Table of Contents
Letter to the Editor
Advances in Decision Sciences
Volume 2011, Article ID 495392, 17 pages
http://dx.doi.org/10.1155/2011/495392
Research Article

Rough Multisets and Information Multisystems

1TIFAC Core in Cyber Security, Amrita Vishwa Vidyapeetham, Tamilnadu, Coimbatore 641 112, India
2Department of Mathematics, National Institute of Technology Calicut, Kerala, Calicut 673 601, India

Received 3 August 2011; Accepted 1 November 2011

Academic Editor: Graham Wood

Copyright © 2011 K. P. Girish and Sunil Jacob John. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Rough set theory uses the concept of upper and lower approximations to encapsulate inherent inconsistency in real-world objects. Information multisystems are represented using multisets instead of crisp sets. This paper begins with an overview of recent works on multisets and rough sets. Rough multiset is introduced in terms of lower and upper approximations and explores related properties. The paper concludes with an example of certain types of information multisystems.

1. Introduction

The advance in science and technology has given rise to a wide range of problems, where the objects under analysis are characterized by many diverse features (attributes), which may be quantitative and qualitative. Furthermore, the same objects may exist in several copies with different values of attributes, and their convolution is either impossible or mathematically incorrect. Examples of such problems are the classification of multicriteria alternatives estimated by several experts, the recognition of graphic symbols, text document processing, and so on. A convenient mathematical model for representing multiattribute objects is a multiset or a set with repeating elements. The most essential property of multisets is the multiplicity of the elements that allows us to distinguish it from a set and consider it as a qualitatively new mathematical concept.

Huge chunks of information are acquired and stored every day in computer systems for various global applications. It is hard to imagine a computer application that does not interface with a database or one that does not dump megabytes of information into a database every day. Information is constantly generated by a variety of basic day-to-day applications. The information that is being acquired is growing exponentially every minute. It gives rise to incomplete, uncertain, or vague information. Extracting useful information from such huge chunks of information is a difficult task. The convenient and effective tools to make this process easier are the theories of rough sets.

In any information system, some situations may occur, where the respective counts of objects in the universe of discourse are not single. In such situations we have to deal with collections of information in which duplicates are significant. In such cases multisets play an important role in processing the information. The information system dealing with multisets is said to be an information multisystem. Thus, information multisystems are more compact when compared to the original information system.

The authors have given a new dimension to Pawlak’s rough set theory, replacing its universe by multisets. This is called a rough multiset and is a useful structure in modelling information multisystem. The process involved in the intermediate stages of reactions in chemical systems is a typical example of a situation which gives rise to multiset relations. Information multisystem is represented using rough multisets and is more convenient than ordinary rough sets. An illustrative example supporting this claim is provided in Section 4.

Rough multisets are defined in terms of lower mset approximation and upper mset approximation with the help of equivalence mset relations introduced by the authors in [1]. Grzymala-Busse introduced the concept of rough multisets using multirelations in [2]. A multirelation is a relation connecting two objects in which a pair is repeated more than once. But a multiset relation according to the authors is in entirely different concept when compared to the concept multirelation introduced by Grzymala-Busse. Multiset relation is a relation connecting two objects in which each object and the pair are repeated more than once; that is, in a multirelation “ is related to ” with the pair () repeated more than once, but in the case of a multiset relation “ repeated m times” is related to “ repeated n times” (i.e., is related to ) with the pair () repeated more than once. Thus, a multirelation is a relation connecting the elements in the sets. At the same time multiset relation is a relation connecting the elements in the multisets.

This paper proposes a systematic approach to rough multisets, related properties, and information multisystems. It begins with the introduction to multisets and a brief survey of Yager’s theory of bags [3]. The concept of mset relations, knowledge mset base, and classifications are given in Section 3. Section 4 includes the discussion of the underlying concepts of the rough set theory, introduction to rough multisets and related properties, and information multisystems and concludes with information and decision tables of rough multisets. The last section contains the conclusion and the scope of future work.

2. Theory of Multisets

This section begins with the origin of multisets (mset) and a brief survey of the notion of the theory of multisets introduced by Yager [3].

In mathematics, a multiset (or bag) is the generalization of a set. A member of a multiset can have more than one membership [46], while each member of a set has only one membership. The use of multisets in mathematics predates the name “multiset” by nearly 90 years. Richard Dedekind used the term multisets in a paper published in 1888. Knuth [7] also lists other names that were proposed for multisets, such as list, bunch, bag, and weighted set. Zermelo-Fraenkel’s set theory, commonly known as ZF theory, is the classical theory based on first-order logic and is undoubtedly the foundation of mathematics. In first-order logic, a formal theory MST (multiset theory) that contains ZF as a special case has been formulated. MST was introduced by Cerf et al. [8] in 1972.

