Multisets and Fuzzy Multisets as a Framework of Information Systems

Multisets and Fuzzy Multisets as a Framework of Information Systems Sadaaki Miyamoto Department of Risk Engineering School of Systems and Information Engineering University of Tsukuba, Ibaraki 305-8573, Japan [email protected]


Abstract. Multisets are now a common tool and a fundamental framework in information processing. Their generalization to fuzzy multisets has also been studied. In this paper the basics of multisets and fuzzy multisets are reviewed, fundamental properties of fuzzy multisets are proved, and advanced operations are defined. Applications to rough sets, fuzzy data retrieval, and automatic classification are moreover considered.

1

Introduction

Recently many studies discuss multisets and their applications such as databases, information retrieval, and new computing paradigm [5]. Multisets have sometimes been called bags. Indeed, while the well-known book by Knuth [9] uses the term of multisets, another book by Manna and Waldinger [11] devotes a chapter to bags. The terms of multiset and bag can thus be used interchangeably. Although the author prefers to use the term multisets, some readers may interpret them to be bags instead. This paper discusses a generalization of multisets, that is, fuzzy multisets [28, 7, 24, 25, 8, 14, 15, 16, 18, 23, 10]. Hence the ordinary nonfuzzy multisets are sometimes called crisp multisets by the common usage in fuzzy systems theory. A characteristic of fuzzy multisets is that definitions of elementary operations require a nontrivial data handling of sorting membership sequences. We will see why such sorting is essential by introducing the α-cut for fuzzy multisets. Another cut operation for a multiset is called here ν-cut that corresponds to the α-cut for fuzzy sets. The ν-cut is generalized to fuzzy multisets and commutative properties between these cuts and elementary operations are proved. Both theoretical and real-world applications of fuzzy multisets are considered. As a theoretical application, rough approximations [21] of fuzzy multisets are discussed. Moreover fuzzy database systems and information retrieval are mentioned. Lastly, methods of data classification and clustering are briefly discussed.


This research has partially been supported by the Grant-in-Aid for Scientific Research, the Ministry of Education, Sports, Culture, Science and Technology, Japan, No.16650044.

V. Torra and Y. Narukawa (Eds.): MDAI 2004, LNAI 3131, pp. 27–40, 2004. c Springer-Verlag Berlin Heidelberg 2004


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2

Multisets and Fuzzy Multisets

Before considering fuzzy multisets, a brief review of crisp multisets [9, 11, 2] is given. We assume finite sets and multisets for simplicity. 2.1

Crisp Multisets

Let us begin by a simple example. Example 1. Assume X = {x, y, z, w} is a finite set of symbols. Suppose we have a number of objects but they are not distinguishable except their labels x, y, or z. For example, we have two balls with the label x and one ball with y, three with 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 call 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 of 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: Count M : X → {0, 1, 2, . . .} (cf. [2, 9, 11]). 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 . The collection of all crisp multisets of X is denoted by C(X) here. Let us 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 followings are basic relations and operations for crisp multisets. 1. 2. 3. 4. 5.

(inclusion): M ⊆ N ⇔ Count M (x) ≤ Count N (x), ∀x ∈ X. (equality): M = N ⇔ Count M (x) = Count N (x), ∀x ∈ X. (union): Count M∪N (x) = max{Count M (x), Count N (x)}. (intersection): Count M∩N (x) = min{Count M (x), Count N (x)}. (addition): Count M⊕N (x) = Count M (x) + Count N (x).

Readers should note that the operations resemble those for fuzzy sets, but the upper bound for Count(·) is not assumed.

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Since the upper bound is not specified, the complement of multiset is difficult to be studied. Instead, an operation of nonstandard difference M ∼ N is defined as follows.  Count M (x), (Count N (x) = 0), Count M∼N (x) = 0 (Count N (x) > 0). 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 = {3/z}. 2.2

Fuzzy Multisets

Yager [28] first discussed fuzzy multisets where 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 we have x with the membership 0.2, x with 0.3, y with the membership 1, and two y’s with 0.5 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 [28]. The collection of all fuzzy multisets is denoted by F M(X), while the family of all (ordinary) fuzzy sets is denoted F (X). For x ∈ X, a 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). Hence we can write A = {(µ1A (x), . . . , µpA (x))/x}x∈X

