ISSN 0973-533X Volume 1 Number 2 (2006) pp. -
© Research India Publications
http://www.ripublication.com/afm.htm
* This paper is an extended version of [11].
Review of Fixed Points for Fuzzy IF-Then Rules
∗)Khosro Soleimani1 and M. Mashinchi2
1Mathematicts Department, Engineering School,
Islamic Azad University of Najaf Abad Branch, Najaf Abad, Iran.
2Faculty of Mathematics and Computer Sciences, Shahid Bahonar University of Kerman, Kerman, Iran.
Abstract
In this paper we use an interpretation of fuzzy if-then rules to consider the notion of fixed points that are also called fix mundis in the literature invented by Fathi/Temme. We review some known results on fixed points of fuzzy if- then rules and add more new ones and bring up the notion of fixed points in fuzzy chaining syllogism and we obtain new and interesting conclusions.
Finally we bring up a new algorithm by using fixed points for optimization problems with fuzzy rule base constraints.
Keywords: Fuzzy If-Then Rules, Functional Operators, Fixed Points, Fix Mundis, Fuzzy Chaining Syllogism, Optimization.
1. Introduction
Fixed points for fuzzy if-then rules are studied by some authors (see the references of [1]). But this is recently that they are introduced as fix mundis to be applied in real applications by Fathi/Temme and et al. [1]. A fuzzy if-then rule is interpretable by a functional operator [2], by which the notion of a fixed point is introduced. Fixed points have a major approach for physical modeling in term of classical functional analysis. In fuzzy control there is no need to look for fixed points, but in other application areas of fuzzy logic there are a lot of examples to apply fixed points [1].
In this note we review some known results on fixed points where more new conclusions are added.
It is well known that fuzzy chaining syllogism inference is a very vital part of fuzzy logic [3] and hence in fuzzy logic applications. Therefore in this note we will bring up the notion of fixed points to fuzzy chaining syllogism inference and then we will prove some new results.
2. Interpretation of fuzzy IF-Then rule
Definition 1. Let F and G be fuzzy sets on U, i.e. F,G:U →[0,1]. A rule R such as if F then G, is called a fuzzy if-then rule.
Definition 2. Let F and G be fuzzy sets on U. The standard fuzzy relation S based on F and G is defined by
S(x,y)=min( F(x),G(y)).
Definition 3. Let R be a fuzzy if-then rule if F then G , and S be the standard fuzzy relation based on F and G. For an arbitrary input F′:U →[0,1] for R, we define the inferred output fuzzy set G′ by
. )},
) , ( ), ( {min(
sup )
(y F x S x y y U
G
U x
′ ∈
′ =
∈
Moreover, let ΦR :FP(U)→FP(U)be a functional operator such that ΦR(F′)=G′, where FP(U) is the set of all fuzzy sets on U. Then ΦR(F′) is called the compositional rule of inference (CRI) of R in standard form, for a given input F′.
Theorem 1 [1]. Let R be a fuzzy if-then rule and F′be in FP(U), and ΦR be a CRI of R. Suppose c∈[0,1] be such that c≤hgt(F∩G), where hgt(.) is the height of a fuzzy set A in FP(U) defined by hgt(F) sup{F(x)}
x∈U
= .
If H(x)=min(c,G(x)); x∈U, then H is a fixed point for ΦR, that is ΦR(H)=H. Definition 4. Let Fi and Gi; i=1,2,..,n; n≥2, be fuzzy sets on U, then
=
n n
nbesuchas if F thenG
R
G then F if as such be R
G then F if as such be R RB #
2 2
2
1 1
1
is called a fuzzy if-then rule base.
Definition 5. Let RB be a fuzzy if-then rule base. The functional operator
) ( ) (
,..., :
, 2
1R Rn FPU FPU
R →
Φ ,
ΦR1,R2,...,Rn(F′)=max(ΦR1(F′),...,ΦRn(F′)),
where ΦRi(F′)=Gi′; i=1,...,n, interprets the rule base RB based on CRI according to first inferred then aggregate (FITA) principle for a given input F′in FP(U).
