Fuzzy Relations and Fuzzy Functions in Fuzzy Partial

Fuzzy Relations and Fuzzy Functions in Fuzzy Partial Set Theory Martina Daˇ nková Centre of Excellence IT4Innovations, division of the University of Ostrava, Institute for Research and Applications of Fuzzy Modeling, 30. dubna 22, 701 03 Ostrava 1, Czech Republic [email protected], WWW home page: http://irafm.osu.cz

Abstract. In this paper, we will study partially-defined fuzzy relations i.e., fuzzy relations with membership functions that are not necessarily defined everywhere. We will handle them in a suitable framework that is a fuzzy partial set theory. It provides tools for dealing with undefined values of the membership by means of special operations based on the connectives and quantifiers of a background fuzzy logic. We analyze a suitability of operations of a fuzzy partial set theory for a meaningful definition of the functionality property. This property determines fuzzy functions and we will be concerned with their properties. Keywords: Fuzzy function, Partial function, Undefined values, Functional relation

1

Introduction

Fuzzy partial set theory (FPST) that has been introduced in [1], formalizes fuzzy sets that can have undefined membership degrees and offers basic fuzzy set operations and properties for their handling. Consequently, it provides tools for dealing with undefined values of the membership. The background logic for FPST is a fuzzy partial propositional logic that has been proposed in [2]. We will touch only a few aspects of the theory. Our purpose is to present first steps towards analysis of fuzzy functions in FPST. In our approach, fuzzy function is a fuzzy relation fulfilling the functionality property – a direct generalization of the classical property that specifies functions out of relations. In the fuzzy community the functionality property has been studied by many authors [3–6] and it is also known as the unique mapping [7]. We will propose the notion of functionality for fuzzy relations with variable domains in agreement with our intuitive expectations. Further, we will study properties of fuzzy functions w.r.t. fuzzy set operations and a relational composition. We stem from [8] where the functionality has been explored in the framework of Fuzzy Class Theory [9]. We define the semantics of a simple first-order extension of fuzzy partial propositional logic and a simple theory of fuzzy partial

2

Martina Daˇ nkov´ a

sets of the first order. We next introduce a selection of basic fuzzy partial set– theoretic notions and present a few results about these notions. Because of space limitation, we omit all proofs; they will be given in the upcoming full paper.

2

Fuzzy Partial Sets

We call fuzzy sets that can have undefined membership degrees—fuzzy partial sets. We identify fuzzy partial sets with pairs A = (XA , µA ), where XA ⊆ U is a crisp domain of A, U is a universe of discourse, and µA : XA → L is a membership function from XA to a suitable structure L of membership degrees. We denote this fact by A ⊂ ∼ XA . Whenever it is clear from the context, we refer to fuzzy partial sets simply as fuzzy sets. But a reader should keep in mind that from the point of view of the universe U a fuzzy set A ⊂ ∼ XA is partial. 2.1

The Representation of Fuzzy Partial Sets

The main idea of the representation of fuzzy partial sets is to replace undefined membership values by a dummy element • that stands outside the scale for truth values L and is incomparable with any a ∈ L. Consequently, original partial membership functions to L of fuzzy sets (with undefined membership values) are replaced by total functions to the extended scale L ∪ {•} that represent fuzzy partial sets. Let A = (XA , µA ) be a fuzzy set, where XA is a crisp domain and µA : XA → L is a membership function from XA to a suitable structure L of membership degrees. We introduce a new dummy index • ∈ / L as a new membership degree, designed for undefined membership values. And we set µA (x) = • for all x ∈ / XA . Definition 1. Let L ̸= ∅ and L• = L ∪ {•}. We shall say that a fuzzy set A = (XA , µA ) in a universe U ⊇ XA is represented by a L• -valued membership function µ˙ A on X, defined for each x ∈ U as: { µA (x) if x ∈ XA µ˙ A (x) = (1) • if x ∈ U \ XA . This representation is clearly one to one correspondence between fuzzy partial sets on the universe U and L• -valued functions on U . Note that the original fuzzy set A (Figure 1(a)) can be recovered from µ˙ A (Figure 1(b)) by setting XA = {x ∈ U | µ˙ A (x) ̸= •} and defining µA as the restriction of µ˙ A to XA . Therefore we may handle fuzzy partial sets by means of total L• -valued functions on a common unverse U . 2.2

Operations with Undefined Degrees

Then, a natural question arise: how do we define usual fuzzy set-theoretical operations such as the unions or intersections of fuzzy sets for fuzzy sets on different

Fuzzy Functions

(a) The fuzzy set A = (XA , µA ).