(1)

In order to define an operation between two fuzzy multisets 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 therefore append an appropriate number of zeros for this purpose. The resulting length for A and B is denoted by

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L(x; A, B) = max{p, p }: it depends on each x ∈ X. We sometimes write L(x) instead of L(x; A, B) when no ambiguity arises. If we define the length L(x; A) by L(x; A) = max{j : µjA (x) = 0 }. we have L(x; A, B) = max{L(x; A), L(x; B)}. 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 put 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}. Basic relations and operations for fuzzy multisets are as follows [14]. 1. [Inclusion] A ⊆ B ⇔ µjA (x) ≤ µjB (x), j = 1, . . . , L(x),

∀x ∈ X.

A = B ⇔ µjA (x) = µjB (x), j = 1, . . . , L(x),

∀x ∈ X.

2. [Equality] 3. [Addition] A ⊕ B is defined by the addition operation in X × [0, 1] for crisp multisets [28]: if A = {(xi , µi ), . . . , (xk , µk )} and B = {(xp , µp ), . . . , (xr , µr )} are two fuzzy multisets, A ⊕ B = {(xi , µi ), . . . , (xk , µk ), (xp , µp ), . . . , (xr , µr )}. 4. [Union] µjA∪B (x) = µjA (x) ∨ µjB (x), j = 1, . . . , L(x). 5. [Intersection] µjA∩B (x) = µjA (x) ∧ µjB (x), j = 1, . . . , L(x). 6. [t-norm and conorm] Let a t-norm and conorm operations for two fuzzy sets F, G be F TG and F SG, respectively; they are given by µF TG (x) = t(µF (x), µG (x)),

µF TG (x) = t(µF (x), µG (x)).

A well-known operation is the algebraic product TA for which t(a, b) = ab and hence µF TA G (x) = µF (x)µG (x). Their extensions to fuzzy multisets are straightforward: µjATB (x) = t(µjA (x), µjB (x)), j = 1, . . . , L(x), µjASB (x) = s(µjA (x), µjB (x)), j = 1, . . . , L(x).

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7. [α-cut] The α-cut (α ∈ (0, 1]) for a fuzzy multiset A, denoted by [A]α , is defined as follows. µ1A (x) < α ⇒ Count [A]α (x) = 0, µjA (x) ≥ 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 defined as follows. µ1A (x) ≤ α ⇒ Count ]A[α (x) = 0, µjA (x) > α, µj+1 A (x) ≤ α ⇒ Count ]A[α (x) = j, j = 1, . . . , L(x). 8. [Cartesian product] Given two fuzzy multisets A = {(x, µ)} and B = {(y, ν)}, the Cartesian product is defined:   (x, y, µ ∧ ν) A ×B = The combination is taken for all (x, µ) in A and (y, ν) in B. 9. [Difference] The nonstandard difference A ∼ B is defined as follows.  µjA (x), (µ1B (x) = 0) j µA∼B (x) = 0, (µ1B (x) > 0) where j = 1, . . . , L(x). 10. [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.

(2)

(The latter includes the former as a special case.) The following propositions are valid. The proofs are immediate and therefore omitted here. Proposition 1. Assume 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 . When the strong cut is used, we have the same results.

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Proposition 2. Assume 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 3. Assume 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 A, B, and C are fuzzy multisets of X. The followings 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). The class of all fuzzy multisets of a particular universe thus forms a distributive lattice. The Number of Elements. The number of elements, or cardinality of a fuzzy multiset A is given by  L(x;A)  j µA (x). | A |= j=1

x∈X

Moreover we define



L(x;A)

| A |x =

µjA (x).

j=1

We thus have | A |= x∈X | A |x . We moreover introduce an L2 -norm for A for later use:

 A  = | A TA A |x . x∈X

It is easily seen that  L(x;A)  A  = {µjA (x)}2 . 2

x∈X

j=1

We easily have | A |2 ≤| X | A  using the Schwarz inequality.