Definition 6. Let RB be a fuzzy if-then rule base. The functional operator
( )
{
min ( ), ( , )}
sup ) )(
.., (
, 2
1 F y F x S x y
U x R
R
R n ′ = ′
Ψ ∈
where S(x,y)=max(S1(x,y),..,Sn(x,y)) and Si(x,y)=min(Fi(x),Gi(y)) i=1,2,..,n interprets the rule base RB based on CRI according to first aggregate then inferred (FATI) principle for a given input F′in FP(U).
Theorem 2 [1]. Let RB be a fuzzy if-then rule base, where n=2. If 1. ΦR1,R2 interprets RB based on the CRI according to FITA principle , 2. Let c1≤hgt(F1∩G1) and c2 ≤hgt(F2∩G2) where c1,c2∈[0,1],
3.Let c1 sup{min(F1(x),min(c2,G2(x)))}
x∈U
≥ , c2 sup{min(F2(x),min(c1,G1(x)))}
x∈U
≥ ,
4. H(x)=max(min(c1,G1(x),min(c2,G2(x))); x∈U,then H is a fixed point for ΦR1,R2, i.e.
. ) (
2 1,
H
R H
R =
Φ
Lemma 1. Let A , B are fuzzy sets on U, then the following equality holds
} ) ( sup , ) ( sup max{
))}
( , ) ( {max(
sup Ax B x Ax B x
U x U x U
x∈ = ∈ ∈
Proof: It is clear that U x x A x
A
U x
∈
∀
≤sup∈ ( ) )
( and B x B x x U
U x
∈
∀
≤sup∈ ( ) )
( then
U x x
B x
A x
B x A
U x U
x
∈
∀
≤max{sup∈ ( ), sup∈ ( )} )
) ( , ) (
max( .
Therefore A x B x A x B x x U
U x U
x U
x
∈
∀
≤ ∈ ∈
∈ {max( ( ), ( ))} max{sup ( ), sup ( )}
sup (1)
And therefore A x B x A x B x A x x U
U x
∈
∀
≥
∈ {max( ( ), ( ))}≥max{ ( ), ( )} ( )
sup and by the same
way
U x x
B x B x A x
B x A
U x
∈
∀
≥
∈ {max( ( ), ( ) )}≥ max{ ( ) , ( )} ( )
sup .
Thus supA(x) sup{max(A(x),B(x))}
U x U
x∈ ≤ ∈ and supB(x) sup{max(A(x),B(x))}
U x U
x∈ ≤ ∈ , therefore
sup{max( ( ) , ( ) )} max{sup ( ) , sup ( )}
x U x U x U
A x B x A x B x x U
∈ ≥ ∈ ∈ ∀ ∈ (2)
By (1) and (2) we have
U x x
B x
A x
B x A
U x U
x U
x
∈
∀
= ∈ ∈
∈ {max( ( ), ( ))} max{sup ( ), sup ( )}
sup .
Lemma 2. Let Fi,GiandHi;i=1,..,n be fuzzy sets on U, where H is fixed point for i rule i in Definition 4 and ci∈[0,1];i=1,..,n . If c Fi x Hj x j i
U x
i ≥ ∀ ≠
∈ {min( ( ), ( ))};
sup ,
then
i j y x S x H y
H j i
U x
i ≥ ∀ ≠
∈ {min( ( ), ( , ))};
sup )
( .
Proof : Suppose that c Fi x Hj x j i
U x
i ≥ ∀ ≠
∈
))};
( ), ( {min(
sup , then
. )]};
( ), , ( min[
{ sup )]}
( )), ( ), ( min[min(
{ sup
)]}
( )), ( ), ( min[min(
{ sup )) ( ))}, ( ), ( {min(
sup min(
)) ( , min(
i j x H y x S x
H y G x F
y G x H x F y
G x H x F y
G c
j i
U x j
i i U
x
i j i U
x i
j i U
x i
i
≠
∀
=
=
=
≥
∈
∈
∈
∈
Thus H y Hj x Si x y j i
U x
i ≥ ∀ ≠
∈ {min( ( ), ( , ))};
sup )
( . Theorem 3 [6]. Let RB be a fuzzy if-then rule base, according to Definition 4 and let
n i
x G c x
Hi( )=min( i, i( )); =1,2,.., . If
1. ΦR ,.,1 Rn interprets RB based on the CRI according to FITA principle, 2. ci ≤hgt(Fi∩Gi) where ci∈[0,1]; i=1,…,n .
