Applying Fuzzy Sets and Rough Sets as Metric for Vagueness and Uncertainty in Information Retrieval Systems

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1 Applying Fuzzy Sets and Rough Sets as Metric for Vagueness and Uncertainty in Information Retrieval Systems Nancy Mehta,Neera Bawa Lect. In CSE, JCDV college of Engineering. Abstract-This paper reviews and compares theories of fuzzy sets and rough sets applying information retrieval. Vagueness and uncertainty have attracted the attention of philosophers and logicians for many years. The aim of this paper is to synthetically present the rough set and fuzzy set approach to the modeling of flexibility with respect to vagueness and uncertainty in the specification of users information needs. The two theories model different types of uncertainty.the rough set theory takes into consideration the indiscernibility between objects; typically characterized by an equivalence relation. Rough sets are the results of approximating crisp sets using equivalence classes. The fuzzy set theory deals with the ill-definition of the boundary of a class through a continuous generalization of set characteristic functions. The indiscernibility between objects is not used in fuzzy set theory. So the membership relation is best theory for categorization of two theories. Keywords: Vagueness, Uncertainty, Fuzzy sets, Information Retrieval, Classical sets, Rough sets. I. INTRODUCTION Where, for a system, there exists an incomplete knowledge of the underlying logistics and a limited data base upon which to make predictive assessments, a degree of human judgement is necessitated. This, of course, is accompanied always by the imprecision and vagueness that characterize human rationale and, consequently, it is in such circumstances that a quantitative and qualitative technique that incorporates the concept of vagueness, may provide a useful perspective. The two theories of fuzzy sets and rough sets are generalizations of classical set theory for modeling vagueness and uncertainty [1,3]. A fundamental question concerning both theories is their connections and differences [2,5]. There have been many studies on this topic. While some authors argued that one theory is more general than the other [2,7], it is generally accepted that they are related but distinct and complementary theories [4,6,13]. The two theories model different types of uncertainty [8]. In the rough-set model, a given set is represented by a pair of ordinary sets called the lower and upper approximations [9,10]. The approximation space is constructed based on an equivalence relation defined by a set of attributes. There are two views for the interpretation of the rough-set model. Under one view, the approximation space can be understood in terms of two additional set-theoretic operators [12]. They assign for each subset of the universe a lower approximation and an upper approximation [9]. We regard this interpretation to be the operator-oriented view. The other interpretation is a set-oriented view, which considers a rough set as the family of sets having the same lower and upper approximations. The rough-set model is useful in the study of information system, classification and machine learning [11]. In the fuzzy-set model, it is assumed that the available information is insufficient to define a set precisely. Instead, a pair of sets referred to as the lower and upper bounds is used to define the range of the unknown set. In other words, any member of the family of sets bounded by the lower and upper bounds can in fact be the set. Such a framework is similar to the interval-numeric algebra [14]. In this paper we explore this topic from another facet, which is the system view. II. MEMBERSHIP RELATION IN CLASSICAL SET THEORY A. Membership Relation and Inclusion Relation Generally speaking, classical set theory is built on the definition and boundary of a set. That is to say as soon as the definition of a set is finished, the membership relation between an element and the set is certain. If given an element and a set, we can discuss whether the element is a member of the set. In classical set theory, inclusion relation is one of the important and basic relations or operations. It describes the relations between sets and sets. And membership relation is used to describe the relations between elements and sets. In the following discussion, we will see the connections between the two relations. Let U be a non-empty set called universe. There are a subset A U and a subset B U. If any element x A then x B, we can say A B. So we can find that the inclusion relation ( ) is induced by membership relation ( ). In fact, membership relation not only induces the inclusion relation but also induces the body-shadow relation [15] and other operations of classical set theory. B. Membership Relation and Body-Shadow Relation What is body-shadow relation? In classical set theory, function and mapping are considered an especial kind of binary relation. Let f AXB be a binary relation on A and B, where A and B are both non-empty sets. If for each ISSN: ISBN:

