Restrictions imposed by the fuzzy extension of ... - Semantic Scholar
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Journal of Intelligent & Fuzzy Systems 11 (2001) 9–22 IOS Press
Restrictions imposed by the fuzzy extension of relations and functions Babak N. Araabia,∗, Nasser Kehtarnavaz b and Caro Lucas a a
b
Department of Electrical and Computer Engineering, University of Tehran, Tehran 14395, Iran Department of Electrical Engineering, University of Texas at Dallas, Richardson, TX 75083, USA
Abstract. The fuzzy extension principle has been widely used to extend the domain of mathematical functions and relations from elements of a referential set to fuzzy subsets of that referential set. However, there are restrictions associated with the fuzzy extension principle. This paper addresses the question how restricted is the family of “fuzzy set to fuzzy set mappings” obtained by the fuzzy extension of non-fuzzy functions and relations, as compared to the general family of all possible fuzzy set to fuzzy set mappings. A theorem is presented, with the necessary and sufficient conditions, to determine this restriction for the fuzzy extension of non-fuzzy relations and point-valued functions. It is shown that the fuzzy extension principle would impose a restriction on extended fuzzy set to fuzzy set mappings, which is similar to the linear restriction for point-valued functions. Moreover, the extension of mappings from a set-valued domain to a fuzzy set-valued domain is discussed. It is shown that this extension is well-behaved only for those mappings which preserve subsethood order. Another theorem with the necessary and sufficient condition has been proved to determine the imposed restriction during the fuzzy extension of subsethood order preserving set to set mappings. The two extensions have been compared and several examples are provided. Keywords: Fuzzy extension principle, set to set mapping, fuzzy set to fuzzy set mapping, fuzzy relational equation, subsethood order preserving mapping
1. Introduction The Fuzzy Extension Principle (FEP) was introduced by Zadeh [14] to extend the domain of crisp mathematical functions and relations from the elements of a referential set to fuzzy subsets of that referential set. The FEP has been widely used as an essential tool to fuzzify classical point-valued concepts and theories. The scope of applications of the FEP is quite vast, since it can fuzzify point-valued functions, and functions virtually can be found anywhere in engineering, science, and mathematics. The FEP has been utilized to extend point-valued concepts in fuzzy arithmetic [5], fuzzy control [3], and fuzzy decision making [16], as well as more abstract areas like fuzzy topology [7], and fuzzy metric spaces [4]. ∗ Corresponding
author: Dr. N. Kehtarnavaz, Department of Electrical Engineering, EC33, University of Texas at Dallas, PO Box 830688, Richardson, TX 75083-0688, USA. Tel.: +1 972 883 6838; Fax: +1 972 883 2710; E-mail: [email protected]. 1064-1246/01/$8.00 2001 – IOS Press. All rights reserved
Before starting with the definition of the FEP, the precise meaning of few expressions and notations used throughout the paper are in order: Let U and V be two finite referential sets. Sets of all crisp subsets of U and V are indicated by P U and PV , respectively, while sets of all fuzzy subsets of U and V are denoted by P˜U and P˜V . The domain and co-domain of mappings are asserted for the sake of clarity of theorems and discussions, that is, point to point mapping (PP mapping) is a crisp function from U to V ; set to set mapping (SS mapping) is a crisp function from P U to PV , and fuzzy set to fuzzy set mapping (FF mapping) is a crisp function from P˜U to P˜V . The meaning of point to set and set to point mappings should be clear by analogy. A relation is a crisp subset of U ×V , that is, a member of PU×V , and a fuzzy relation is a fuzzy subset of U × V , that is, a member of P˜U×V . PP mappings and relations are represented with small and capital italic letters, respectively (e.g. g and G). SS mappings are
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B.N. Araabi et al. / Restrictions imposed by the fuzzy extension of relations and functions
denoted by a bar over the mappings names (e.g. g), and FF mappings and fuzzy relations are indicated by a tilde over the mappings and relations names, respectively ˜ (e.g. g˜ and G). A fuzzy set A ∈ P˜U is defined by its membership function µA : U → [0, 1]. The α-cut of the fuzzy set A is denoted by Aα = {u|µA (u) α}, a ∈ [0, 1]. For a fuzzy set A ∈ P˜U and for a ∈ [0, 1], the scalar multiplication α · A ∈ P˜U is defined by µα·A (u) = min{α, µA (u)}
∀u ∈ U
(1)
The resolution identity [14] is a representation for a fuzzy set A ∈ P˜U based on its α-cuts as follows A= ∪
0α1
α · Aα
(2)
For two fuzzy sets A, B ∈ P˜U , the union of A and B is denoted by A ∪ B, where µA∪B (u) = max{µA (u), µB (u)} ∀u ∈ U
(3)
˜ ∈ P˜U×V , then A ◦ G ˜ ∈ P˜V Suppose A ∈ P˜U and G is defined by µA◦G˜ (v) = max min{µA (u), µG˜ (u, v)} u∈U
(4)
∀v ∈ V
Equation (4) is a realization of the celebrated Compositional Rule of Inference (CRI) [15]. It is easy to show that “◦” is a binary operator having the distributive property with respect to union and the associative property with respect to scalar multiplication [10], that is ˜ = (A1 ◦ G) ˜ ∪ (A2 ◦ G) ˜ (A1 ∪ A2 ) ◦ G
(5)
˜ = α · (A ◦ G) ˜ (α · A) ◦ G
(6)
and ˜ ∈ P˜U×V , and a ∈ [0, 1]. where A, A1 , A2 ∈ P˜U , G The FEP is applied to PP mappings and relations to extend them to FF mappings g˜ : P˜U → P˜V µB (v) =
B = g˜(A)
max
{u|(u,v)∈G}
∀A ∈ P˜U
µA (u) ∀v ∈ V
(7) (8)
where G ⊂ U × V is a relation over U × V , and g˜ is the fuzzy extension of G based on the FEP. The extension of a PP mapping is a special case of the extension of a relation; therefore, it is not addressed separately. In Eqs (7) and (8), if one represents relation G via its membership function µ G (·, ·), Eq. (8) will change to
µB (v) = max min{µA (u), µG (u, v)} u∈U
(9)
∀v ∈ V
Equation (9) conveys the idea of a projection. This interpretation has been shown in Fig. 1. In fact, for a ˜ = G, Eqs (9) and (4) are equivanon-fuzzy relation G lent. Moreover, as an alternative for the FEP, Zadeh in [14] suggests a paradigm to extend SS mappings to FF mappings. Consider a SS mapping g : P U → PV , ∀A ∈ PU , the fuzzy extension of g, g˜, is defined as g˜(A) = ∪
0α1
α · g(Aα ) ∀A ∈ P˜U
(10)
where Aα = {u|µA (u) α} is the α-cut of fuzzy set A, a ∈ [0, 1]. This paper deals with two issues: 1. How restricted is the FEP? i.e. Can one obtain, in general, all FF mappings with the unrestricted definition g˜ : P˜U → P˜V
B = g˜(A)
∀A ∈ P˜U (11)
using the FEP? If not, which subfamily of FF mappings in Eq. (11) can be obtained? 2. How suitable is Eq. (10) for extending the domain of mappings from a set valued domain to a fuzzy set valued one? How restricted is this extension of SS mappings to FF mappings, as compared to the extension of PP mappings to FF mappings in Eqs (7) and (8)? Sections 2 and 3 discuss the above two issues, respectively. In Section 2, a theorem is proved that provides the extent of restrictions imposed by the FEP. It is shown that the FEP would impose a restriction on FF mappings, which is similar to the linear restriction for point-valued functions. In Section 3, the extension of SS mappings to FF mappings is discussed. It is shown that this extension is well-behaved only for those mappings that preserve the subsethood order. The restrictions imposed by the latter extension are then addressed in a theorem. It is proved that the extension of subsethood order preserving (SOP) SS mappings restricts the range of possible FF mappings to SOP cutworthy FF mappings. This family of FF mappings is larger than the first family obtained from the FEP, Eqs (7) and (8); however, it still makes up a small portion of all possible FF mappings as defined in Eq. (11). Section 4 contains a few examples to illustrate different situations, and finally, the conclusions are stated in Section 5.