Definition 2.1. An mset drawn from the set is represented by a function Count or defined as , where represents the set of nonnegative integers.

Here is the number of occurrences of the element in the mset . We present the mset drawn from the set as , where is the number of occurrences of the element , in the mset . However, those elements which are not included in the mset have zero count. Let be an mset from with appearing times in . It is denoted by .

Example 2.2. Let be any set. Then is an mset drawn from .
To be clear, a set is a special case of an mset.
Let and be two msets drawn from a set . Then, the following are defined in [3, 9, 10]:(1), (2), (3), (4), (5), (6), where and represent mset addition and mset subtraction, respectively.
Let be an mset drawn from a set . The support set of denoted by is a subset of and ; that is, is an ordinary set. is also called root set.
An mset is said to be an empty mset if, for all .
The cardinality of an mset drawn from a set is denoted by Card () or and is given by Card .

Definition 2.3. A domain is defined as a set of elements from which msets are constructed. The mset space is the set of all msets whose elements are in such that no element in the mset occurs more than times.
The set is the set of all msets over a domain such that there is no limit on the number of occurrences of an element in an mset.
If , then .

Definition 2.4. Let be a support set and the mset space defined over . Then, for any mset , the complement of in is an element of such that

Note 1. Using Definition 2.4, the mset sum can be modified as follows:

3. Mset Relations and Knowledge Mset Base

This section gives the concept of relations, equivalence relations, and partitions in the context of multisets. Knowledge mset base and classifications are introduced instead of knowledge base and classifications in any information system.

A relation in mathematics is defined as an object that has its existence as such within a definite context or setting. It is literally a case wherein any change in this setting will involve a change in the relation that is being defined. The particular type of context that is needed here is formalized as a collection of elements from which are chosen the elements of the relation in question. The larger collection of elementary relations or tuples is constructed by means of the set theoretic product commonly known as the Cartesian product.

In mathematics, especially set theory and logic [11, 12], a relation is a property that assigns truth values to combinations (-tuples) of individuals. Typically, the property describes a possible connection between the components of a -tuple. For a given set of -tuples, a truth value is assigned to each -tuple according to whether the property does or does not hold. An example of a ternary or triadic relation is “ was—connected— by ,” where is a 3-tuple of cities; for example, “Kerala is connected to Delhi via Mumbai by road” is true, while “India is connected to USA via Pakistan by road” is false. The information Table 1 can represent this relation.

tab1
Table 1: Relation: —connected by roads—to —via .

The data given in the table are equivalent to the following ordered triples: The table for relation is an extremely simple example of a relational database. Theoretical aspects of databases are the specializations of a branch of computer science. Their practical impacts have become all too familiar to our everyday life. Computer scientists, logicians, and mathematicians, however, tend to see things differently when they look at these concrete examples and samples of the more general concept of a relation. According to Augustus De Morgan, “when two objects, qualities, classes or attributes, viewed together by the mind, are seen under some connection, that connection is called a relation.”

In any information multisystem, duplicates occur in the intermediate stages of processing information. While processing, each information can be associated within the elements of collections of information or with elements of another collection of information. This type of association is called an mset relation. For example, in chemical systems the chemical compounds are formed by interactions which take place in a systematic manner. Some molecules in the set of molecules might occur in more than a single copy. From the mathematical point of view, these sets of molecules are called msets and the rules of reactions are called mset relations.

3.1. Knowledge Mset Base and Classifications

Knowledge is deep-seated in the classificatory abilities of human beings and other species. For example, knowledge about the environment is primarily manifested as an ability to classify a variety of situations from their point of survival in the real world.

Among the given finite mset (the universe) of objects any submset of the universe will be called an mset concept (i.e., concepts with repetition) or an mset category in and any family of mset concepts in will be referred to as abstract knowledge with repetition. Empty mset is admitted as an mset concept.

Some families of classifications with repetitions are more dealt with, like mset classifications over . A family of mset classifications over will be called a knowledge mset base over denoted by . Thus, a knowledge mset base represents a variety of mset classification skills of an intelligent agent or group of agents which constitute the fundamental equipments of the agent needed to define its relation to the environment or itself.

For mathematical reasons equivalence mset relations are used, instead of classifications about knowledge mset base, since these two concepts are actually interchangeable and mset relations are easier to deal with.

