Fuzzy Multisets and Their Generalizations Sadaaki Miyamoto Institute of Engineering Mechnics and Systems University of Tsukuba, Ibaraki 305-8573, Japan phone/fax:+81-298-53-5346
[email protected] Abstract. Fuzzy multisets with infinite membership sequences and their generalization using set-valued memberships are considered. Two metric spaces of the infinite fuzzy multisets are defined in terms of cardinality. One is the completion of fuzzy multisets of finite membership sequences; the other is derived from operations among fuzzy multisets and realvalued multisets. Theoretical properties of these infinite fuzzy multisets are discussed.
1
Introduction
Recently, many studies discuss multisets and their applications such as relational databases, information retrieval, and new computing paradigms. Multisets have sometimes been called bags. While the well-known book by Knuth [5] uses the term of multisets, another book by Manna and Waldinger [7] devotes a chapter to bags. The terms of multiset and bag can thus be used interchageably. The author prefers to use multisets but some readers may interpret them to be bags instead. This paper is concerned with a generalization of multisets, that is, fuzzy multisets [16,3,14,4,13,15,8,9,10,6]. The ordinary nonfuzzy multisets are called crisp multisets by the common usage in fuzzy systems theory. We discuss theoretical aspects of fuzzy multisets; in particular, the focus is on infinite features of multisets. Namely, infinite sequences of memberships and a distance between fuzzy multisets are discussed. Further generalizations of fuzzy multisets are defined using a closed set in a plane as the membership range. All arguments are elementary, and hence we omit proofs of the propositions.
2
Finite Multisets and Fuzzy Multisets
Before considering fuzzy multisets, a brief review of crisp multisets is useful. In the preliminary consideration we assume finite sets and multisets for simplicity. 2.1
Crisp Multisets
Let us begin by a simple example. Example 1. Assume that X = {x, y, z, w} is a set of symbols and suppose we have a number of objects but they are not distinguishable except their labels of C.S. Calude et al. (Eds.): Multiset Processing, LNCS 2235, pp. 225–235, 2001. c Springer-Verlag Berlin Heidelberg 2001
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x, y, z, or w. For example, we have two balls with the label x and one ball with y, three with label z, but no ball with the label w. Moreover we are not allowed to put additional labels to distinguish two x’s. Therefore a natural representation of the situation is that we have a collection {x, x, y, z, z, z}. We can also write
{2/x, 1/y, 3/z, 0/w}
to show the number for each element of the universe X, or {2/x, 1/y, 3/z} by ignoring zero of w. We will say that there are three occurrences of x, two occurrences of y, and so on. We proceed to general definitions. Assume X = {x1 , . . . , xn } is a finite set called the universe or the basis set. A crisp multiset M of X is characterized by the function Count M (·) whereby a natural number (including zero) corresponds to each x ∈ X, that is [1,5,7]: Count M : X → {0, 1, 2, . . . }. For a crisp multiset, different expressions such as M = {k1 /x1 , . . . , kn /xn } and
k1
kn
M = {x1 , . . . , x1 , . . . , xn , . . . , xn } are used. An element of X may thus appear more than once in a multiset. In the above example x1 appears k1 times in M , hence we have k1 occurrences of x1 . Consider the first example: {2/x, 1/y, 3/z}. We have Count M (x) = 2, Count M (y) = 1, Count M (z) = 3, Count M (w) = 0. The following are basic relations and operations for crisp multisets. (inclusion): (equality):
M ⊆ N ⇔ Count M (x) ≤ Count N (x),
∀x ∈ X.
(1)
M = N ⇔ Count M (x) = Count N (x),
∀x ∈ X.
(2)
(union): Count M ∪N (x) = max{Count M (x), Count N (x)} = Count M (x) ∨ Count N (x).
(3)
(intersection): Count M ∩N (x) = min{Count M (x), Count N (x)} = Count M (x) ∧ Count N (x).
(4)
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Count M ⊕N (x) = Count M (x) + Count N (x).