B.N. Araabi et al. / Restrictions imposed by the fuzzy extension of relations and functions
11
B
A
V
U
Fig. 1. G is a relation over U × V , that is, G = {(u, v)|uGv} ⊂ U × V . The FEP provides a mechanism to project fuzzy subsets of U to max µA (u)). Gray scale is used to provide a visualization of a fuzzy set. fuzzy subsets of V , based on G (i.e. B = g˜(A), µB (v) = {u|(u,v)∈G}
˜ (A1 ∪ A2 ) ◦ G
2. Imposed restriction by the FEP To determine the restrictions imposed by the FEP, given by Eqs (7) and (8), on the extended FF mappings, first the restriction of fuzzy relational equations [10] is derived. The restrictions imposed by the FEP are then addressed in a subsequent theorem. Theorem 1 Restrictions imposed by fuzzy relational equations. Consider a general FF mapping g˜ : P˜U → P˜V as defined in Eq. (11). A necessary and sufficient condition for g˜ to have the following representation in the form of fuzzy relational equations ˜ G ˜ ∈ P˜U×V g˜(A) = A ◦ G
∀A ∈ P˜U
(12)
is given by g˜(A1 ∪ A2 ) = g˜(A1 ) ∪ g˜(A2 ) ∀A1 , A2 ∈ P˜U g˜(α · A) = α · g˜(A) ∀A ∈ P˜U , ∀α ∈ [0, 1]
(14)
˜ = α · (A ◦ G) ˜ → g˜(α · A) (α · A) ◦ G = α · g˜(A)
(19)
“◦” and “·” are associative [10], therefore Eq. (14). ˜ → µg˜({u }) (v) g˜({µ0 }) = {u0 } ◦ G 0 = max min{µ{u0 } (u), µG˜ (u, v)}
(20)
u∈U
⇒ µg˜({u0 }) (v) = µG˜ (u0 , v)
(21)
Hence Eq. (15), and this concludes the necessity. Now, sufficiency: (22)
Considering Eqs (13) and (14) g˜(A) = g˜ ∪ µA (u) · {u} u∈U
= ∪ µA (u) · g˜({u})
(23)
u∈U
(15)
Proof – Necessity: ˜ g˜(A) = A ◦ G
(16)
µA◦G˜ (v) = max min{µA (u), µG˜ (u, v)}
(17)
u∈U
“◦” is distributive over “∪” [10], hence Eq. (13).
u∈U
(13)
(18)
= g˜(A1 ) ∪ g˜(A2 )
A = ∪ µA (u) · {u}
˜ in Eq. (12) is then defined as follows G µG˜ (u, v) = µg˜({u}) (v) ∀(u, v) ∈ U × V
˜ ∪ (A2 ◦ G) ˜ → g˜(A1 ∪ A2 ) = (A1 ◦ G)
˜ in Eqs (15) and (23) is equivawith the definition of G lent to Eq. (12). Theorem 1 indicates that fuzzy relational equations consist of a small family out of all possible FF mappings. Because of properties Eqs (13) and (14), fuzzy linear functions appears to be an appropriate alternative name for fuzzy relational equations as stated in Eq. (12).
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B.N. Araabi et al. / Restrictions imposed by the fuzzy extension of relations and functions
Equation (9) shows that the FEP is a special case of the CRI. On the other hand, Eq. (12) as defined in Eq. (4) is a realization of the CRI. Comparing Eqs (9) and (4), it can be seen that Eq. (9) is a special case of Eq. (4) for ˜ Hence, the results of Theorem 1 can be non-fuzzy G. used to determine the restrictions imposed by the FEP.
g : PU → PV
B = g(A)
∀A ∈ PU
(26)
In this case, the SS mapping itself conveys the uncertainty that is related to non-specificity. Therefore, a fuzzy extension of g, in general, can be defined as g˜ g˜ : P˜U → P˜V
B = g˜(A)
∀A ∈ P˜U
(27)
Theorem 2 Imposed restrictions by the fuzzy extension principle.
where
Consider a general FF mapping g˜ : P˜U → P˜V as defined in Eq. (11). If g˜ satisfies Eqs (13) and (14), and
The value of g˜(A) for A ∈ P˜U − PU originates from the definition of fuzzy extension of g. Dempster-Shafer (evidence) theory is a good example of a set-valued theory that has been extended to the fuzzy domain utilizing a proper fuzzy extension [1,8]. A relation G ⊂ U × V can be represented by a set to point mapping g 1 : PU → V or a point to set mapping g2 : U → PV , as defined by
g˜({u}) ∈ PV
∀u ∈ U
(24)
then g˜ is the fuzzy extension of a relation G via the FEP, with the following representation g˜(A) = A ◦ G µA◦G (v) = max min{µA (u), µG (u, v)}
(25)
u∈U
where relation G in Eq. (25) is defined with µ G (u, v) = µg˜({u}) (v), ∀(u, v) ∈ U × V . Conversely, if a FF mapping ˜g is the extended version of a relation G via the FEP, then g˜ satisfies Eqs (13), (14) and (24). Proof – Straightforward consequence of Theorem 1. Based on Theorem 2, the FEP is more restrictive than it seems at the first glance. Confining FF mappings to what is extended by the FEP means confining all possible FF mappings to a subfamily of fuzzy relational equations (fuzzy linear functions). Three Eqs (13), (14) and (24) completely determine the restrictions imposed by the FEP on the family of all possible FF mappings defined in Eq. (11).