(3)

Multisets and Fuzzy Multisets as a Framework of Information Systems

2.3

33

Images of Fuzzy Multisets

Let us consider two images f A=



{f (x)}

(4)

x∈A

and f (A) =



{f (x)}

(5)

x∈A

where A is a fuzzy multiset of X. The author has studied (4) which uses the addition ⊕ instead of the union (5) (cf. [15, 16]). The image (4) is not an ordinary one, since f  A  is generally a multiset even when A is an ordinary set. It has been noted that such a multiset image arises in linear data handling in information processing [15, 16]. In contrast, the image by (5) is compatible with the ordinary image by the extension principle. Let us remind that a fuzzy set is represented by A = {(xi , µi )}i=1,...,q . We then have f  A  = {(f (xi ), µi )}i=1,...,q

(6)

Concerning (5), we use the membership sequence (1) for a fuzzy multiset A. We hence obtain (7) µif (A) (y) = max µiA (x) x∈f −1 (y)

(If f −1 (y) = ∅, then µ1f (A) (y) = 0). It is immediately to see that if A is an ordinary fuzzy set, the above relation implies the extension principle. 2.4

ν-cut

In contrast to the α-cut of a fuzzy set, another operation of ν-cut for multiset is defined. Let ν be a given natural number. A ν-cut for a crisp multiset M , denoted by ν M  , is defined as follows. M ν = {x ∈ X : Count M (x) ≥ ν}. Proposition 5. Let M , N be crisp multisets of X. Take an arbitrary ν ∈ {0, 1, 2, . . .}. We then have M ∪ N ν = M ν ∪ N ν , ν

ν

ν

M ∩ N  = M  ∩ N  . Notice also that, for the addition, M ⊕ N ν = M ν ⊕ N ν in general.

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Sadaaki Miyamoto

We see that the ν-cut corresponds to the α-cut for ordinary fuzzy sets. It thus is naturally defined. We proceed to generalize the ν-cut to fuzzy multisets. Let A be a fuzzy multiset of X and ν be a given natural number. The ν-cut ν of A, denoted by A , is a fuzzy set whose membership is given by x ∈ X.

µA ν (x) = µνA (x),

(8)

ν

In other words, the membership of A is the νth value in the membership sequence of A. We have the following propositions which show the validity of the definition. Proposition 6. Let A be an arbitrary fuzzy multiset of X. Then, [A]α ν = [Aν ]α holds. Namely, an α-cut and a ν-cut are commutative. Proof. x ∈ [A]α ν ⇔ Count [A]α (x) ≥ ν ⇔ µνA (x) ≥ α ⇔ µA ν (x) ≥ α ν

⇔ x ∈ [A ]α   Proposition 7. Let A and B be arbitrary fuzzy multisets of X. Take any ν ∈ {0, 1, 2, . . .}. We then have ν

ν

ν

A ∪ B = A ∪ B , ν ν ν A ∩ B = A ∩ B . ν

ν

ν

Note also that for ⊕, A ⊕ B = A ⊕ B in general. 2.5

Cuts and Images

While the α-cut is commutative with each of the union, intersection, and addition, the ν-cut does not commute with the addition. This implies that for a fuzzy multiset A, ν ν f  A  = f  A  in general. In contrast, it is easily seen that ν

ν

f (A) = f (A ). We thus have the following. Proposition 8. The image f ( · ) defined by (5) or by (7) commutes with the α-cut, ν-cut, and the union. In contrast, f  ·  by (4) or (6) commutes with α-cut and the addition. Namely, f ([A]α ) = [f (A)]α ν

ν

f (A ) = f (A) f (A ∪ B) = f (A) ∪ f (B) f  [A]α  = [f  A ]α f  A ⊕ B  = f  A  ⊕ f  B .

Multisets and Fuzzy Multisets as a Framework of Information Systems

3

35

Application to Rough Sets, Information Retrieval, and Automatic Classification

We consider three applications of fuzzy multisets to rough sets, information retrieval, and automatic classification problems. 3.1

Rough Fuzzy Multisets

Dubois and Prade [3] have generalized rough sets [21] and defined rough fuzzy sets. Assume that a classification X/R of X is given. The upper approximation of a fuzzy set A which is written as R∗ [A], and the lower approximation written as R∗ [A] are defined by the following: µR∗ [A] (Y ) = max µA (x), x∈Y