(5)
Count M N (x) = 0 ∧ (Count M (x) − Count N (x)).
(6)
(subtraction):
The symbols ∨ and ∧ are the infix notations of max and min, respectively. Readers should note that the operations resemble those for fuzzy sets, but the upper bound for Count(·) is not assumed. Example 2. Consider the multiset M in Example 1 and N = {1/x, 4/y, 3/w}. Then, M ⊕ N = {3/x, 5/y, 3/z, 3/w}, M ∪ N = {2/x, 4/y, 3/z, 3/w}, M ∩ N = {1/x, 1/y}, M N = {1/x, 3/z}. Real-valued multisets Blizard [2] generalized multisets to real-valued multisets, from which we remark nonnegative real-valued multiplicity. Thus, the Count function takes nonnegative real-values: (Count : X → R+ ∪ {0}). The basic relations and set operations use (1)–(6). 2.2
Fuzzy Multisets
Yager [16] first discussed fuzzy multisets, although he uses the term of fuzzy bag; an element of X may occur more than once with possibly the same or different membership values. Example 3. Consider a fuzzy multiset A = {(x, 0.2), (x, 0.3), (y, 1), (y, 0.5), (y, 0.5)} of X = {x, y, z, w}, which means that x with the membership 0.2, x with 0.3, y with the membership 0.5, and two y’s with 0.5 are contained in A. We may write A = {{0.2, 0.3}/x, {1, 0.5, 0.5}/y} in which the multisets of membership {0.2, 0.3} and {1, 0.5, 0.5} correspond to x and y, respectively. Count A (x) is thus a finite multiset of the unit interval [16]. The collection of all finite fuzzy multisets of X is denoted by FM(X).
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For x ∈ X, the membership sequence is defined to be the decreasingly-ordered sequence of the elements in Count A (x). It is denoted by (µ1A (x), µ2A (x), . . . , µpA (x)), where µ1A (x) ≥ µ2A (x) ≥ · · · ≥ µpA (x). When we define an operation between two fuzzy multisets, say A and B, the lengths of the membership sequences µ1A (x), µ2A (x), . . . , µpA (x), and µ1B (x), µ2B (x), . . . , µpB (x), should be set to be equal. We hence define the length L(x; A), the length of µjA (x): L(x; A) = max{j : µjA (x) = 0 }, and define
L(x; A, B) = max{L(x; A), L(x; B)}.
When no ambiguity arises, we write L(x) = L(x; A, B) for simplicity. Example 4. Let A = {{0.2, 0.3}/x, {1, 0.5, 0.5}/y}, B = {{0.6}/x, {0.8, 0.6}/y, {0.1, 0.7}/w}. For the representation of the membership sequence, we get L(x) = 2,
L(y) = 3,
L(z) = 0,
L(w) = 2
and we have A = {(0.3, 0.2)/x, (1, 0.5, 0.5)/y, (0, 0)/w}, B = {(0.6, 0)/x, (0.8, 0.6, 0)/y, (0.7, 0.1)/w}. The following are basic relations and operations for fuzzy multisets [8]: 1. [inclusion] A ⊆ B ⇔ µjA (x) ≤ µjB (x), j = 1, . . . , L(x),
∀x ∈ X.
(7)
A = B ⇔ µjA (x) = µjB (x), j = 1, . . . , L(x),
∀x ∈ X.
(8)
2. [equality] 3. [addition] A⊕B is defined by the addition operation in X ×[0, 1] for crisp multisets [16]. Namely, if A = {(xi , µi ), . . . , (xk , µk )} and
B = {(xp , µp ), . . . , (xr , µr )}
are two fuzzy multisets, then A ⊕ B = {(xi , µi ), . . . , (xk , µk ), (xp , µp ), . . . , (xr , µr )}.
(9)
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4. [union]
229
µjA∪B (x) = µjA (x) ∨ µjB (x), j = 1, . . . , L(x).
(10)
µjA∩B (x) = µjA (x) ∧ µjB (x), j = 1, . . . , L(x).