g˜(A) = g(A)
g1 (A) = v
∀A ∈ PU
if and only if
(u, v) ∈ G, ∀u ∈ A g2 (u) = B
(28)
(29)
if and only if
(u, v) ∈ G, ∀v ∈ B
(30)
Consequently, the FEP in its original form is also capable of extending set to point and point to set mappings [13]. However, it is not sufficient to extend SS mappings in the general form Eq. (26). Zadeh in [14] offers an extension paradigm from SS mappings to FF mappings as well, i.e. from Eq. (26) to Eq. (27) g˜(A) = g˜ ∪ α · Aα 0α1 (31) ∆ = ∪ α · g(Aα ) ∀A ∈ P˜U 0α1
3. Extension of SS mappings to FF mappings The FEP can extend the domain of functions from the elements of a referential set, U , to the fuzzy subsets of that referential set. Through this extension, in fact, two types of uncertainties are injected into input/output pairs of observations simultaneously, namely, nonspecificity and vagueness [6]. Non-specificity can be modeled by sets [2], and relates to situations where input/output observations are only clear as sets of data points (SS mappings). Vagueness, on the other hand, relates to the non-sharp boundaries of sets, and can be modeled by fuzzy sets (FF mappings). In many applications, functions are naturally defined over a set valued domain as SS mappings, as follows
where Aα is the α-cut of fuzzy set A, α ∈ [0, 1]. The idea behind the extension Eq. (31) stems from the resolution identity Eq. (2), and it supports Eq. (28). In [14] it is mentioned that Eqs (9) and (31) are equivalent. This interpretation of equivalence is not completely clear at the first glance due to the fact that (9) extends PP mappings to FF mappings, while Eq. (31) extends SS mappings to FF mappings. The relation of Eqs (9) and (31) cab be shown as follows G⊂U ×V → (8) (30) −→ g : PU → PV −→ g˜1 : P˜U → P˜V (32) (8) −→ −→ g˜2 : P˜U → P˜V → g˜1 = g˜2
B.N. Araabi et al. / Restrictions imposed by the fuzzy extension of relations and functions
As shown in Eq. (32), extending a relation G via Eq. (9) to a FF mapping is equivalent to first extending G via Eq. (9) to a SS mapping and then extending that SS mapping to a FF mapping via Eq. (31). Either path results in the same extended FF mapping. A proof of the above equivalence can be found in [11]. Equivalence of g˜1 and g˜2 justifies Eq. (31) as a suitable extension, but the question that naturally arises is: what if g in Eq. (32) is not the extended version of a non-fuzzy relation G? Would the extension Eq. (31) be rational yet? In what follows we attempt to answer this question. Based on Eq. (32) the FEP can be decomposed into two parts: the extension of a PP mapping or a relation to a SS mapping via Eq. (9), and the extension of a SS mapping to a FF mapping via Eq. (31). Now, the question becomes: does g in Eq. (32) have any specific feature that helps Eq. (31) to be a rational, well-behaved extension? In fact, the answer is yes. g in Eq. (32) has the following SOP property A1 ⊂ A2 ⇒ g(A1 ) ⊂ g(A2 ) ∀A1 , A2 ∈ PU
(33)
i.e. g in Eq. (32) preserves the order of subsethood as an input/output transformation. The extension Eq. (31) extends the property Eq. (33) to fuzzy subsets A1 ⊂ A2 ⇒ g˜(A1 ) ⊂ g˜(A2 ) ∀A1 , A2 ∈ P˜U
(34)
The following theorem exhibits the effect of the SOP in Eq. (33) on the extension Eq. (31). Theorem 3 Extension of a SOP SS mapping to a FF mapping. Consider a SS mapping g in Eq. (26), and assume g is a SOP mapping as stated in property Eq. (33). The extension Eq. (31) is equivalent to the following extension [˜ g (A)]α = g(Aα ) ∀A ∈ P˜U ∀α ∈ [0, 1] (35) Proof – Consider
=
∪
α · [˜ g(A)]α
∪
α · g(Aα )
0α1 0α1
(36)
and g˜(A) =
∪
α · g(Aα ) → [˜ g (A)]β = ∪ α · g(Aα ) 0α1
0α1
0α 0 2∆ v−∆
Consider the following set-valued glitch removing filter, g : PU → PV µg(A) (v) v+∆ 1 1 2∆ v−∆ µA (u)du = 0 otherwise
1 2
∆>0
where for discrete U , the integral is replaced by a summation. g is a SOP SS mapping so its fuzzy extension,
B.N. Araabi et al. / Restrictions imposed by the fuzzy extension of relations and functions
A1
1
17
A2
membership value
0 .9 0 .8 0 .7 0 .6 0 .5 0 .4 0 .3 0 .2 0 .1 0 0
0 .5
1
1 .5
2
2 .5
3
3 .5
4
u
(a) A1
1
A2
membership value
0 .9 0 .8 0 .7 0 .6 0 .5 0 .4 0 .3 0 .2 0 .1 0 0
0 .5
1
1 .5
2
2 .5
3
3 .5
4
u
(b) Fig. 5. g˜ in example 3 is a SOP FF mapping. Shaded areas in (a) and (b) indicate g˜(A1 ) ∪ g˜(A2 ) and g˜(A1 ∪ A2 ), respectively. They are not equal to each other. g˜ is not a cutworthy extension of a SOP SS mapping.
g˜, via Eq. (31) would be a cutworthy well-behaved one. Neither g(A1 ∪ A2 ) is equal to g(A1 ) ∪ g(A2 ), nor g˜(A1 ∪ A2 ) is equal to g˜(A1 ) ∪ g˜(A2 ). Figure 6 shows samples of this filtering scheme for both fuzzy and non-
fuzzy cases. The difference between g˜(A 1 ∪ A2 ) and g˜(A1 ) ∪ g˜(A2 ) has also been shown. This example shows that the converse of Corollary 1 does not hold, and justifies why two sets of conditions are not equiv-
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B.N. Araabi et al. / Restrictions imposed by the fuzzy extension of relations and functions
A1
1
A2
membership value
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0
0.5
1
1.5
2
2.5
3
3.5
(a) A1
1
4
u
A2
membership value
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0
0.5
1
1.5
2
2.5
3
3.5
4
u
(b)
Fig. 6. Shaded areas in (a) and (b) indicate g(A1 ) ∪ g(A2 ) and g(A1 ∪ A2 ), respectively, where g is a SOP SS mapping defined in Example 4. Shaded areas in (c) and (d) indicate g˜(A1 ) ∪ g˜(A2 ) and g˜(A1 ∪ A2 ), respectively, where g˜ is a cutworthy fuzzy extension of a SOP SS mapping. However, neither g(A1 ) ∪ g(A2 ) is equal to g(A1 ∪ A2 ), nor g˜(A1 ) ∪ g˜(A2 ) is equal to g˜(A1 ∪ A2 ). This example shows that the converse of Corollary 1 does not hold.
alent. In Fig. 6, ∆ is chosen to be 0.1. Example 5 Eroding filter Consider the following set-valued eroding filter, g : PU → PV
µg(A) (v) =
1 0
1 2∆
v+∆
µ! (u)du ε otherwise
v−∆
1 < ε < 1, ∆ > 0 2
B.N. Araabi et al. / Restrictions imposed by the fuzzy extension of relations and functions
A1
1
19
A2
membership value
0 .9 0 .8 0 .7 0 .6 0 .5 0 .4 0 .3 0 .2 0 .1 0 0
0 .5
1
1 .5
2
2 .5
3
3 .5
4
u
(c) A1
1
A2
membership value
0 .9 0 .8 0 .7 0 .6 0 .5 0 .4 0 .3 0 .2 0 .1 0 0
0 .5
1
1 .5
2
2 .5
3
3 .5
4
u
(d) Fig. 6, continued.
where for discrete U , the integral is replaced by a summation. g is a SOP SS mapping so its fuzzy extension, g˜, via Eq. (31) would be a cutworthy and consistent one. Basic features are similar to Example 4. Figure 7 shows the difference between g˜(A 1 ∪ A2 ) and g˜(A1 ) ∪ g˜(A2 ). In Fig. 7, ε and ∆ are chosen to be 0.6 and 0.1, respectively.