3

(b) The representation µ˙ A for A.

Fig. 1. The representation of the fuzzy set A = (XA , µA ) on a universe U ⊇ XA .

domains? There are several meaningful options that follows from suitable extensions of connectives used in definitions of the intended operations. In this paper, we will recall only two families of extended connectives descended from classical 3-valued connectives [?,?,10]. For more explanation and other extensions see [2]. In the sequel, let us assume L be an MTL-algebra of the form L = ⟨L, ∨, ∧, ∗, ⇒, 0, 1⟩.

(2)

We will call the operation ∗ product and ⇒ residuum. Convention: To reduce the number of parenthesis used in mathematical expressions we set that ∗ has the highest priority and ⇒ the lowest priority out of all operations that are at the disposal. Definition 2. Let L be an MTL-algebra of the form (2). – The Bochvar operation cB ∈ {∧B , ∨B , ∗B , ⇒B }, cB : L• × L• → L• is interpreted by the following truth table for all binary operations of L (and similarly for higher and lower arities): cB α •

•

β

αcβ • •

(3)

•

– The Soboci´ nski operation cS ∈ {∧S , ∨S , ∗S }, cS : L• × L• → L• , which treat • as the neutral element; and the Soboci´ nski-style residuum ⇒S residuated with ∗S : cS

β

•

⇒S

α αcβ α

α

•

•

β

•

β

•

α ⇒ β ¬α β

•

(4)

4


The names Bochvar and Soboci´ nski in the above definitions of operations have been chosen according to the interpretation of three-valued connectives (namely, Bochvar and Soboci´ nski conjunction and disjunction, see, e.g., [10]) with which they coincide on the three values {0, 1, •}. Definition 3. Let αi ∈ L• for each i ∈ I (where I is an arbitrary index set). Then we define: { ∧ inf αi if αi ̸= • for each i ∈ I – The Bochvar infimum αi = i∈I B • otherwise i∈I   sup αi if αi ̸= • for each i ∈ I ∨ – The Bochvar supremum αi = i∈I B • otherwise i∈I   inf αi if αi ̸= • for some i ∈ I ∧ i∈I nski infimum αi = αi ̸=• – The Soboci´ S  i∈I • otherwise    sup αi if αi ̸= • for some i ∈ I ∨ – The Soboci´ nski supremum αi = αi∈I i ̸=• S  • i∈I otherwise. Observe that all four operators coincide with the usual supremum and infimum ∧ ∨on L if all of their operands are defined. Otherwise, the Bochvar operators B , B yield the undefined ∨ • as soon as any operands are undefined, while ∧ value the Soboci´ nski operators S , S ignore the undefined values and only yield • if all of their operands are undefined. Remark 1. The Bochvar (Soboci´ nski) infimum and supremum correspond with the Bochvar (Soboci´ nski) universal and existential quantifiers, respectively, see [1]. 2.3

Unions and Intersections of Fuzzy Partial Sets

Definition 4. Let us consider two fuzzy sets A = (XA , µA ) and B = (XB , µB ): – We define A ∪B B = (XA ∩ XB , µA∪B ) A ∩B B = (XA ∩ XB , µA∩B ) A eB B = (XA ∩ XB , µAeB )

Bochvar union of A and B Bochvar intersection of A and B Bochvar strong intersection of A and B

where µA∪B B (x) = µA (x) ∨ µB (x), for all x ∈ XA ∩ XB µA∩B B (x) = µA (x) ∧ µB (x), for all x ∈ XA ∩ XB µAeB B (x) = µA (x) ∗ µB (x), for all x ∈ XA ∩ XB Bochvar union is denoted by ∪B , Bochvar intersection is denoted by ∩B and Bochvar strong intersection is denoted by eB .