µR∗ [A] (Y ) = min µA (x). x∈Y

where Y ∈ X/R. Notice that the above rough approximations are commutative with the α-cut. Namely, the followings are valid. R∗ [[A]α ] = [R∗ [A]]α , R∗ [[A]α ] = [R∗ [A]]α . The upper approximation can be described in terms of the extension principle. Namely, let g be the natural mapping of X onto X/R. That is, for an arbitrary x ∈ X, there exists Y ∈ X/R such that x ∈ Y , whereby we define g(x) = Y . Then R∗ [A] satisfies µR∗ [A] (Y ) = µg(A) (Y ). The above argument implies that the upper approximation of a fuzzy multiset is defined by using the same mapping g(A). Assume that A is a fuzzy multiset of X. The upper approximation of A, denoted by R∗ [A], is defined by g(A). (g(·) is the natural mapping of X onto X/R.) Thus, µjR∗ [A] (Y ) = µjg(A) (Y ) = max µjA (x), x∈Y

j = 1, . . . , p. In contrast, the lower approximation cannot be defined using an image. Assume that A is a fuzzy multiset of X. The lower approximation of A, denoted by R∗ [A] is defined by µjR∗ [A] (Y ) = min µjA (x), x∈Y

j = 1, . . . , p.

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Sadaaki Miyamoto

Proposition 9. Let A be an arbitrary fuzzy multiset of X. The following equations then hold. [R∗ [A]]α = R∗ [[A]α ]

R∗ [A] = R∗ [A ] [R∗ [A]]α = R∗ [[A]α ] ν

ν

ν

ν

R∗ [A] = R∗ [A ] Proof. It is easy to see that the first two equations are valid, observing properties of the image. For the third equation, we note

Count [A]α (x) = n Count R∗ [[A]α ] (Y ) = n ⇔ x∈Y

⇔ ∀x ∈ Y, µjA (z) ≥ α,

j = 1, . . . , n,

∃z ∈ Y, < α k = n + 1, . . . ,

j ⇔ µA (x) ≥ α, j = 1, . . . , n, µkA (z)

x∈Y

µkA (x) < α, k = n + 1, . . .

x∈Y

⇔ Count [R∗ [A]]α (Y ) = n. ν

ν

R∗ [A] = R∗ [A ] is shown in a similar manner. We omit the detail. 3.2

 

Fuzzy Database and Information Retrieval

Fuzzy database systems have been considered by many researchers (e.g., [22]). Let us see how fuzzy multisets arise from a simple operation to fuzzy database. Although there are different frameworks for fuzzy database, a most simple form is a set of tuples t = (a1 , a2 , . . . , aN , µ) with the degree of relevance µ. Let us consider a simple example of fuzzy database F D of two tuples: F D = {(a1 , a2 , µ), (a1 , a2 , µ )} where a1 = a1 . Thus F D is not a fuzzy multiset but an ordinary fuzzy set. A simple operation of SELECT of the second column (attribute) leads to {(a2 , µ), (a2 , µ )} which is a fuzzy multiset. Another interpretation is the application of f ((a1 , a2 )) = (a2 ). Then, f  {(a1 , a2 , µ), (a1 , a2 , µ )}  = {(a2 , µ), (a2 , µ )}. Generally, the SELECT operation of an ordinary database uses such f  ·  and hence its extension to fuzzy databases produces fuzzy multisets. The reason why f  ·  instead of the ordinary f (·) is used is that the former is a simple sequential operation in which no check whether two elements are equal or not is necessary. Thus f  ·  is far more efficient than f (·). Since a database system should have union and intersection operations, those for fuzzy multisets should be implemented in fuzzy database systems. Researches of information retrieval systems on the web is now very active. For developing advanced search capabilities, consideration of an appropriate model of information retrieval is absolutely necessary.