The authors introduced new notations [1] for the purpose of defining Cartesian product of msets, mset relations, and their domain and codomain. The entry of the form denotes that is repeated -times, is repeated -times, and the pair () is repeated -times. The counts of the members of the domain and codomain vary in relation to the counts of the coordinate and coordinate in . For this purpose the notations and are introduced. denotes the count of the first coordinate in the ordered pair , and denotes the count of the second coordinate in the ordered pair .

Definition 3.1. Let and be two msets drawn from a set , then the Cartesian product of and is defined as .

We can define the Cartesian product of three or more nonempty msets by generalizing the definition of the Cartesian product of two msets.

Definition 3.2. A submset of is said to be an mset relation on if every member of has a count of the product and . One denotes related to by .
The Domain and Range of the mset relation on is defined as follows:
Dom ,
Ran .

Example 3.3. Let be an mset. Then ,,,,,,, is an mset relation defined on . Here and .

Definition 3.4. Let be an mset relation defined on and . One defines , the -relative mset of , the set of all in such that there exists some such that ; that is, .
Similarly, if , then , the -relative mset of , is the mset of all in with the property that there exists some such that . From the preceding definitions, we see that is the union of all msets .

Remark 3.5. If two msets have the same elements with distinct multiplicity, then their -relative msets are the same.

Example 3.6. From Example 3.3 we see that , andNow, if , then .

Definition 3.7. (i) An mset relation on an mset is reflexive if for all in .
(ii) An mset relation on an mset is symmetric if implies .
(iii) An mset relation on an mset is transitive if , then .
An mset relation on an mset is called an equivalence mset relation if it is reflexive, symmetric, and transitive.

Example 3.8. Let be an mset. Then the mset relation given by , , ,, is an equivalence mset relation.

Definition 3.9. A partition of a nonempty mset is a collection of nonempty submsets of such that(1)each element of belongs to one of the msets in ,(2)if and are distinct elements of , then .The msets in are called the blocks or cells of the partition.

Example 3.10. Let be an mset, and consider the partition of the mset . Then the blocks of are and and each element in a block is only related to every other’s element in the same block. Thus, the equivalence mset relation determined by is .

If is an equivalence mset relation over , then is the family of all -equivalence classes of (or classification of ) referred to as mset categories or mset concepts of . or denotes the mset category in containing an element .

By a knowledge mset base, we can understand a relational multisystem , where is a nonempty finite mset called the universe and is a family of equivalence mset relations over .

If and , the (intersection of all equivalence mset relations belonging to ) is also an equivalence mset relation and will be denoted by IND() and will be called an indiscernibility mset relation over .

Moreover, Thus, (i.e., the family of all -equivalence classes of the equivalence mset relation IND()) denotes knowledge mset base associated with the family of equivalence mset relations , called -basic knowledge mset base about in .-equivalence classes of IND() are called basic mset categories (mset concepts) of knowledge .

In fact -basic mset categories are those properties of the universe which can be expressed employing knowledge . In other words, they are fundamental building blocks of our knowledge or basic properties of the universe which can be expressed employing knowledge .

Let be a knowledge mset base. denotes the family of all equivalence mset relations defined in as Thus, is the maximal mset of equivalence mset relations containing all elementary mset relations of .

Finally the family of all mset categories in the knowledge mset base will be referred to as mset categories.

4. Rough Multiset and Information Multisystems

In 1982 Pawlak [13] introduced a new mathematical tool “Rough Sets” that is devised to deal with vagueness and uncertainty. Surprisingly, this is again a contribution to humanity from one belonging to the field of computer science—during the same period, the same community that gifted several other elegant creations, like the fuzzy set theory by Lotfi Zadeh in 1965. It is also interesting to note that both of the theories address basically the same issue, namely, “vagueness” and this fact is not merely a coincidence. That “vagueness” in general is different from “probability” which is now gaining acceptance after the long, fierce debates that took place during the years immediately following the advent of the fuzzy set theory in 1965. So Pawlak did not have to fight that battle. Yet he had to utter this warning, which is an excellent distinctive criterion, namely, “vagueness is the property of sets whereas uncertainty is the property of an element” [14]. In the introduction to his short communication [15], Pawlak declares that “we compare this concept with that of the fuzzy set and we show that these two concepts are different.” However, later in Pawlak a probable change in opinion is observed as reflected in the following categorical remark: “Both fuzzy and rough set theory represent two different approaches to vagueness. Fuzzy set theory addresses gradualness of knowledge, expressed by the fuzzy membership—whereas rough set theory addresses granularity of knowledge expressed by indiscernibility relation” [16].