5. Summary and conclusion The FEP in its original version is a tool to extend the domain of mappings from the elements of a referential set to the fuzzy subsets of that referential set. In general, this approach towards producing fuzzy mappings leads to a restricted family of FF mappings. Theorem 2
20
B.N. Araabi et al. / Restrictions imposed by the fuzzy extension of relations and functions
A1
1
A2
membership value
0 .9 0 .8 0 .7 0 .6 0 .5 0 .4 0 .3 0 .2 0 .1 0 0
0 .5
1
1 .5
2
2 .5
3
3 .5
4
u
(a) A1
1
A2
membership value
0 .9 0 .8 0 .7 0 .6 0 .5 0 .4 0 .3 0 .2 0 .1 0 0
0 .5
1
1 .5
2
2 .5
3
3 .5
4
u
(b) Fig. 7. g˜ in example 5 is a SOP FF mapping, and a cutworthy fuzzy extension of a SOP SS mapping. Shaded areas in (a) and (b) indicate g˜(A1 ) ∪ g˜(A2 ) and g˜(A1 ∪ A2 ), respectively. They are not equal to each other; however, g˜(A1 ) ∪ g˜(A2 ) ⊂ g˜(A1 ∪ A2 ).
together with Theorem 1 state that only a subfamily of fuzzy relational equations (fuzzy linear functions) can be reached via the FEP. Extending SS mappings leads to a larger family of FF mappings. Theorem 3 justifies Zadeh’s approach to extend SS mappings, which stems from the resolution identity; however, at the same time it shows that
this extension is much more reasonable when the SS mapping preserves subsethood order. In this case, the extension is a well-behaved cutworthy and consistent one. Theorem 4 is analogous to Theorem 2 for the extension of SS mappings with the necessary and sufficient conditions. Corollary 1 together with Theorem 5 show that the latter extension is more relaxed compared
B.N. Araabi et al. / Restrictions imposed by the fuzzy extension of relations and functions
21
Table 1 Paper summary g˜ is a fuzzy set to fuzzy set mapping, i.e., P˜U → P˜V , which maps A to g˜(A), ∀A ∈ P˜U
Starting non-fuzzy relation or mapping
Imposed restrictions on extended fuzzy set to fuzzy set mapping g~ (necessary and sufficient conditions)
Reconstruction
Subsethood order preserving set to set mapping g : PU → PV g~( A) =
µ g~ ( A) (v) = max µ A (u ) {u | ( u ,v )∈G }
0≤α ≤1
↓ g~( A1
A2 ) = g~ ( A1 )
α ⋅ g ( Aα )
⊂
Extension scheme
Crisp relation G ⊂ U ×V
↓ g~ ( A2 )
g~ (α ⋅ A) = α ⋅ g~ ( A) g~({u}) ∈ PV , ∀u ∈ U
⇒ / ⇐
[ g~( A)]α = g~ ( Aα ) g~( A) ∈ PV , ∀A ∈ PU With the above three restrictions g can be reconstructed uniquely from g~ using g ( A) = g~ ( A) ∀A ∈ PU
With the above three restrictions G can be reconstructed uniquely from g~ using µ G (u , v) = µ g~ ({u }) (v) ∀(u, v) ∈ U × V
to the original FEP. Therefore, a larger family of FF mappings is accessible via the latter extension. Table 1 provides a summary of the results reported in this paper. As seen in Table 1, the extension of relations to FF mappings generates a subfamily of all possible fuzzy relational equations. Considering the imposed constraints, it is too restrictive if we confine FF mappings to this subfamily. On the other hand, the extension of SS mappings via Eq. (31) is rational only for those SS mappings that preserve subsethood order. For this family of extended FF mappings, the imposed restrictions are merely preserving the subsethood order in the fuzzy domain and cutworthyness. Hence, for SOP SS mappings, the extension Eq. (31) is completely rational generating consistent results. This extension does not appear too restrictive when the SOP property is transported directly from set domain to fuzzy domain. The combination of cutworthyness and SOP property is similar to the smoothness condition during the interpolation of point-valued functions. However, it should be emphasized that the entire family of FF mappings, which is obtained in this manner, is still small and restricted as compared to the family of all possible FF mappings. In essence, the materials presented in this paper provide a better understanding of the nature and limita-
A1 ⊂ A2 ⇒ g~ ( A1 ) ⊂ g~ ( A2 )
tions of FF mappings that are accessible through extension of the non-fuzzy function and relations. How such restrictions affect the applications of the fuzzy set theory – where fuzzy extension has been utilized – can be explored in future research. References [1]
[2] [3] [4] [5]
[6] [7] [8]
[9]
B.N. Araabi and C. Lucas, On the fuzzy generalization of Dempster-Shafer theory, Proc. Int’l Conf. Intelligent and Cognitive Systems, (Vol. 1), ICICS-96, Tehran, Iran, 1996, pp. 161–166. H.B. Enderton, Elements of set theory, Academic press, New York, NY, 1977. M. Jamshidi, N. Vadiee and T.J. Rose, eds, Fuzzy logic and control, Prentice Hall, Englewood Cliffs, NJ, 1993. O. Kaleva and S. Seikkala, On fuzzy metric spaces, Fuzzy Sets and Systems 12 (1984), 215–229. A. Kaufmann and M.M. Gupta, Introduction to fuzzy arithmetic: Theory and applications, Van Nostrand, New York, NY, 1985. G.J. Klir and B. Yuan, Fuzzy sets and fuzzy logic: Theory and applications, Prentice Hall, Upper Saddle River, NJ, 1995. R. Lowen, Fuzzy topological spaces and fuzzy compactness, J. Math. Anal. Applications 56 (1976), 621–633. C. Lucas and B.N. Araabi, Generalization of the DempsterShafer theory: A fuzzy-valued measure, IEEE Trans. Fuzzy Systems 7 (1999), 255–270. H.T. Nguyen, A note on the extension principle for the fuzzy sets, J. Math. Anal. Applications 64 (1978), 369–380.
22 [10] [11] [12] [13] [14]
B.N. Araabi et al. / Restrictions imposed by the fuzzy extension of relations and functions W. Pedricz, Fuzzy control and fuzzy systems, Research Studies Press, Taunton, England, 1993. M.J. Wierman, Total extension of set functions and relations, Proc. Annual Meeting NAFIPS, New York, 1997, pp. 292–294. M.J. Wierman, Extending set functions and relations, Int’l J. of General Systems 26 (1997), 91–96. R.R. Yager, A characterization of extension principle, Fuzzy Sets and Systems 18 (1986), 205–217. L.A. Zadeh, The concept of linguistic variable and its
application to approximate reasoning-I, Inform. Sci. 8 (1975), 199–249. [15] L.A. Zadeh, Fuzzy sets and information granularity, in: Advances in Fuzzy Set Theory and Applications, M.M. Gupta et al., eds, North-Holland, Amsterdam, Nederland, 1979, pp. 3– 18. [16] H.J. Zimmermann, L.A. Zadeh and B.R. Gaines, eds, Fuzzy sets and decision analysis, North-Holland, New York, NY, 1984.
