17-Sept-2009
Properties of preference operator
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
42
17-Sept-2009
Properties of preference operator
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
43
17-Sept-2009
Properties of preference operator
7. Bisymmetric property:
8. Common basis property: for all z
II. Preference and negation operator
1.
2.
3.
),(),,(),(),,( 21212211 yypxxppyxpyxpp
),(),,(),( yzpxzppyxp
),(),( xypyxp
)(),(),( xypyxp
)(),(),( yxpyxp
44
17-Sept-2009
Properties of preference operator
III. Preference and aggregation
1. Transitivity with aggregation:
2. Common basis principles
3. Inverse property:
4. Neutrality:
),(),(),,( zxpzypyxpa
),(),,(),( xzpzypayxp
),(, zypxay
),(),,(0 xypyxpa
45
17-Sept-2009
Properties of preference operator
5. Exchangeability:
6. Preference of aggregation:
yzxpazyaxp ),,(),(,
),(),,(),(),,(
:casevariable2
),,(),...,,(),,()(),()
22112211
2211
yxpyxpayxayxap
yxpyxpyxpayaxapa nn
),,(),...,,(),,()(),() 2211 nnwww yxpyxpyxpayaxapb
),(),,()),,(
:casevariable2
),(),...,,(),,(),() 2211
zypzxpazyxap
yxpyxpyxpaayxapc nnww
46
17-Sept-2009
Properties of preference operator
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
47
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
48
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
49
17-Sept-2009
Distending function
50
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.
51
17-Sept-2009
Distending function instead of membership function
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)(
52
17-Sept-2009
Distending function instead of membership function
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
53
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
54
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
55
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
56
17-Sept-2009
Distending function as preference
57
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
58
17-Sept-2009
Distending function as preferences on the real line
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
59
17-Sept-2009
Distending function as preferences on the real line
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
60
17-Sept-2009
Distending function as preferences on the real line
7. Translation property:
II. Preference and negation operator
III. Preference and aggregation
1. Transitivity with aggregation:
2. Common basis principles
),(),( )()( yxPzyzxP
),(),( )()( xyPyxP
),(),(),,( )()()( zxPzyPyxPa
),(),,(),( )()()( yzPzxPayxP
61
17-Sept-2009
Distending function as preferences on the real line
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
62
17-Sept-2009
Distending function as preferences on the real line
3. Antisymmetric:
0
)()(0
)()(
0)()(
0)()(
),(),,(),(),,(
),(),,(),(),,(
xyPyxPdxyPyxPd
xyPyxPcxyPyxPc
63
17-Sept-2009
Animation
64
17-Sept-2009
Animation
65
17-Sept-2009
Animation
66
17-Sept-2009
Animation
67
17-Sept-2009
Animation
Animation17-Sept-2009
69
Thank you for your attention!
17-Sept-2009
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