Rough set theory is a powerful tool for dealing with the uncertainty, granularity, and incompleteness of knowledge in information systems. When there is a huge amount of data, it is very difficult to extract useful information from them, mainly because of the incompleteness of the data; that is, there might be some critical pieces of data that is missing. Vagueness and uncertainty lead to difficulty in extracting useful information from very large databases. Rough set theory differs from others by being a fundamental approach to several disciplines of artificial intelligence like expert systems, machine learning, and knowledge discovery from databases.

Definition 4.1 (information system). An information system is a pair , where is a nonempty finite set of objects called the universe of discourse and is a nonempty finite set of attributes.

Information multisystems are represented using multisets instead of crisp sets. Multisets are defined as those sets in which an element may have several occurrences as against one in a crisp set. The very definition of multisets implies the omission of object identifiers. Number of occurrences of each object is denoted in an additional column named (counts or multiplicity). Table 3 is the multiset representation of the information multisystem. Formally, an information multisystem can be defined as a triple, , where is an mset of objects, is the set of attributes, and is an mset relation defined on . For example, the chemical system can be defined as , where is the mset of all possible molecules, is an algorithm describing the reaction vessel or domain and how the rules are applied to the molecules inside the vessel, and is the set of “collision” rules, representing the interaction among the molecules.

In the Pawlak rough set model, an equivalence relation on the universe of objects is defined based on their attribute values. After the introduction of Pawlak’s rough set, many mathematicians introduced different rough set models based on various types of binary relations instead of equivalence binary relations. Based on these studies, Yao and Lin [17] introduced a new type of rough set model called nonstandard rough set model induced by various binary relations. Pawlak’s standard rough set is considered and extended to the context of multisets.

Let , and is an equivalence mset relation. We will say that is an -definable mset if is the union of some mset categories; otherwise, is an -undefinable mset.

The -definable msets are those submsets of the universe which can be exactly defined in the knowledge mset base , whereas the -undefinable msets cannot be defined in this knowledge mset base.

The -definable msets will also be called -exact msets, and -undefinable msets will also be called -inexact mset or -rough mset.

Mset will be called exact mset in if there exists an equivalence mset relation such that is -exact mset and is said to be a rough mset in , if is -rough mset for any .

A rough mset is a formal approximation of a conventional mset in terms of a pair of msets which give the lower approximation and upper approximation of the original mset.

4.1. Approximations of an Mset

Let be an mset and an equivalence mset relation on . generates a partition on , where are the -equivalence classes generated by the equivalence mset relation .

An -equivalence class in containing an element is denoted by . The pair is called an mset approximation space. For any , we can define the lower mset approximation and upper mset approximation of by respectively.

The pair is referred to as the rough mset of . The rough mset gives rise to a description of under the present knowledge, that is, the classification of .

Suppose that is an arbitrary mset relation on . With respect to , we can define a neighborhood of an element in as follows: The mset is also called an -relative mset.

Definition 4.2. Let be an arbitrary mset relation on the universal mset , and let be the -relative mset (i.e., neighborhood of ) in . For any , a pair of lower and upper mset approximations and , are defined by The pair is referred to as rough mset of .

4.2. Properties of Rough Multisets

For any equivalence mset relation on a nonempty mset , the following conditions hold:(i)for every ;(ii); (iii)if with , then ;(iv)for any submsets and of , ;(v)if , then ;(vi)for any submsets and of , ;(vii)for any submsets and of , ;(viii)for any submsets and of , ;(ix)for every ;(x)for every ;(xi)for every ;(xii)for any submsets and of ;(xiii)for any submsets and of .

Proof. (i) One has
(ii) Since, for every . Thus, . Also since , hence .
(iii) Since is an equivalence mset relation and , there is no equivalence class in the submset of . Hence, .
(iv) One has
(v) If and , then and so ; hence, . Thus, .
(vi) Since and and .
(vii) One has
(viii) One has
(ix) Since and , to prove that , it is enough to prove that .
Let
(x) Let , we want to show that , that is, for all . Since is an equivalence mset relation, then or for all , then, for all and ; that is, for all , then ; thus, and so .
(xi) Let ; we want to show that . Since , then ; also is an equivalence relation; hence, for all , then for all ; that is, ; thus, . Hence, .
(xii) One has
(xiii) One has

The following example shows that the equalities in the properties (vi), (vii), and (x) does not always hold.

Example 4.3. From Example 3.8, the equivalence classes are , and . Let and be two submsets of . Then . Therefore, . But and . Thus, .

Also and . But . Therefore, .

Furthermore, and . Therefore, .

Theorem 4.4 (see [1]). For any submsets and of M(1), (2).

Theorem 4.5. For any submsets and of (1), (2).