Fuzzy Functions

5

– We define Soboci´ nski union of A and B is defined by A ∪S B = (XA ∪ XB , µA∪S B ) A ∩S B = (XA ∪ XB , µA∩S B )

Soboci´ nski union of A and B Soboci´ nski intersection of A and B

A eS B = (XA ∪ XB , µAeS B )

Soboci´ nski strong intersection of A and B

where µA∪S B (x) = µA (x) ∨ µB (x),

for x ∈ XA ∩ XB ,

µA∩S B (x) = µA (x) ∧ µB (x), µAeS B (x) = µA (x) ∗ µB (x),

for x ∈ XA ∩ XB , for x ∈ XA ∩ XB ,

µA∪S B (x) = µA∩S B (x) = µAeS B (x) = µA (x), µA∪S B (x) = µA∩S B (x) = µAeS B (x) = µB (x),

for x ∈ XA \ XB , for x ∈ XB \ XA .

That is, the membership degree of x in A ∪B B is considered to be defined only if the membership degrees of x in both A and B are defined. By means of operations given in Definition 2, the representation of Bochvar and Soboci´ nski unions and intersections can be expressed in a uniform way: µ˙ A∩B B (x) = µ˙ A (x) ∧B µ˙ B (x) µ˙ AeB B (x) = µ˙ A (x) ∗B µ˙ B (x)

(5) (6)

µ˙ A∪B B (x) = µ˙ A (x) ∨B µ˙ B (x) µ˙ A∩S B (x) = µ˙ A (x) ∧S µ˙ B (x) µ˙ AeS B (x) = µ˙ A (x) ∗S µ˙ B (x)

(7) (8) (9)

µ˙ A∪S B (x) = µ˙ A (x) ∨S µ˙ B (x),

(10)

for x ∈ U . Thus, Bochvar and Soboci´ nski operations with fuzzy sets can be defined straightforwardly by means of Bochvar and Soboci´ nski operations with their representations. 2.4

Characteristics of Fuzzy Partial Sets

Characteristics that have been introduced for fixed-domain fuzzy sets can be modified for fuzzy partial sets in a straightforward manner. Since, we have two types of operations at a disposal, consequently, more than one meaningful modification is available. It appears, there are also several characteristics which are meaningful for fuzzy partial sets. Definition 5. Let A = (XA , µA ) be a fuzzy set, XA ⊆ U , and µ˙ A the representation of A due to Definition 1. – We say that A is total on U and write TotX (A) (or simply Tot(A) if the universe U is fixed) if dom A = U , i.e., if µ˙ A (x) ̸= • for all x ∈ U .

6


– We say that A is crisp and write Crisp(A) if µA (x) ∈ {0, 1} for all x ∈ XA , i.e., if µ˙ A (x) ∈ {0, 1, •} for all x ∈ U . Alongside, we will use binary relations between fuzzy partial sets, such as equality and inclusion. Definition 6. Let A = (XA , µA ) and B = (XB , µB ) be fuzzy sets, where µA : XA → L and µB : XB → L, and let U ⊇ XA ∪ XB . Then we say that: 1. A and B are strongly equal, written A = B, if XA = XB and µA (x) = µB (x) for all x ∈ XA . 2. A is a subfunction of B, written A =sub B, if XA ⊆ XB and µA (x) = µB (x) for all x ∈ XA . 3. A is strongly included in B, written A ⊆ B, if XA = XB and µA (x) ≤ µB (x) for all x ∈ XA . 4. A is subincluded in B, written A ⊆sub B, if XA ⊆ XB and µA (x) ≤ µB (x) for all x ∈ XA . Remark 2. The relations introduced in Definition 6 are bivalent (yes/no). Graded notions of inclusion and equality ∧ [11], [12, Sect. 18.2.2]) can be defined too, ∧ (cf. e.g., by means of the operators B , S and ⇒B , ⇒S of Section 2.2. For instance, the degree of Bochvar–Soboci´ nski inclusion might be defined as: ∧ ( ) (A ⊆BS B) = µ˙ A (x) ⇒S µ˙ B (x) . B x∈U

In this paper, though, we shall leave graded relations between fuzzy partial sets aside.

3

Fuzzy Relations in FPST

Let us start with the Cartesian product of (two) fuzzy sets. In our apparatus, we set µ˙ A×B (x, y) = • for (x, y) outside the domain of A × B. As can be easily observed, it is the Bochvar extension ∗B of the product ∗, which should then be used to make the domain of A × B equal to XA × XB . Therefore we define: Definition 7. Let A = (XA , µA ) and B = (YB , µB ) be fuzzy sets, where µA : XA → L and µB : YB → L. Let U ⊇ XA ∪ YB and let µ˙ A , µ˙ B : U → L• be the representations of A, B. The Bochvar Cartesian product A ×B B is defined by the representation µ˙ A×B B : U × U → L• as follows: µ˙ A×B B (x, y) = µ˙ A (x) ∗B µ˙ B (y)

for each (x, y) ∈ U × U .