Journal of Intelligent & Fuzzy Systems 11 (2001) 9–22 IOS Press
Restrictions imposed by the fuzzy extension of relations and functions Babak N. Araabia,∗, Nasser Kehtarnavaz b and Caro Lucas a a
b
Department of Electrical and Computer Engineering, University of Tehran, Tehran 14395, Iran Department of Electrical Engineering, University of Texas at Dallas, Richardson, TX 75083, USA
Abstract. The fuzzy extension principle has been widely used to extend the domain of mathematical functions and relations from elements of a referential set to fuzzy subsets of that referential set. However, there are restrictions associated with the fuzzy extension principle. This paper addresses the question how restricted is the family of “fuzzy set to fuzzy set mappings” obtained by the fuzzy extension of non-fuzzy functions and relations, as compared to the general family of all possible fuzzy set to fuzzy set mappings. A theorem is presented, with the necessary and sufficient conditions, to determine this restriction for the fuzzy extension of non-fuzzy relations and point-valued functions. It is shown that the fuzzy extension principle would impose a restriction on extended fuzzy set to fuzzy set mappings, which is similar to the linear restriction for point-valued functions. Moreover, the extension of mappings from a set-valued domain to a fuzzy set-valued domain is discussed. It is shown that this extension is well-behaved only for those mappings which preserve subsethood order. Another theorem with the necessary and sufficient condition has been proved to determine the imposed restriction during the fuzzy extension of subsethood order preserving set to set mappings. The two extensions have been compared and several examples are provided. Keywords: Fuzzy extension principle, set to set mapping, fuzzy set to fuzzy set mapping, fuzzy relational equation, subsethood order preserving mapping
1. Introduction The Fuzzy Extension Principle (FEP) was introduced by Zadeh [14] to extend the domain of crisp mathematical functions and relations from the elements of a referential set to fuzzy subsets of that referential set. The FEP has been widely used as an essential tool to fuzzify classical point-valued concepts and theories. The scope of applications of the FEP is quite vast, since it can fuzzify point-valued functions, and functions virtually can be found anywhere in engineering, science, and mathematics. The FEP has been utilized to extend point-valued concepts in fuzzy arithmetic [5], fuzzy control [3], and fuzzy decision making [16], as well as more abstract areas like fuzzy topology [7], and fuzzy metric spaces [4]. ∗ Corresponding
author: Dr. N. Kehtarnavaz, Department of Electrical Engineering, EC33, University of Texas at Dallas, PO Box 830688, Richardson, TX 75083-0688, USA. Tel.: +1 972 883 6838; Fax: +1 972 883 2710; E-mail: [email protected]. 1064-1246/01/$8.00 2001 – IOS Press. All rights reserved
Before starting with the definition of the FEP, the precise meaning of few expressions and notations used throughout the paper are in order: Let U and V be two finite referential sets. Sets of all crisp subsets of U and V are indicated by P U and PV , respectively, while sets of all fuzzy subsets of U and V are denoted by P˜U and P˜V . The domain and co-domain of mappings are asserted for the sake of clarity of theorems and discussions, that is, point to point mapping (PP mapping) is a crisp function from U to V ; set to set mapping (SS mapping) is a crisp function from P U to PV , and fuzzy set to fuzzy set mapping (FF mapping) is a crisp function from P˜U to P˜V . The meaning of point to set and set to point mappings should be clear by analogy. A relation is a crisp subset of U ×V , that is, a member of PU×V , and a fuzzy relation is a fuzzy subset of U × V , that is, a member of P˜U×V . PP mappings and relations are represented with small and capital italic letters, respectively (e.g. g and G). SS mappings are
10
B.N. Araabi et al. / Restrictions imposed by the fuzzy extension of relations and functions
denoted by a bar over the mappings names (e.g. g), and FF mappings and fuzzy relations are indicated by a tilde over the mappings and relations names, respectively ˜ (e.g. g˜ and G). A fuzzy set A ∈ P˜U is defined by its membership function µA : U → [0, 1]. The α-cut of the fuzzy set A is denoted by Aα = {u|µA (u) α}, a ∈ [0, 1]. For a fuzzy set A ∈ P˜U and for a ∈ [0, 1], the scalar multiplication α · A ∈ P˜U is defined by µα·A (u) = min{α, µA (u)}
∀u ∈ U
(1)
The resolution identity [14] is a representation for a fuzzy set A ∈ P˜U based on its α-cuts as follows A= ∪
0α1
α · Aα
(2)
For two fuzzy sets A, B ∈ P˜U , the union of A and B is denoted by A ∪ B, where µA∪B (u) = max{µA (u), µB (u)} ∀u ∈ U
(3)
˜ ∈ P˜U×V , then A ◦ G ˜ ∈ P˜V Suppose A ∈ P˜U and G is defined by µA◦G˜ (v) = max min{µA (u), µG˜ (u, v)} u∈U
(4)
∀v ∈ V
Equation (4) is a realization of the celebrated Compositional Rule of Inference (CRI) [15]. It is easy to show that “◦” is a binary operator having the distributive property with respect to union and the associative property with respect to scalar multiplication [10], that is ˜ = (A1 ◦ G) ˜ ∪ (A2 ◦ G) ˜ (A1 ∪ A2 ) ◦ G
(5)
˜ = α · (A ◦ G) ˜ (α · A) ◦ G
(6)
and ˜ ∈ P˜U×V , and a ∈ [0, 1]. where A, A1 , A2 ∈ P˜U , G The FEP is applied to PP mappings and relations to extend them to FF mappings g˜ : P˜U → P˜V µB (v) =
B = g˜(A)
max
{u|(u,v)∈G}
∀A ∈ P˜U
µA (u) ∀v ∈ V
(7) (8)
where G ⊂ U × V is a relation over U × V , and g˜ is the fuzzy extension of G based on the FEP. The extension of a PP mapping is a special case of the extension of a relation; therefore, it is not addressed separately. In Eqs (7) and (8), if one represents relation G via its membership function µ G (·, ·), Eq. (8) will change to
µB (v) = max min{µA (u), µG (u, v)} u∈U
(9)
∀v ∈ V
Equation (9) conveys the idea of a projection. This interpretation has been shown in Fig. 1. In fact, for a ˜ = G, Eqs (9) and (4) are equivanon-fuzzy relation G lent. Moreover, as an alternative for the FEP, Zadeh in [14] suggests a paradigm to extend SS mappings to FF mappings. Consider a SS mapping g : P U → PV , ∀A ∈ PU , the fuzzy extension of g, g˜, is defined as g˜(A) = ∪
0α1
α · g(Aα ) ∀A ∈ P˜U
(10)
where Aα = {u|µA (u) α} is the α-cut of fuzzy set A, a ∈ [0, 1]. This paper deals with two issues: 1. How restricted is the FEP? i.e. Can one obtain, in general, all FF mappings with the unrestricted definition g˜ : P˜U → P˜V
B = g˜(A)
∀A ∈ P˜U (11)
using the FEP? If not, which subfamily of FF mappings in Eq. (11) can be obtained? 2. How suitable is Eq. (10) for extending the domain of mappings from a set valued domain to a fuzzy set valued one? How restricted is this extension of SS mappings to FF mappings, as compared to the extension of PP mappings to FF mappings in Eqs (7) and (8)? Sections 2 and 3 discuss the above two issues, respectively. In Section 2, a theorem is proved that provides the extent of restrictions imposed by the FEP. It is shown that the FEP would impose a restriction on FF mappings, which is similar to the linear restriction for point-valued functions. In Section 3, the extension of SS mappings to FF mappings is discussed. It is shown that this extension is well-behaved only for those mappings that preserve the subsethood order. The restrictions imposed by the latter extension are then addressed in a theorem. It is proved that the extension of subsethood order preserving (SOP) SS mappings restricts the range of possible FF mappings to SOP cutworthy FF mappings. This family of FF mappings is larger than the first family obtained from the FEP, Eqs (7) and (8); however, it still makes up a small portion of all possible FF mappings as defined in Eq. (11). Section 4 contains a few examples to illustrate different situations, and finally, the conclusions are stated in Section 5.