Proof. (1) One has
(2) One has

The following example gives the analysis of information multisystems with a given number of objects and attributes.

Example 4.6. Let us consider mset of balls (objects) which are of different colors (attributes) , different types , and different prices (attributes) . Thus, the information system consists of objects and 5 attributes.

The balls can be classified as costly black cricket ball, cheap blue football, and so forth. Thus, the mset of balls in can be classified according to color, type, and cost. The example follows.

Balls are black, are blue, and are red.

Among the balls are foot balls, are cricket balls and are volleyballs which are of that are costly and that are cheap.

In other words by these classifications three equivalence mset relations , and are defined with the following -equivalence classes: These -equivalence classes are mset concepts (mset categories) in our knowledge mset base .

Basic mset categories are mset intersections of elementary mset categories. For example, msets are basic mset categories of black volleyballs, blue cricket balls, and red volleyballs, respectively.

Msets are elementary basic mset categories of black volleyball of cheap cost, blue cricket ball of high cost, and red volleyball of cheap cost, respectively.

Msets are mset categories black or blue (not red), blue or red (not black), and black or red (not blue), respectively.

Note that some mset categories are not available in this knowledge mset base. For example, msets are empty which means that mset categories of blue footballs and black cricket balls do not exist in our knowledge mset base (i.e., empty mset categories).

4.3. Information Tables and Decision Tables

From the information system as in Example 4.6, the information table representing the universe of discourse is constructed.

Table 2 can be analyzed in the following way. The first 5-tuple can be interpreted as number of black football with cheap cost, similar way for the rest of the tuples.

tab2
Table 2
tab3
Table 3

Consider the information Table 3 with respect to the given information system .

Table 3 can be analyzed in the following way. The table contains three blocks with the block containing black footballs and black volleyballs with cheap cost and black volleyballs with high cost. Similarly block contains blue cricket balls with high cost and blue volleyballs with cheap cost, and block contains red footballs and red cricket balls with cheap cost and red volleyballs with high cost.

With respect to the information Table 3, the decision table can be constructed in the following way. The first column of the decision table represents object identifiers with multiplicity. Identifiers are used to uniquely identify each row in the table. The decision table has six attributes . The attribute is the decision attribute with values 1, 2, 3, and is the count (multiplicity) of the object. The other attributes are color and cost—the condition attributes. In Table 2 the universe of discourse consists of objects. The partition of based on Table 3, the objects of decision attribute , is With respect to this decision, the decision Table 4, presented above, follows.

tab4
Table 4

Considering the target set , the lower approximations and upper approximations are given as follows:

From the above-discussed information multisystem, decision tables (Tables 5 and 6), associated with each block, there exist two nonnegative integers representing the minimum (lower) and maximum (upper) number of copies. The minimum and maximum values may depend upon some statistical analysis of the data concerning the necessity and utility of each item and the number of items needed for the relevant programmes. The decision is taken about the number of items that are needed according to their demand and other factors involved.

tab5
Table 5: Lower approximation decision.
tab6
Table 6: Upper approximation decision.

5. Conclusion and Future Work

Pawlak introduced the concept of rough sets which have a wide range of applications in various fields like artificial intelligence, cognitive sciences, knowledge discovery from databases, machine learning expert systems, and so forth [1826]. These types of situations deal with objects and attributes. Many situations may occur, where the counts of the objects in the universe of discourse are not single. In information systems we come across situations which involve the concept of a multiset. This paper begins with Yager’s theory of multisets. After presenting the theoretical study, the concepts mset relations, equivalence mset relations, partitions, and knowledge mset base have been established. Finally the concept of rough multisets and related properties with the help of lower mset approximation and upper mset approximations have been introduced. It is observed that rough multisets are important frameworks for certain types of information multisystems.

Chakrabarty [27, 28] introduced two types of bags called IC bags and bags, which are suitable for situations, where the counts of the objects in information system are not fixed and are represented in the form of intervals of positive integers and power set of positive integers . These kinds of problems appear, for instance, during a nuclear fission, when a nucleus (consisting of protons and neutrons) is split into multiple nuclei, each of them with its own number of protons and neutrons. Thus, the information multisystem can be associated with IC bags or bags by the help of lower mset approximations and upper mset approximations. This association could be used for certain types of decision analysis problems and could prove useful as mathematical tools for building decision support systems.

Acknowledgments

The authors would like to thank the anonymous referees and Editor-in-Chief, for carefully examining the paper and providing very helpful comments and suggestions. They are also grateful to Mrs. K. Sreelatha for improving the linguistic quality of the paper.

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