(11)
5. [intersection] 6. [α-cut] The α-cut (α ∈ (0, 1]) for a fuzzy multiset A, denoted by [A]α , is a crisp multiset defined as follows: µ1A (x) < α ⇒ Count [A]α (x) = 0,
(12)
µjA (x)
(13) (14)
≥ α,
µj+1 A (x)
< α ⇒ Count [A]α (x) = j, j = 1, . . . , L(x).
Moreover the strong α-cut (α ∈ [0, 1)), denoted ]A[α , is a crisp multiset defined as follows: µ1A (x) ≤ α ⇒ Count ]A[α (x) = 0,
(15)
µjA (x)
(16)
> α,
µj+1 A (x)
≤ α ⇒ Count ]A[α (x) = j,
j = 1, . . . , L(x).
(17)
7. [Cartesian product] Given two fuzzy multisets A = {(x, µ)} and B = {(y, ν)}, the Cartesian product is defined by: A×B = (x, y, µ ∧ ν) (18) The combination is taken for all (x, µ) in A and (y, ν) in B. 8. [Multirelation] Notice that a crisp relation R on X is a subset of X × X. Given a fuzzy multiset A of X, a multirelation R obtained from R is a subset of A × A: for all (x, µ), (y, ν) ∈ A, (x, y, µ ∧ ν) ∈ R ⇐⇒ (x, y) ∈ R When R is a fuzzy relation on X, then (x, y, µ ∧ ν ∧ R(x, y)) ∈ R.
(19)
(The latter includes the former as a special case.) The following propositions are valid. The proofs are not difficult and hence omitted. Proposition 1. Assume that A and B are fuzzy multisets of X. The necessary and sufficient condition for A ⊆ B is that for all α ∈ (0, 1], [A]α ⊆ [B]α . Moreover, the condition for A = B is that for all α ∈ (0, 1], [A]α = [B]α . Proposition 2. Assume that A and B are fuzzy multisets of X. The necessary and sufficient condition for A ⊆ B is that for all α ∈ [0, 1), ]A[α ⊆ ]B[α . Moreover, the condition for A = B is that for all α ∈ [0, 1), ]A[α = ]B[α .
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Proposition 3. Assume that A and B are fuzzy multisets of X. Take an arbitrary α ∈ (0, 1). We then have: [A ∪ B]α = [A]α ∪ [B]α ,
[A ∩ B]α = [A]α ∩ [B]α , [A × B]α = [A]α × [B]α
[A ⊕ B]α = [A]α ⊕ [B]α , ]A ∪ B[α = ]A[α ∪ ]B[α , ]A ⊕ B[α = ]A[α ⊕ ]B[α ,
]A ∩ B[α = ]A[α ∩ ]B[α , ]A × B[α = ]A[α × ]B[α .
Proposition 4. Assume that A, B, and C are fuzzy multisets of X. The followings equalities are valid: A ∪ B = B ∪ A, A ∩ B = B ∩ A, A ∪ (B ∪ C) = (A ∪ B) ∪ C, A ∩ (B ∩ C) = (A ∩ B) ∩ C, (A ∩ B) ∪ C = (A ∪ C) ∩ (B ∪ C), (A ∪ B) ∩ C = (A ∩ C) ∪ (B ∩ C). Therefore, the class of all fuzzy multisets of a particular universe forms a distributive lattice. The cardinality and a metric space of fuzzy multisets The cardinality of a fuzzy multiset A is given by |A| =
L(x;A) x∈X
Moreover, we define
j=1
L(x;A)
|A|x =
j=1
µjA (x).
µjA (x).
We thus have |A| = x∈X |A|x . We can define a metric space of fuzzy multisets. Namely, given two multisets A, B ∈ FM(X), the distance between A and B is defined by L(x;A,B)
|µjA (x) − µjB (x)|.