B.N. Araabi et al. / Restrictions imposed by the fuzzy extension of relations and functions
11
B
A
V
U
Fig. 1. G is a relation over U × V , that is, G = {(u, v)|uGv} ⊂ U × V . The FEP provides a mechanism to project fuzzy subsets of U to max µA (u)). Gray scale is used to provide a visualization of a fuzzy set. fuzzy subsets of V , based on G (i.e. B = g˜(A), µB (v) = {u|(u,v)∈G}
˜ (A1 ∪ A2 ) ◦ G
2. Imposed restriction by the FEP To determine the restrictions imposed by the FEP, given by Eqs (7) and (8), on the extended FF mappings, first the restriction of fuzzy relational equations [10] is derived. The restrictions imposed by the FEP are then addressed in a subsequent theorem. Theorem 1 Restrictions imposed by fuzzy relational equations. Consider a general FF mapping g˜ : P˜U → P˜V as defined in Eq. (11). A necessary and sufficient condition for g˜ to have the following representation in the form of fuzzy relational equations ˜ G ˜ ∈ P˜U×V g˜(A) = A ◦ G
∀A ∈ P˜U
(12)
is given by g˜(A1 ∪ A2 ) = g˜(A1 ) ∪ g˜(A2 ) ∀A1 , A2 ∈ P˜U g˜(α · A) = α · g˜(A) ∀A ∈ P˜U , ∀α ∈ [0, 1]
(14)
˜ = α · (A ◦ G) ˜ → g˜(α · A) (α · A) ◦ G = α · g˜(A)
(19)
“◦” and “·” are associative [10], therefore Eq. (14). ˜ → µg˜({u }) (v) g˜({µ0 }) = {u0 } ◦ G 0 = max min{µ{u0 } (u), µG˜ (u, v)}
(20)
u∈U
⇒ µg˜({u0 }) (v) = µG˜ (u0 , v)
(21)
Hence Eq. (15), and this concludes the necessity. Now, sufficiency: (22)
Considering Eqs (13) and (14) g˜(A) = g˜ ∪ µA (u) · {u} u∈U
= ∪ µA (u) · g˜({u})
(23)
u∈U
(15)
Proof – Necessity: ˜ g˜(A) = A ◦ G
(16)
µA◦G˜ (v) = max min{µA (u), µG˜ (u, v)}
(17)
u∈U
“◦” is distributive over “∪” [10], hence Eq. (13).
u∈U
(13)
(18)
= g˜(A1 ) ∪ g˜(A2 )
A = ∪ µA (u) · {u}
˜ in Eq. (12) is then defined as follows G µG˜ (u, v) = µg˜({u}) (v) ∀(u, v) ∈ U × V
˜ ∪ (A2 ◦ G) ˜ → g˜(A1 ∪ A2 ) = (A1 ◦ G)
˜ in Eqs (15) and (23) is equivawith the definition of G lent to Eq. (12). Theorem 1 indicates that fuzzy relational equations consist of a small family out of all possible FF mappings. Because of properties Eqs (13) and (14), fuzzy linear functions appears to be an appropriate alternative name for fuzzy relational equations as stated in Eq. (12).
12
B.N. Araabi et al. / Restrictions imposed by the fuzzy extension of relations and functions
Equation (9) shows that the FEP is a special case of the CRI. On the other hand, Eq. (12) as defined in Eq. (4) is a realization of the CRI. Comparing Eqs (9) and (4), it can be seen that Eq. (9) is a special case of Eq. (4) for ˜ Hence, the results of Theorem 1 can be non-fuzzy G. used to determine the restrictions imposed by the FEP.
g : PU → PV
B = g(A)
∀A ∈ PU
(26)
In this case, the SS mapping itself conveys the uncertainty that is related to non-specificity. Therefore, a fuzzy extension of g, in general, can be defined as g˜ g˜ : P˜U → P˜V
B = g˜(A)
∀A ∈ P˜U
(27)
Theorem 2 Imposed restrictions by the fuzzy extension principle.
where
Consider a general FF mapping g˜ : P˜U → P˜V as defined in Eq. (11). If g˜ satisfies Eqs (13) and (14), and
The value of g˜(A) for A ∈ P˜U − PU originates from the definition of fuzzy extension of g. Dempster-Shafer (evidence) theory is a good example of a set-valued theory that has been extended to the fuzzy domain utilizing a proper fuzzy extension [1,8]. A relation G ⊂ U × V can be represented by a set to point mapping g 1 : PU → V or a point to set mapping g2 : U → PV , as defined by
g˜({u}) ∈ PV
∀u ∈ U
(24)
then g˜ is the fuzzy extension of a relation G via the FEP, with the following representation g˜(A) = A ◦ G µA◦G (v) = max min{µA (u), µG (u, v)}
(25)
u∈U
where relation G in Eq. (25) is defined with µ G (u, v) = µg˜({u}) (v), ∀(u, v) ∈ U × V . Conversely, if a FF mapping ˜g is the extended version of a relation G via the FEP, then g˜ satisfies Eqs (13), (14) and (24). Proof – Straightforward consequence of Theorem 1. Based on Theorem 2, the FEP is more restrictive than it seems at the first glance. Confining FF mappings to what is extended by the FEP means confining all possible FF mappings to a subfamily of fuzzy relational equations (fuzzy linear functions). Three Eqs (13), (14) and (24) completely determine the restrictions imposed by the FEP on the family of all possible FF mappings defined in Eq. (11).