3. c F x H x j i
j i U
x
i ∀ ≠
≥ ∈
; ) ( ), ( min
sup .
4. Let H(x)=max(H1(x),...,Hn(x)) x∈U.
Then H is a fixed point for ΦR ,..,1 Rn, i.e. ΦR1,..,Rn(H)=H. Proof: By Definition 5, we have
)) ( ),.., ( max(
) )(
( 1
1,..,
H H
y
H n
n R R
R
R = Φ Φ
Φ , (3)
where
( )
{
max min( ( ), ( , )),..,min( ( ), ( , ))}
sup
))}
, ( )), ( ),.., ( {min(max(
sup ))}
, ( ), ( {min(
sup ) )(
(
1
1
y x S x H y
x S x H
y x S x H x H y
x S x H y
H
i n i
U x
i n U
x i
U x Ri
∈
∈
∈
=
=
= Φ
By Lemma 1 we have
( ) ( )
( ) ( )
1
1 1
1
( )( ) max sup{min( ( ), ( , ))},.., sup{min( ( ), ( , ))}
sup{min ( ) , ( , ) },..., sup{min ( ) , ( , ) }, ( ), max sup{min ( ) , ( , ) },..., sup{min ( ) , ( , ) }
Ri
i n i
x U x U
i i i i
x U x U
i i n i
x U x U
H y H x S x y H x S x y
H x S x y H x S x y H y
H x S x y H x S x y
∈ ∈
∈ ∈ −
∈ + ∈
Φ =
=
.
(4)
By lemma 2 and (4) we have:
) ( ) )(
(H y Hi y
Ri =
Φ (5)
Therefore by (3) and (5) we have
).
( )) ( ,..., ) ( max(
)) ( ),.., ( max(
) )(
( 1
,.., 1
1 R H y R H R H H y Hn y H y
R n = Φ Φ n = =
Φ
Theorem 4 [6]. Let RB be a fuzzy if-then rule base, according to Definition 4 and let
n i
x G c x
Hi( )=min( i, i( )); =1,2,.., . If
1. ΨR ,.,1 Rn interprets RB based on the CRI according to FATI principle, 2. ci≤hgt(Fi∩Gi) where ci∈[0,1]; i=1,…,n .
3. c F x H x j i
j i U
x
i ∀ ≠
≥sup∈ min ( ), ( ) ; . 4. Let H(x)=max(H1(x),...,Hn(x)) x∈U.
Then H is a fixed point for ΨR ,..,1 Rn, i.e. ΨR1,..,Rn(H)=H. Proof : Let S(x,y)=max(S1(x,y),..,Sn(x,y)).
By Definition 6
))}
, ( )), ( ),.., ( {min(max(
sup ))}
, ( ), ( {min(
sup ) )(
( 1
1,.., H y H x S x y H x Hn x S x y
U x U
x R
R n
∈
∈ =
= Ψ
( )
{
1}
1
sup max min( ( ), ( , )),.., min( ( ), ( , ))
max sup{min( ( ), ( , ))},..,sup{min( ( ), ( , ))} .
n x U
n
x U x U
H x S x y H x S x y
H x S x y H x S x y
∈
∈ ∈
=
=
(6)
But
=
=
=
∈
∈
∈
∈
∈
))}
, ( ), ( {min(
sup
.., ))}, , ( ), ( {min(
sup )) max
, ( ), ( min(
)),.., , ( ), ( max min(
sup
))}
, ( ),.., , ( max(
), ( {min(
sup ))}
, ( ), ( {min(
sup
1 1
1
y x S x H
y x S x H y
x S x H
y x S x H
y x S y x S x
H y
x S x H
n i U
x
i U
x n
i i U
x
n i
U x i
U x
1 1
1
sup{min( ( ), ( , ))},..,sup{min( ( ), ( , ))}, ( ) max ,sup{min( ( ), ( , ))},.., sup{min( ( ), ( , ))}
i i i i
x U x U
i i i n
x U x U
H x S x y H x S x y H y
H x S x y H x S x y
∈ ∈ −
∈ + ∈
=
(7)
Since H (y) sup{min(Hi(x),Si(x,y))}.