(20)
d(A, B) ≥ 0; d(A, B) = 0 ⇐⇒ A = B, d(A, B) = d(B, A),
(21) (22)
d(A, C) ≤ d(A, B) + D(B, C),
(23)
d(A, B) =
x∈X
j=1
We can immediately have
hence the set of fuzzy multisets becomes a metric space.
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Infinite Features of Fuzzy Multisets
So far all discussions are concerned with finite multisets. It is possible to introduce the infinity {+∞} into the consideration of crisp multisets: Count M (x):X→ {0, 1, . . . , ∞}. However, for the purpose of simple and effective calculations, we consider Count M (x) as being always finite. This assumption of finite crisp multisets is useful for focusing the infinite features of fuzzy multisets. 3.1
Infinite Membership Sequences
Infinite fuzzy multisets means that for x ∈ X, Count A (x) may be an infinite set of the unit interval I = [0, 1]. We note that α-cuts of a fuzzy multiset A gives well-defined crisp multisets [A]α and ]A[α for α ∈ (0, 1). Since we assume a crisp multiset of finite cardinality (Count M (x) < ∞), an infinite subset of I does not necessarily provide a well-defined crisp multiset. Hence, instead of an infinite set, we use a sequence (ν 1 , ν 2 , . . . ) as the Count function. Namely we assume Count A (x) = (ν 1 , ν 2 , . . . )
(24)
where ν j → 0 as j → 0. Basic relations and operations for infinite fuzzy multisets should also be defined in terms of membership sequences using (7)–(19) with the associated sequences being infinite (e.g., L(x) → ∞). We assume that, regarding fuzzy multisets, all sorting of the infinite membership sequence in (24) can be performed by an effective computational procedure. The class of all infinite fuzzy multisets of X is denoted here by FM0 (X). We immediately have the next proposition. Proposition 5. For A, B ∈ FM0 (X), (i) A ⊕ B ∈ FM0 (X), (ii) A ∪ B ∈ FM0 (X), (iii) A ∩ B ∈ FM0 (X). 3.2
The Metric Space of Infinite Fuzzy Multisets
We consider the subset of FM0 (X) in which the cardinality is finite: |A| =
∞ x∈X j=1
µjA (x) < ∞.
(25)
This subset is denoted by FM1 (X): FM1 (X) = { |A| < ∞ : A ∈ FM0 (X) }. For A, B ∈ FM1 (X), the distance is naturally defined by d(A, B) =
∞ x∈X j=1
|µjA (x) − µjB (x)|.
(26)
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The axioms of a metric, (21)–(23), are satisfied for d(A, B). The next proposition is similar to Proposition 5. Proposition 6. For A, B ∈ FM1 (X), (i) A ⊕ B ∈ FM1 (X), (ii) A ∪ B ∈ FM1 (X), (iii) A ∩ B ∈ FM1 (X). The space FM1 (X) of infinite fuzzy multisets is justified as follows. Suppose we have an infinite Cauchy sequence of finite fuzzy multisets A1 , A2 , · · · ∈ FM(X): d(Ai , Aj ) → 0, as i, j → ∞. Then, the sequence converges to a fuzzy multiset A ∈ FM1 (X). On the other hand, for an arbitrary A ∈ FM1 (X), there exists a sequence of finite fuzzy multisets A1 , A2 , · · · ∈ FM(X) such that Aj → A as j → ∞. The metric space of infinite fuzzy multisets is complete, whereas that of finite fuzzy multisets is not.