g˜(A) = g(A)
g1 (A) = v
∀A ∈ PU
if and only if
(u, v) ∈ G, ∀u ∈ A g2 (u) = B
(28)
(29)
if and only if
(u, v) ∈ G, ∀v ∈ B
(30)
Consequently, the FEP in its original form is also capable of extending set to point and point to set mappings [13]. However, it is not sufficient to extend SS mappings in the general form Eq. (26). Zadeh in [14] offers an extension paradigm from SS mappings to FF mappings as well, i.e. from Eq. (26) to Eq. (27) g˜(A) = g˜ ∪ α · Aα 0α1 (31) ∆ = ∪ α · g(Aα ) ∀A ∈ P˜U 0α1
3. Extension of SS mappings to FF mappings The FEP can extend the domain of functions from the elements of a referential set, U , to the fuzzy subsets of that referential set. Through this extension, in fact, two types of uncertainties are injected into input/output pairs of observations simultaneously, namely, nonspecificity and vagueness [6]. Non-specificity can be modeled by sets [2], and relates to situations where input/output observations are only clear as sets of data points (SS mappings). Vagueness, on the other hand, relates to the non-sharp boundaries of sets, and can be modeled by fuzzy sets (FF mappings). In many applications, functions are naturally defined over a set valued domain as SS mappings, as follows
where Aα is the α-cut of fuzzy set A, α ∈ [0, 1]. The idea behind the extension Eq. (31) stems from the resolution identity Eq. (2), and it supports Eq. (28). In [14] it is mentioned that Eqs (9) and (31) are equivalent. This interpretation of equivalence is not completely clear at the first glance due to the fact that (9) extends PP mappings to FF mappings, while Eq. (31) extends SS mappings to FF mappings. The relation of Eqs (9) and (31) cab be shown as follows G⊂U ×V → (8) (30) −→ g : PU → PV −→ g˜1 : P˜U → P˜V (32) (8) −→ −→ g˜2 : P˜U → P˜V → g˜1 = g˜2
B.N. Araabi et al. / Restrictions imposed by the fuzzy extension of relations and functions
As shown in Eq. (32), extending a relation G via Eq. (9) to a FF mapping is equivalent to first extending G via Eq. (9) to a SS mapping and then extending that SS mapping to a FF mapping via Eq. (31). Either path results in the same extended FF mapping. A proof of the above equivalence can be found in [11]. Equivalence of g˜1 and g˜2 justifies Eq. (31) as a suitable extension, but the question that naturally arises is: what if g in Eq. (32) is not the extended version of a non-fuzzy relation G? Would the extension Eq. (31) be rational yet? In what follows we attempt to answer this question. Based on Eq. (32) the FEP can be decomposed into two parts: the extension of a PP mapping or a relation to a SS mapping via Eq. (9), and the extension of a SS mapping to a FF mapping via Eq. (31). Now, the question becomes: does g in Eq. (32) have any specific feature that helps Eq. (31) to be a rational, well-behaved extension? In fact, the answer is yes. g in Eq. (32) has the following SOP property A1 ⊂ A2 ⇒ g(A1 ) ⊂ g(A2 ) ∀A1 , A2 ∈ PU
(33)
i.e. g in Eq. (32) preserves the order of subsethood as an input/output transformation. The extension Eq. (31) extends the property Eq. (33) to fuzzy subsets A1 ⊂ A2 ⇒ g˜(A1 ) ⊂ g˜(A2 ) ∀A1 , A2 ∈ P˜U
(34)
The following theorem exhibits the effect of the SOP in Eq. (33) on the extension Eq. (31). Theorem 3 Extension of a SOP SS mapping to a FF mapping. Consider a SS mapping g in Eq. (26), and assume g is a SOP mapping as stated in property Eq. (33). The extension Eq. (31) is equivalent to the following extension [˜ g (A)]α = g(Aα ) ∀A ∈ P˜U ∀α ∈ [0, 1] (35) Proof – Consider
=
∪
α · [˜ g(A)]α
∪
α · g(Aα )
0α1 0α1
(36)
and g˜(A) =
∪
α · g(Aα ) → [˜ g (A)]β = ∪ α · g(Aα ) 0α1
0α1
0α 0 2∆ v−∆
Consider the following set-valued glitch removing filter, g : PU → PV µg(A) (v) v+∆ 1 1 2∆ v−∆ µA (u)du = 0 otherwise
1 2
∆>0
where for discrete U , the integral is replaced by a summation. g is a SOP SS mapping so its fuzzy extension,
B.N. Araabi et al. / Restrictions imposed by the fuzzy extension of relations and functions
A1
1
17
A2
membership value
0 .9 0 .8 0 .7 0 .6 0 .5 0 .4 0 .3 0 .2 0 .1 0 0
0 .5
1
1 .5
2
2 .5
3
3 .5
4
u
(a) A1
1
A2
membership value
0 .9 0 .8 0 .7 0 .6 0 .5 0 .4 0 .3 0 .2 0 .1 0 0
0 .5
1
1 .5
2
2 .5
3
3 .5
4
u
(b) Fig. 5. g˜ in example 3 is a SOP FF mapping. Shaded areas in (a) and (b) indicate g˜(A1 ) ∪ g˜(A2 ) and g˜(A1 ∪ A2 ), respectively. They are not equal to each other. g˜ is not a cutworthy extension of a SOP SS mapping.
g˜, via Eq. (31) would be a cutworthy well-behaved one. Neither g(A1 ∪ A2 ) is equal to g(A1 ) ∪ g(A2 ), nor g˜(A1 ∪ A2 ) is equal to g˜(A1 ) ∪ g˜(A2 ). Figure 6 shows samples of this filtering scheme for both fuzzy and non-
fuzzy cases. The difference between g˜(A 1 ∪ A2 ) and g˜(A1 ) ∪ g˜(A2 ) has also been shown. This example shows that the converse of Corollary 1 does not hold, and justifies why two sets of conditions are not equiv-
18
B.N. Araabi et al. / Restrictions imposed by the fuzzy extension of relations and functions
A1
1
A2
membership value
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0
0.5
1
1.5
2
2.5
3
3.5
(a) A1
1
4
u
A2
membership value
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0
0.5
1
1.5
2
2.5
3
3.5
4
u
(b)
Fig. 6. Shaded areas in (a) and (b) indicate g(A1 ) ∪ g(A2 ) and g(A1 ∪ A2 ), respectively, where g is a SOP SS mapping defined in Example 4. Shaded areas in (c) and (d) indicate g˜(A1 ) ∪ g˜(A2 ) and g˜(A1 ∪ A2 ), respectively, where g˜ is a cutworthy fuzzy extension of a SOP SS mapping. However, neither g(A1 ) ∪ g(A2 ) is equal to g(A1 ∪ A2 ), nor g˜(A1 ) ∪ g˜(A2 ) is equal to g˜(A1 ∪ A2 ). This example shows that the converse of Corollary 1 does not hold.
alent. In Fig. 6, ∆ is chosen to be 0.1. Example 5 Eroding filter Consider the following set-valued eroding filter, g : PU → PV
µg(A) (v) =
1 0
1 2∆
v+∆
µ! (u)du ε otherwise
v−∆
1 < ε < 1, ∆ > 0 2
B.N. Araabi et al. / Restrictions imposed by the fuzzy extension of relations and functions
A1
1
19
A2
membership value
0 .9 0 .8 0 .7 0 .6 0 .5 0 .4 0 .3 0 .2 0 .1 0 0
0 .5
1
1 .5
2
2 .5
3
3 .5
4
u
(c) A1
1
A2
membership value
0 .9 0 .8 0 .7 0 .6 0 .5 0 .4 0 .3 0 .2 0 .1 0 0
0 .5
1
1 .5
2
2 .5
3
3 .5
4
u
(d) Fig. 6, continued.
where for discrete U , the integral is replaced by a summation. g is a SOP SS mapping so its fuzzy extension, g˜, via Eq. (31) would be a cutworthy and consistent one. Basic features are similar to Example 4. Figure 7 shows the difference between g˜(A 1 ∪ A2 ) and g˜(A1 ) ∪ g˜(A2 ). In Fig. 7, ε and ∆ are chosen to be 0.6 and 0.1, respectively.