Multisets and Fuzzy Multisets as a Framework of Information Systems

37

Items of information on the web arise redundantly and with degrees of relevance. The same information content may appear many times by a search; some of them are judged to be more relevant and others not. Such state of information is best be captured using fuzzy multisets with multiple occurrences and memberships for relevance. We will briefly mention some issues in information retrieval based on fuzzy multisets here, but further discussion of them except classification problems studied in the next section is omitted to save the space. First, the retrieval operation has the form of f  ·  of the sequential processing of information on the web. Note again the above remark that f  ·  is more natural and efficient than the mathematical f (·). Advanced retrieval function should use dictionaries whereby associative retrieval can be carried out [12, 13]. Such retrieval should employ fuzzy multirelations defined above, and then it is straightforward to extend methods of associative retrieval to the case of fuzzy multisets. Another method is retrieval of similar documents in which a measure of similarity between two documents or elements of information should be defined. We can use metrics discussed in the next section for this purpose. 3.3

Classification and Clustering

Although most studies in automatic classification in engineering concentrates pattern recognition [4], classification problems have also been discussed in relation to information retrieval frequently. There are two major categories of automatic classification problems: supervised classification and unsupervised classification. As a major part of classification problems is supervised, the former is sometimes simply called classification, while the latter, unsupervised one is called clustering. Supervised Classification. For supervised classification, strong theory such as the Bayesian technique [4] are available. However, most techniques are based on the assumption of the Euclidean space and we have a weaker structure of fuzzy multisets, such parametric techniques are unusable. On the other hand, nonparametric methods such as the nearest neighbor (NN) and K-nearest neighbor (KNN) are applicable, if a metric D(A, B) is defined between an arbitrary pair of fuzzy multisets (cf. [4], Chapter 4). For simplicity, suppose fuzzy multisets B1 , . . . , BN are given and they are classified into two classes C and C  . Moreover suppose a new fuzzy multiset A should be classified into one of these two classes. ˆ = arg NN. Find nearest element to A: B

min

B1 ,...,BN 

ˆ ∈ C, clasd(A, Bi ) and if B

ˆ ∈ C  , classify A into C . sify A into C; if B KNN. Find K nearest elements to A, that is, those Bi ’s that have K smallest distances from A. Classify A into the class which has the majority out of the K nearest elements.

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Sadaaki Miyamoto

For the purpose of applying these techniques, we require a metric space of fuzzy multisets. There are different types of metric spaces for fuzzy multisets. We consider two different metrics D1 and D2 : D1 (A, B) = | A ∪ B | − | A ∩ B |,  D2 (A, B) = | A TA A | + | BTA B | −2 | A TA B |.

(9) (10)

It is easy to see these metrics correspond to the L1 and L2 metrics by observing  j | µA (x) − µjB (x) |, D1 (A, B) = x∈X

D2 (A, B)2 =

j



x∈X

| µjA (x) − µjB (x) |2 .

j

We hence have | A |= D1 (A, ∅) and A  = D2 (A, ∅). The nearest neighbor and K-nearest neighbor methods of classification based on these metrics are now straightforward, as seen above. Clustering. Document clustering has also been studied [26, 12] and recently there are search engines employing clusters [6]. Fuzzy clustering of documents using D1 and D2 is studied in [19] where it is shown that the major difference is in calculating cluster centers in fuzzy cmeans [1]. Further studies on fuzzy clustering include the use of the kernel-trick in support vector machines [27] whereby nonlinearities of cluster boundaries are handled. For the detail, see [20]. Note. The space of fuzzy multisets is essentially infinite dimensional, even when the underlying space X is finite and we are interested in finite multisets, i.e., those having finite occurrences in X × [0, 1]. The infinity comes from the limit of a sequence of finite fuzzy multisets is an infinite fuzzy multiset. This fact can be ignored in most applications of fuzzy multisets, but such fuzzy multisets may be of interest in future theoretical studies. When handling infinite fuzzy multisets, such inequalities as (3) are important.

4

Conclusion

We have overviewed fuzzy multisets and considered two types of applications: an application is theoretical and rough sets are discussed; another includes information retrieval and automatic classification of objects. Recent methods of information retrieval include rough set-based retrieval [17], where the use of fuzzy multisets should further be studied. As noted in the introduction, areas of multiset applications are becoming broader. Moreover we are encountering fuzzy multisets unconsciously. For example, populations in genetic algorithms are considered to be fuzzy multisets,

Multisets and Fuzzy Multisets as a Framework of Information Systems

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although current theory may not be very useful to genetic algorithms yet. However, new operations may be added to fuzzy multisets and on the other hand genetic algorithms can include some features of fuzzy multisets. Future studies encompassing these fields seem to be promising.

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