4
A Set-Valued Membership: A Generalization of Fuzzy Multisets
The membership sequence can be identified with a real-valued step function with the nonnegative real variable y ∈ R+ ∪ {0} µ1A (x), (y = 0), fA (y; x) = (27) µjA (x), (j − 1 < y ≤ j, j = 1, 2, . . . ). A crisp multiset M is represented by a simple function of fM (y; x) = 1,
0 ≤ y ≤ Count M (x),
(28)
which is a particular case of Equation (27) by putting µjM (x) = 1 (j = 1, . . . , Count M (x)). Notice that the last function, (28), is naturally extended to the case of positive real-valued multisets using the same definition (28) except that Count M (x) is real-valued. We thus assume that, instead of the membership sequences, fuzzy multisets and positive-real valued multisets are characterized by these functions. Let us consider these functions in a unified framework, namely, we allow operations between fA (y; x) by (27) and fM (y; x) by (28). The operations of union and intersection are immediately defined by using fA (y; x) ∨ fM (y; x) and fA (y; x) ∧ fM (y; x), respectively. On the other hand, the addition, which is characterisctic to the multisets, cannot directly be defined from these functions. For the operation of the addition, we use a closed set νA (x) in the (y, z)-plane naturally derived from fA (y; x). Namely, we define (29) νA (x) = (y, z) ∈ R2 : 0 ≤ y ≤ supp fA (·; x), 0 ≤ z ≤ fA (y; x) ,
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where supp fA (·; x) = { y : fA (y; x) = 0 }. We moreover define a function gA (·; x) of the variable z: gA (z; x) = sup y : (y, z) ∈ νA (x) . (30) We can now define the addition operation by using gA⊕M (z; x) = gA (z; x) + gM (z; x), whereby the set νA⊕B (x) is defined by νA⊕B (x) = (y, z) ∈ R2 : 0 ≤ z ≤ supp gA⊕M (·; x), 0 ≤ y ≤ gA⊕B (z; x) .
(31)
(32)
We now proceed to generalize the class of functions fA (y; x). This generalization is necessary when we consider fuzzy multisets and real-valued multisets in the same framework. We easily note that the arbitrary monotone non-increasing step functions are included in this set since they are derived from operations between fuzzy multisets and real-valued multisets. Notice also that the cardinality is extended to the norm of the function fA (y; x); thus we define
∞ fA (y; x)dy, (33) |fA (·; x)| = 0 fA = |fA (·; x)|. (34) x∈X
An appropriate class of fA is the set of monotone nonincreasing and uppersemicontinuos functions whose norms are finite. This class is denoted by GFM1 (X) = fA : fA < ∞ . This class is justified from the fact that for an arbitrary f ∈ GFM1 (X), we can take a sequence of monotone non-increasing step functions that converges to f . This class is moreover the completion of the class of all monotone nonincreasing step functions whose cardinality is finite. By abuse of terminology we write A ∈ GFM1 (X). Since fA and the associated νA (x) is interchangeable, we are considering a generalized fuzzy multiset A characterized by the closed set νA (x) for each x ∈ X. Operations on the generalized fuzzy multisets are now summarized. (i) inclusion (ii) equality (iii) union
A ⊆ B ⇐⇒ νA (x) ⊆ νB (x),
∀x ∈ X.
A = B ⇐⇒ νA (x) = νB (x),
∀x ∈ X.
νA∪B (x) = νA (x) ∪ νB (x),
∀x ∈ X.
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(iv) intersection νA∩B (x) = νA (x) ∩ νB (x),
∀x ∈ X.
(v) addition Define gA (z; x) and gB (z; x) by (30), and calculate gA⊕B (z; x) using (31), whereby νA⊕B (x) is obtained from (32). It is easily seen that the GFM1 is a distributive lattice. This class is a metric space where the metric is defined by ∞ |fA (y; x) − fB (y; x)|dy. d(A, B) = fA − fB = x∈X
5
0
Conclusion
We have reviewed fuzzy multisets and discussed features of infiniteness in fuzzy multisets. It should be noticed that an α-cut of such a fuzzy multiset with an infinite membership sequence will produce a finite crisp multiset. Further generalization using a closed set in a plane has been considered. We thus observe analytical features in fuzzy multisets in addition to algebraic properties from the discussion of metric spaces and analytical completion. Future studies include further development of the theory, e.g., t-conorms in fuzzy multisets and further generalization of the multisets. Moreover, the relation between fuzzy multisets and rough sets [12] should be investigated. Applications of fuzzy multisets include modeling of information retrieval systems [11], which we will discuss elsewhere.
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