2 element x A there exists the one and only element y B, which make (x, y) f true, then f is called function. We put it as the following format: f: A B or y =f (x). In the formula: f (A) = {f (x) x A}, A is called definition field and it is the body. B is the value field, and f (A) is the shadow of A. The body-shadow relations include functions, mappings, projections and the reverse relations of them and so on. In a word, body-shadow relation is a generalized concept. From the following example we can see the connections between membership relation and body-shadow relation. Example 1. Let U be a non-empty and finite set called universe, U={x 1, x 2, x 3, x 4, x 5, x 6 }.R is an equivalence relation on U, then we can get the equivalence classes:[x 1 ] R ={x 1, x 2 } and [x 3 ] R ={x 3, x 4, x 5, x 6 }. According to the membership relation between elements and equivalence classes, we can get the following figure (Figure 1.). From the above example we can see that the body-shadow relation is also the inducement of membership relation. The importance of the membership relation has been shown above. And mathematics can be formally built on the two main relations, i.e. part-whole relation and body-shadow relation. In the sense we can say that math is built on set theory and symbol logic. Through the discussion of this section we can find the membership relation is one of the most basic and important relation of set theory. In the above discussion we have known that membership relation in classical set theory is the most primary relation. It has the dualistic properties: the membership relation between an element and a set is sureeither yes or no. If an element belongs to a set, the value of this membership function is "1",or else the value is "0". In the classical set theory there are only two cases about the relation between an element and a set. We say this membership relation is absolute in a sense. The absolute property looks like the two-valued logic. Usually we can define a concept exactly by a set only if we know the concept completely. But in practice we often face the insufficient and incomplete information, which leads to that we have difficulties to define a crisp concept. So the concept of a set is not crisp, either. Thus there are fuzzy sets existing. From the definition of a fuzzy set we can find the membership relation is not dualistic. The membership relation allows partial membership, which denotes the degree of membership relation. That is to say, the value of membership function is not in {0, 1}, but in an interval [0, 1]. We consider the generalization which extends the value of membership function from {0, 1 } to [0, 1] is quantitative. We also can consider the membership relation in fuzzy sets as a many-valued logic. In the following example we will unify the expression of crisp sets and fuzzy sets by membership function. Example 2. Let U be the universe, U={1, 2,3,4,5,6,7,8}, there are two crisp subsets of U: S 1 ={2,3,4}, S 2 ={1,2,3,4,5}, and there are two fuzzy subsets of U Figure 1. III. FUZZY SET THEORY: THE QUANTITATIVE GENERALIZATION OF MEMBERSHIP RELATION The notion of fuzzy sets provides a convenient tool for representing vague concepts by allowing partial membership [16]. Here we should notice the relation between the vague concepts and partial membership relation. In [17] Lin says fuzzy theories have been driven by application, so there is no uniformly accepted formal theory for fizzy notion. And he thinks a fuzzy set is not a set. As pointed out recently by Zadeh [18], fuzzy logic has many facets: the logical facets, the set-theoretic facet, the relational facets, and the epistemic facet [16]. In this section we still study the same topic but in different viewpoint. From the system view, we consider the fuzzy set theory the generalization of classical set theory. Here we start from the primary relation-membership relation to study fuzzy sets. A. Membership Relation in Fuzzy Sets We use unified functions T 1, T 2, T 3, and T 4 to show S 1, S 2, MS 1, and MS 2 as follows: S 1 (TI: U {0, 1}) T 1 (1) = 0, T 1 (2) =1, T 1 (3)=1, T 1 (4)=1,T 1 (5)= 0, T 1 (6)= 0, T 1 (7)=0, T 1 (8)=0 S 2 (T 2 : U {0, l}) T 2 (1)= 1, T 2 (2)= 1, T 2 (3) = 1, T 2 (4) = 1, T 2 (5)= 1, T 2 (6)=0, T 2 (7) = 0, T 2 (8) = 0 MS 1 (T 3 : U [0, 1]) T 3 (1) = 0.1, T 3 (2) = 1, T 3 (3) = 0.5, T 3 (4) = 0.5,T 3 (5)= 0.6, T 3 (6) = 0, T 3 (7) = 0.4,T 3 (8) = 1 MS 2 (T 4 : U [0, 1]) T 4 (1)= 0.8, T 4 (2)= 0.5, T 4 (3)= 0.2, T 4 (4)= 0.7,T 4 (5) = 0.4, T 4 (6) = 0.6, T 4 (7) = 0, T 4 (8) = 0.5 B. The Connections between Operations and Membership Relation in Fuzzy Sets Earlier we have discussed the membership relation is the primary relation in classical set theory; in fact it's true of fuzzy set theory. Membership relation can induce many relations and operations in fuzzy sets. We say the fuzzy ISSN: ISBN:

3 sets are the generalization of classical sets. And the generalization also can be considered a mapping which is from set to real number system R=<R, {+, *, <, 0, }>. We can use the known operators of R system to extend the operators of sets. Example 3. Based on example 2, we use the membership functions and operations of R system to express the relations or operations of fuzzy sets and classical sets. Inclusion or party-whole relation: S 1 S 2 T 1 (x) < T 2 (x), T 1 (x) < T 2 (x) iff x U, T 1 (x) T 2 (x) MS 1 MS 2 (T 3 (x) T 4 (x): U [0,1]), where T 3 (x) T 4 (x) = min {T 1 (x), T 2 (x)} According to the above condition there are S S2, MS1 MS2 and MS2 MS1. Intersection operation: S 1 S 2 (T 1 (x) T 2 (x): U->{0, 1}), where T 1 (x) T 2 (x) = min {T1 (x), T 2 (x)} MS 1 MS 2 (T 1 (x) T 2 (x): U [0, 1]), where T 1 (x) T 2 (x) = min {T 1 (x), T 2 (x)} Union operation: S 1 S 2 (T 1 (x) T 2 (x): U {0, 1}), where T 1 (x) T 2 (x) =max {T1 (x), T 2 (x)} MS 1 MS 2 (T 1 (x) T 2 (x): U [0, 1]), where T 1 (x) T 2 (x) = max {T1 (x), T 2 (x)} Complementation operation: S 1 ((1-T 1 )(x)): U {0, 1} S 2 ((1-T 2 )(x)): U {0, 1} MS 1 ((l-t 3 )(x)): U [0, 1] MS 2 ((1-T 4 )(x)): U [0, 1], where (l-t)(x) = 1-T (x). In example 3 we discussed one basic relation (inclusion) and three basic operations. Other operations are induced by these operations. From the above example we can find the relation between fuzzy sets and classical sets. Fuzzy sets can be considered as the development of classical set theory based on the quantitative generalization of membership relation. Observer-Observed Model Rough set theory is considered the integration of classical sets theory and the observer-observed model [19]. In rough sets model the membership relation is relative to different observers. Here observer-observed model is a visual description of the character of rough sets and the model reflects the connections between two systems (the knowledge and subset of U). Knowledge of rough sets can be considered as an observer, and the subset of universe is considered as an observed object. Here we show the model by the corresponding figures (Figure 2, Figure 3, and Figure 4). In rough sets the membership relation between an element and a subset of universe is not doubtless, for it is relative to the knowledge of the observer. In this sense we think that the membership relation of rough sets is the qualitative generalization of classical membership relation. Let's look at an example to help us understand this kind of generalization. Example 4. Universe U= {O 1, O 2, O 3, O 4, O 5, O 6, O 7 }, a subset S= {O 1, O 3, O 4, O 6, O 7 }, an equivalence relation R= {< O 1, O 1 >, < O 1, O 2 >, < O 1, O 3 >, < O 2, O 1 >, < O 2, O 2 >, < O 2, O3>, < O 3, O 1 >, < O 3, O 2 >, < O 3, O 3 >,< O 4, O 4 >, < O 4, O 6 >, <O 6, O 4 >, < O 6, O 6 >, < O 5, O 5 >, < O 5, O 7 >, < O 7, O 5 >, <O 7,O 7 >}}. Figure 2. THE RELATION OF SUBSET S AND OBJECT FIELD U IN CLASSICAL SETS IV. ROUGH SET THEORY: THE QUALITATIVE GENERALIZATION OF MEMBERSHIP RELATION Theory of rough sets is motivated by practical need to deal with vague concepts caused by indiscernible and incomplete information. The basis of rough sets is classical set theory. Comparing rough sets with classical sets we can find that the important difference is an observer-observed model existing in rough sets. The importing of observer-observed model leads to the change of membership relation. Next we see how the membership of rough sets generalized from the membership of classical set theory and to mine the connections between them. A. Membership Relation in Rough Sets and the Figure 3. THE KNOWLEDGE OF U RELATIVE TO SOME OBSERVER IN ROUGH SETS Figure 4. THE OBSERVER-OBSERVED MODEL OF S RELATIVE TO KNOWLEDGE OF U IN ROUGH SETS ISSN: ISBN:

4 In classical sets, for each x U, the relation between x and S (S U) has only two possible results: x S or x S. But in the condition of R (observer), the relation between x and S changed. The knowledge of U relative to R: U/R= I {{O 1, O 2, O 3 }, {O 4, O6}, {O 5, O7}}, for the same x and S, results are different. In rough sets model, we use a pair of sets, R_(S) and R (S), to show S. We can get the following results: R_(S)={O 4, O 6 }; R (S)= {O 1, O2, O3, O4, O5, O6, O7}=U. Let x 1 =05, then we have x 2 S, x 1 R_(S) ' x 1 R (S) Let x 2 =04, then we have x 2 S, x 2 R_(S), x 2 R (S) In the above example, the relation of x and S is generalized, such as x 1 is from x 1 S to x 1 R (S), but x2 is from x 2 S to x 2 R_(S) and x 2 R (S). That is to say the membership between elements and sets are multiple. The reason is that the membership is based on or relative to the equivalence relation on U. B. Operations in Rough Sets and Membership Relation We have said in the above that rough set theory is the extension of classical set theory, so the operations in classical sets all exist in rough sets. In the following we give out some definitions of rough sets. Lower approximation: apr (A) ={x U [X] R A} (1) Upper approximation: apr (A)= {X U 1 [X] R A 0} (2) Where [X] R is the equivalence class of the element X. In fact the two approximate operations are the key operations in rough sets. From the above definitions we can see what the membership relation and knowledge R define. When the knowledge R is changed, the results of approximations are changed, too. On the other hand membership relations in rough sets and approximation operations are conditions for each other, and they are both relative to knowledge R. So in this sense we say membership relation is one of primary relations of rough sets. In sum, rough set theory generalizes the absolute membership relation from the qualitative facet. V. COMPARATIVE APPLICATION OF FUZZY SET AND ROUGH SET With the expansion of the Internet, searching for information goes beyond the boundary of physical libraries. Millions of documents of various media types, such as text, image, video, audio, graphics, and animation, are available around the world and linked by the Internet. Queries are used to find