5. Summary and conclusion The FEP in its original version is a tool to extend the domain of mappings from the elements of a referential set to the fuzzy subsets of that referential set. In general, this approach towards producing fuzzy mappings leads to a restricted family of FF mappings. Theorem 2
20
B.N. Araabi et al. / Restrictions imposed by the fuzzy extension of relations and functions
A1
1
A2
membership value
0 .9 0 .8 0 .7 0 .6 0 .5 0 .4 0 .3 0 .2 0 .1 0 0
0 .5
1
1 .5
2
2 .5
3
3 .5
4
u
(a) A1
1
A2
membership value
0 .9 0 .8 0 .7 0 .6 0 .5 0 .4 0 .3 0 .2 0 .1 0 0
0 .5
1
1 .5
2
2 .5
3
3 .5
4
u
(b) Fig. 7. g˜ in example 5 is a SOP FF mapping, and a cutworthy fuzzy extension of a SOP SS mapping. Shaded areas in (a) and (b) indicate g˜(A1 ) ∪ g˜(A2 ) and g˜(A1 ∪ A2 ), respectively. They are not equal to each other; however, g˜(A1 ) ∪ g˜(A2 ) ⊂ g˜(A1 ∪ A2 ).
together with Theorem 1 state that only a subfamily of fuzzy relational equations (fuzzy linear functions) can be reached via the FEP. Extending SS mappings leads to a larger family of FF mappings. Theorem 3 justifies Zadeh’s approach to extend SS mappings, which stems from the resolution identity; however, at the same time it shows that
this extension is much more reasonable when the SS mapping preserves subsethood order. In this case, the extension is a well-behaved cutworthy and consistent one. Theorem 4 is analogous to Theorem 2 for the extension of SS mappings with the necessary and sufficient conditions. Corollary 1 together with Theorem 5 show that the latter extension is more relaxed compared
B.N. Araabi et al. / Restrictions imposed by the fuzzy extension of relations and functions
21
Table 1 Paper summary g˜ is a fuzzy set to fuzzy set mapping, i.e., P˜U → P˜V , which maps A to g˜(A), ∀A ∈ P˜U
Starting non-fuzzy relation or mapping
Imposed restrictions on extended fuzzy set to fuzzy set mapping g~ (necessary and sufficient conditions)
Reconstruction
Subsethood order preserving set to set mapping g : PU → PV g~( A) =
µ g~ ( A) (v) = max µ A (u ) {u | ( u ,v )∈G }
0≤α ≤1
↓ g~( A1
A2 ) = g~ ( A1 )
α ⋅ g ( Aα )
⊂
Extension scheme
Crisp relation G ⊂ U ×V
↓ g~ ( A2 )
g~ (α ⋅ A) = α ⋅ g~ ( A) g~({u}) ∈ PV , ∀u ∈ U
⇒ / ⇐
[ g~( A)]α = g~ ( Aα ) g~( A) ∈ PV , ∀A ∈ PU With the above three restrictions g can be reconstructed uniquely from g~ using g ( A) = g~ ( A) ∀A ∈ PU
With the above three restrictions G can be reconstructed uniquely from g~ using µ G (u , v) = µ g~ ({u }) (v) ∀(u, v) ∈ U × V
to the original FEP. Therefore, a larger family of FF mappings is accessible via the latter extension. Table 1 provides a summary of the results reported in this paper. As seen in Table 1, the extension of relations to FF mappings generates a subfamily of all possible fuzzy relational equations. Considering the imposed constraints, it is too restrictive if we confine FF mappings to this subfamily. On the other hand, the extension of SS mappings via Eq. (31) is rational only for those SS mappings that preserve subsethood order. For this family of extended FF mappings, the imposed restrictions are merely preserving the subsethood order in the fuzzy domain and cutworthyness. Hence, for SOP SS mappings, the extension Eq. (31) is completely rational generating consistent results. This extension does not appear too restrictive when the SOP property is transported directly from set domain to fuzzy domain. The combination of cutworthyness and SOP property is similar to the smoothness condition during the interpolation of point-valued functions. However, it should be emphasized that the entire family of FF mappings, which is obtained in this manner, is still small and restricted as compared to the family of all possible FF mappings. In essence, the materials presented in this paper provide a better understanding of the nature and limita-
A1 ⊂ A2 ⇒ g~ ( A1 ) ⊂ g~ ( A2 )
tions of FF mappings that are accessible through extension of the non-fuzzy function and relations. How such restrictions affect the applications of the fuzzy set theory – where fuzzy extension has been utilized – can be explored in future research. References [1]
[2] [3] [4] [5]
[6] [7] [8]
[9]
B.N. Araabi and C. Lucas, On the fuzzy generalization of Dempster-Shafer theory, Proc. Int’l Conf. Intelligent and Cognitive Systems, (Vol. 1), ICICS-96, Tehran, Iran, 1996, pp. 161–166. H.B. Enderton, Elements of set theory, Academic press, New York, NY, 1977. M. Jamshidi, N. Vadiee and T.J. Rose, eds, Fuzzy logic and control, Prentice Hall, Englewood Cliffs, NJ, 1993. O. Kaleva and S. Seikkala, On fuzzy metric spaces, Fuzzy Sets and Systems 12 (1984), 215–229. A. Kaufmann and M.M. Gupta, Introduction to fuzzy arithmetic: Theory and applications, Van Nostrand, New York, NY, 1985. G.J. Klir and B. Yuan, Fuzzy sets and fuzzy logic: Theory and applications, Prentice Hall, Upper Saddle River, NJ, 1995. R. Lowen, Fuzzy topological spaces and fuzzy compactness, J. Math. Anal. Applications 56 (1976), 621–633. C. Lucas and B.N. Araabi, Generalization of the DempsterShafer theory: A fuzzy-valued measure, IEEE Trans. Fuzzy Systems 7 (1999), 255–270. H.T. Nguyen, A note on the extension principle for the fuzzy sets, J. Math. Anal. Applications 64 (1978), 369–380.
22 [10] [11] [12] [13] [14]
B.N. Araabi et al. / Restrictions imposed by the fuzzy extension of relations and functions W. Pedricz, Fuzzy control and fuzzy systems, Research Studies Press, Taunton, England, 1993. M.J. Wierman, Total extension of set functions and relations, Proc. Annual Meeting NAFIPS, New York, 1997, pp. 292–294. M.J. Wierman, Extending set functions and relations, Int’l J. of General Systems 26 (1997), 91–96. R.R. Yager, A characterization of extension principle, Fuzzy Sets and Systems 18 (1986), 205–217. L.A. Zadeh, The concept of linguistic variable and its
application to approximate reasoning-I, Inform. Sci. 8 (1975), 199–249. [15] L.A. Zadeh, Fuzzy sets and information granularity, in: Advances in Fuzzy Set Theory and Applications, M.M. Gupta et al., eds, North-Holland, Amsterdam, Nederland, 1979, pp. 3– 18. [16] H.J. Zimmermann, L.A. Zadeh and B.R. Gaines, eds, Fuzzy sets and decision analysis, North-Holland, New York, NY, 1984.