U x
i = ∈ By (6) and (7) we will have
= Ψ
∈ −
∈
∈
∈
) ( ))}, , ( ), ( {min(
sup ))},.., , ( ), ( {min(
sup max ,
, ))}
, ( ), ( {min(
sup ))},.., , ( ), ( {min(
sup ), ( max max ) (
1 1
1 2
1 1
1,..,
y H y x S x H y
x S x H
y x S x H y
x S x H y
H H
n n
n U
x n
U x
n U
x U
x R
R n
#
) ( )) ( ),.., ( max(
) ( ))}, , ( ), ( {min(
sup ))},.., , ( ), ( {min(
sup max ,
, ))}
, ( ), ( {min(
sup ))},.., , ( ), ( {min(
sup ), ( max max
1
1 1
1 1
2 1
y H y H y H
y H y x S x H y
x S x H
y x S x H y
x S x H y
H
n
n n
n U
x n
U x
n U
x U
x
=
=
=
∈ −
∈
∈
∈
#
Figure 1: Fuzzy sets , c1 & c2
Example 1[1]. For the treatment of a patient the systolic value of blood pressure is of importance, especially it must not change. The doctor intends to give two drugs to cure other symptoms, but he has to take care for the (unwanted) side effects of these drugs on blood pressure.
In respect to side effects the drug producing company has given a rule for each drug:
R1: If drug 1 is applied and blood pressure is low then blood pressure will slightly increase.
R2: If drug 2 is applied and blood pressure is high then blood pressure will stabilize around a medium value.
The semantics of these fuzzy statements can be fixed by the following rules:
IF F1 THEN G1 IF F2 THEN G2
with: F1 = low G1 = low but slightly increased F2 = high G2 = around medium value
Now the doctor asks himself: "Which blood pressure should the patient have (or being brought to by other means) so that I can apply both drugs without danger (without change of blood pressure due to the drugs)?".
Assume, the fuzzy sets F1, G1, F2, G2 have the shape as drawn in Figure 1. By using Theorem 2 on the two rules he gets a lot of fix-mundis (the valid values for c1 and c2 are within the black area) and can check whether his patient fulfills the condition. If yes, he can apply the drugs without danger, if not, he has to change the blood pressure to a suitable value before treatment or must choose combination of two different drugs. Examples for fix-mundis are given in Figure 2.
G1 F2
G2 c2
c1
C1=1 & c2=0.6 C1=0.8 & c2=0.4 c1=0.4 & c2=0.6 c1=c2=0.6
Figure 2: Fixed points
F1
3. Fixed points for Chaining syllogism
By a fuzzy chaining syllogism, we mean the following If F then G, (I) If G then K (II) If F then K (III)
where F,G, K are fuzzy sets on U. It is shown that this reasoning holds [2].
Notation 1. Assume H1,H2 andH3 are fixed points for fuzzy if-then rules I , II and III, respectively, i.e.
)), ( , min(
)
(x c G x
Hi = i H2(x)=min(c2,K(x)) and H3(x)=min(c3,K(x))where )
1 hgt(F G
c = ∩ , c2=hgt(G∩K) and c3=hgt(F∩K).
Theorem 5 [4]. Consider the fuzzy chaining syllogism, where G is normalized.
Suppose the assumptions of Notation 1 hold. Assume H1is the input for rule I, then for the output K′ of rule II, we have
1. If c1≤c2, then K′⊆H2, 2. If c1≥c2, then H2 ⊆K′, 3. If c1=c2, then K′=H2.
Proof : Since the fixed point H is an input for rule I, we have 1 I: if H1 then H1,
II: if H1 then K′, Now according to Definition 2, we have
))) ( , min(
, 1 min(
} )) ( , min(
), ( sup min{
)}
)) ( , min(
), ( {min(
sup ) )) ( , min(
)), ( ), ( {min(min(
sup
)}
)) ( ), ( min(
)), ( , {min(min(
sup )}
) ( ), ( min(
), ( {min(
sup ) (
1 1
1 1
1 1
y K c y
K c x G
y K c x G y
K c x
G x G
y K x G x
G c y
K x G x
H y
K
U x
U x U
x
U x U
x
=
=
=
=
==
′ =
∈
∈
∈
∈
∈
Thus K′(y)=min(c1,K(y)) . Hence
1. If c1≤c2, then min(c1,K(y))≤min(c2,K(y)). Therefore
2 2( ); ..