Remark 3. Expanding the definition of ∗B , we obtain: { µA (x) ∗ µB (y) if x ∈ XA and y ∈ YB µ˙ A×B B (x, y) = • otherwise.

(11)

Fuzzy Functions

7

Thus, the Bochvar Cartesian product thus captures the usual definition of Cartesian product of fuzzy sets. Besides the Bochvar product ∗B , we could also use another extension of ∗ to L• in (11), for instance the Soboci´ nski product ∗S We can define the notion of binary fuzzy relation between two fuzzy sets as follows: Definition 8. Let A = (XA , µA ) and B = (YB , µB ) be fuzzy sets, and U ⊇ XA , YB a common universe. Let R = (XR , µR ), where XR ⊆ U × U and µR : XR → L. We say that R is a fuzzy relation between A and B if R ⊆sub A ×B B. If A = B, we speak of fuzzy relations on A. The notion of binary fuzzy relations between two crisp sets is a special case of the one given in the above definition. Proposition 1. Let R, R1 , R2 be fuzzy relations between fuzzy sets A and B. 1. If A ⊆sub A′ and B ⊆sub B ′ , then R is also a fuzzy relation between A′ and B ′ . 2. The fuzzy relations R1 ∩B R2 , R1 ∩S R2 ,R1 eB R2 , R1 eS R2 , are fuzzy relations between A and B as well. 3. If A and B are crisp, then analogous claims hold also for ∪B , ∪S and dB , dS . 4. λ = (∅, ∅) is the smallest and A ×B B the largest fuzzy relation between A and B with respect to subinclusion ⊆sub .

4

Relational Composition of Fuzzy Relations

Relational composition sup-T of fuzzy relations is intended to have the following values: { µR◦S (x, y) for (x, y) ∈ A × C (12) µ˙ R◦SB S (x, y) = • for (x, y) ∈ (U × U ) \ (A × C) It leads to the following setting of operations: a domain of composition is obtained when using the Bochvar operation (∗B for combining the degrees of both relations and the Soboci´ nski supremum for aggregating them. Hence, we obtain the following generalization of sub-T composition of fuzzy relations: Definition 9. Let R = (A×B, µR ) and S = (B ×C, µS ), where µR : A×B → L and µS : B×C → L be fuzzy relations and U ⊇ A∪B∪C a common universe. Let µ˙ R , µ˙ S : U × U → L• be the representation of R, S as in Definition 1. Then, we define the Soboci´ nski–Bochvar sup-T composition R ◦SB S as the fuzzy relation on U such that for all x, y ∈ U : ∨( ) µ˙ R◦SB S (x, y) = µ˙ R (x, z) ∗B µ˙ S (z, y) . S z∈U

8


5

Fuzzy Functions in FPST

In the sequel, let A and B be crisp sets, U ⊇ A, B be a common universe, and ≈i = (X≈i , µ≈i ), for i = 1, 2, be fuzzy relations on A, B, respectively. Moreover, let R = (XR , µR ) be a fuzzy relation between A and B. We require from the definition of fuzzy function the following: µ≈1 (x, x′ ) ∗ µR (x, y) ∗ µR (x′ , y ′ ) ≤ µ≈2 (y, y ′ ),

(13)

for all (x, x′ ) ∈ X≈1 , (y, y ′ ) ∈ X≈2 and (x, y), (x′ , y ′ ) ∈ XR . This specification leads to the definition using the representation of fuzzy sets which combines Soboci´ nski infimum and Bochvar operations. Definition 10. Let µ˙ R , µ˙ ≈1 , µ˙ ≈2 : U × U → L• be the representation of R, ≈1 , ≈2 as in Definition 1. – We say that R is a fuzzy function between A and B w.r.t. ≈1 , ≈2 if ∧ S x,x′ ,y,y ′ ∈U

(µ˙ ≈1 (x, x′ ) ∗B µ˙ R (x, y) ∗B µ˙ R (x′ , y ′ ) ⇒B µ˙ ≈2 (y, y ′ )) = 1.

(14)

– We say that R is a function between A and B if R is a fuzzy function between A and B w.r.t. =, =, and moreover, Crisp(R). 5.1

Set operations and the relational composition with fuzzy functions

Let us summarize properties of fuzzy functions. Theorem 1. Let R = (XR , µR ) and S = (XS , µS ) be fuzzy functions between A and B, respectively, w.r.t. ≈1 , ≈2 , moreover, let T = (XT , µT ) be a fuzzy function between B and C w.r.t. ≈2 , ≈3 Then – R ∩B S = (XR ∩XS , µR∩B S ), R ∩S S = (XR ∪XS , µR∩S S ) are fuzzy functions between A and B, respectively, w.r.t. ≈1 , ≈2 , – R eB S = (XR ∩ XS , µReB S ) is a fuzzy function between A and B, respectively, w.r.t. ≈1 eB ≈1 , ≈2 eB ≈2 . Figure 3 demonstrates Bochvar and Soboci´ nski intersections of crisp functions. Note that R eS S = (XR ∪ XS , µReS S ) is not a fuzzy function between A and B, respectively, w.r.t. ≈1 eB ≈1 , ≈2 eB ≈2 , because R and S are not fuzzy functions w.r.t. ≈1 eB ≈1 , ≈2 eB ≈2 on XR \ XS and XS \ XR , respectively. We will demonstrate it on a simple example over the standard Lukasiewicz algebra. Put R = ([0, 0.5]2 , µR ), S = ([0, 1]2 , µS ) and ≈1 = ≈2 = S, where µS (x, y) =df (1 − |x − y|) ∨ 0 and µR (x, y) = µS (x, y) for x, y ∈ [0, 0.5]2 and µ≈1 = µ≈2 = µS . Then, R eB R is a fuzzy function between [0, 1] and [0, 1] w.r.t.

Fuzzy Functions

(a) A function R = (XR , µR )

9

(b) A function S = (XS , µS )

Fig. 2. Functions with different domains.

≈1 eB ≈1 , ≈2 eB ≈2 . But R eS R is not a fuzzy function w.r.t. ≈1 eB ≈1 , ≈2 eB ≈2 , take e.g. x, x′ = 0.6, y = 0.6, y ′ = 0.7 then µ˙ ≈1 (x, x′ ) ∗B µ˙ ReS R (x, y) ∗B µ˙ ReS R (x′ , y ′ ) ⇒B µ˙ ≈2 (y, y ′ ) = 1 ∗B 1 ∗B 0.9 ∗B • ⇒ 0.9 ∗B 0.9 < 1. | {z } 0.8

Theorem 2. Let the assumptions of Theorem 1 hold. Moreover, let (y, y ′ ) ∈ ≈2 for all x, x′ , z, z ′ such that (x, y), (x′ , y ′ ) ∈ R and (y, z), (y ′ , z ′ ) ∈ T . Then R ◦SB T is a fuzzy function between A and C w.r.t. ≈1 , ≈3 ,

(a) The function R ∩B S

(b) The function R ∩S S

Fig. 3. Intersections of the functions from Figure 2.

Theorem 3. Let R = (XR , µR ) and S = (XS , µS ) be fuzzy functions between A and B w.r.t. ≈1 , ≈2 . If R ∪S S is a fuzzy function between A and B w.r.t. ≈1 , ≈2 then R and S are fuzzy functions between A and B w.r.t. ≈1 , ≈2 . If R ∪B S is a fuzzy function between A and B w.r.t. ≈1 , ≈2 then R and S need not be fuzzy functions between A and B w.r.t. ≈1 , ≈2 . It is easy to find a crisp

counterexample see Figure 4, where the Bochvar-union R ∪B S is crisp function but R is not.

(a) The function R ∪B S

(b) A relation R

(c) A function S

Fig. 4. Bochvar-union of relations.

6

Conclusions

We have introduced basic notions of the fuzzy set and fuzzy relational calculus in FPST. The main focus was put to the section devoted to fuzzy functions for which we have provided some basic properties and illustrative examples. Let us emphasize the fact that results from the classical fuzzy set theory are not directly transferable to FPST. As an example we recall Theorem 2, where we have to add non-trivial requirements. Moreover, we have shown that there are properties of fuzzy functions that hold in the classical fuzzy set theory but not in FPST for all types of fuzzy partial set operations.

Acknowledgement The work was supported by grant No. 1619170S “Fuzzy partial logic” of the Czech Science Foundation.

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