)
(y H y ie K H
K′ ≤ ′⊆
2. If c1≥c2, then min(c1,K(y))≥min(c2,K(y)). Therefore
2 2( ); . .
)
(y H y ie K H
K′ ≥ ′⊇ ♦
Theorem 6 [4]. Suppose the assumptions of Notation1 hold. Let H be the input for 1 rule III and Kˆ be the output for the input H for rule III. Then 1
1. if c1≤c2≤c3 , then Kˆ ⊆H3, 2. if c1≥c2≥c3 , then Kˆ ⊇H3. Proof :
)) ( ), , min(min(
))) ( , min(
, min(
} )) ( , min(
), sup{
min{
) )) ( , min(
)), ( ), ( {min(min(
sup
)}
)) ( ), ( min(
)), ( , {min(min(
sup )}
) ( ), ( min(
), ( {min(
sup ) ˆ(
2 1 1
2
1 1
1 1
y K c c y
K c c
y K c G
F y
K c x
F x G
y K x F x
G c y
K x F x
H y
K
U x U
x
U x U
x
=
=
∩
=
=
=
=
∈
∈
∈
∈
Thus ˆ( ) min(min( , ), ( ))
2
1 c K y
c y
K = . Hence
1. If c1 ≤c2 ≤c3
then ˆ( ) min(min( , ), ( )) min( , ( )) min( , ( )).
3 1
2
1 c K y c K y c K y
c y
K = = ≤
ThereforeKˆ(y)≤H (y) y∈U
3 i.e. G ⊆ H 1 . 2. If c1 ≥c2 ≥c3,
then ˆ( ) min(min( , ), ( )) min( ( )) min( , ( )) .
3 2
2
1 c G y c G y c G y
c y
K = = ≥
Therefore Kˆ(y)≥H (y) y∈U
3 i.e. Kˆ ⊇ H3 . Theorem 7 [4]. Suppose the assumptions of Notation 1 hold. Let G′ be the output of the input H for rule I. Then 2
1. if c3 ≤c1≤c2,then G′⊆H1, 2. if c3 ≥c2 ≥c1,then G′⊇H1. Proof : Note that we have
. ))) ( , min(
), , min(
min(
))) ( , min(
, min(
))) ( , min(
), (
min(
))) ( , min(
))}, ( ), ( {min(
sup min(
) (
3 2
3 2
3
2 2
y G c c
c y
G c c
y G c K
F hgt y
G c x
K x F y
G
U x
=
=
∩
=
′ =
∈
1) if c3 ≤c1 ≤c2,then
. )) ( , min(
)) ( , min(
))) ( , min(
, min(
))) ( , min(
), , min(
min(
)
(y c3 c2 c3 G y c3 c3 G y c3 G y c1 G y
G′ = = = ≤
Therefore, G′(y)≤H1(y). Hence G′⊆H1. 2) if c3 ≥c2 ≥c1,then
)).
( , min(
)) ( , min(
))) ( , min(
, min(
))) ( , min(
), , min(min(
))) ( , min(
, min(
))) ( , min(
), , min(
min(
) (
1 3
2 2
2 2
3 3
2 3
2 3
y G c y
G c y
G c c
y G c c
c y
G c c
y G c c
c y
G
≥
=
=
=
=
′ =
Therefore G′(y)≥H1(y).Hence G′⊇H1. Theorem 8 [4]. Suppose the assumptions of Notation 1 hold. Let G′ be input for rule II and K′ be its output. Then
1. if c3 ≤c2,then K′=H3, 2. if c3 ≥c2,then K′= H2.
proof : Take G′, witch is obtained in Theorem 7, as an input for rule II, then
))) ( , min(
), , min(
min(
)
(y c3 c2 c3 G y
G′ = and
( )
( )
{ }
[ ]
{ }
{
min[min(min( , ( )),min( , ( ))),min( , ( ))]}
sup
)) ( )), ( , min(min(
, )) ( ), , min(min(
min sup
)) ( ), ( min(
, ))) ( , min(
), , min(min(
min sup
} )) ( ), ( min(
), ( {min sup ) (
3 2
3
3 3
2
3 3
2
y G c z
K c z
K<