information items relevant to an information need. Unfortunately, it is not always possible to get relevant information each time relative to one's query. To address this situation, many researchers have proposed the application of fuzzy sets and rough sets for information retrieval. The section describes the application of rough sets and fuzzy sets in informational retrieval systems. A. Fuzzy sets in information retrieval systems Problems usually can be modeled using the binary paradigm. However there are cases where one needs to consider a broader set of choices instead of the only possible two: true or false. The basic idea of fuzzy set theory is that elements or entities can be assigned to sets of varying degrees. That is, instead of either including an element in a given set or excluding it from the set, a membership function is used to express the degree to which the element is a member of the set. For instance to express that a book is new using only two-valued logic, it is necessary to decide on an exact limit on age. For example, to distinguish "new book" from not new book, one may use the criteria that books published within the last 3 years are new, and books older than that are not new. But the fallacy of this concept is that it classifies a book not new even if the book is published hours after the 3 years criteria mentioned earlier. The books can be better classified if one uses more classes, namely, new, relatively, some what old, very old etc. (more than two values or states) and this is not possible with the classical two-valued logic. The fuzzy set approach can be used to classify the documents into fuzzy affinity classes and also to control the actual retrieval process. For example, consider first a document DOC and a particular term A. If A denotes the concept class of all items dealing with the subject denoted by A, then the membership function of documents DOC in set A may be denoted as f A (DOC). In the usual terminology f A (DOC) represents the weight of term A in document DOC. Given a number of concept classes A, B, C,, Z representing various subject areas, it is now possible to identify each document by giving its membership function with respect to each of the concept classes, that is, D = (f A (DOC), f B (DOC),.., f Z (DOC)) In general, the distance (or similarity) between two documents or between the document and a query may be obtained as a function of the differences in the membership functions of two items in the corresponding concept classes. Specifically, given T different concept classes, the fuzzy distance between documents DOC and DOC might be computed as Ranked retrieval is achieved by retrieving the documents in order of increasing fuzzy distance from the query. One attractive feature of fuzzy set theory is the possibility of extending the definition of the membership function from single terms to combination of terms. Thus, given the membership functions of document DOC with respect to terms A and B, the following rules apply for Boolean combinations of terms. ISSN: ISBN:

5 F(A AND B)(DOC) = min(f A (DOC), f B (DOC)) F(A OR B)(DOC) = max(f A (DOC), f B (DOC)) F(NOT A)(DOC) = 1 f A (DOC) Definition: Given a pair of standard sets B and M, a fuzzy set F based on B is a pair (B, f) where f: B ->M. B is the base set or support, M is the membership space and f is the membership function mapping any element of the support in the corresponding membership value. It is possible to generalize the basic definition of fuzzy set with a recursive use of fuzzy sets in the definition of the membership space. A. Fuzziness in Information Retrieval System Information retrieval (IR) can be considered a process of making connections amongst most existing concepts and structuring of comprehensive knowledge [20]. For example, in a document search, the documents required by the requester can be considered existing concepts for the production of request information, that is, the comprehensive knowledge. When one considers documents to be the existing concepts, the group of keywords can be seen as a group of attributes that describes that concept-that is, the concept of the documents. This phenomenon is interesting in language processing, and by extension in information retrieval, because the assignment of individual words to meaning categories is a fuzzy process; so is the assignment of documents to concept categories, and also the retrieval of documents in answer to certain queries [19]. Therefore, to address this situation, lots of research is going on for adopting fuzzy set theory for information retrieval. Some of the problems that can be successfully addressed by fuzzy logic are i) the clarification and interpretation of information ii) the retrieval of information by querying and reasoning and iii) the utilization of information in decision making, designing and optimization tasks. Fuzzy query is also known as knowledge query and using this query imprecise data such as opinions, judgments and values can be expressed in linguistic terms, can be queried from the database. Thus to make fuzzy queries against a relational database one need to decompose the domain of database column into their underlying term sets. B. Rough Sets applied to information retrieval In the information retrieval context there are different objects, which may be partitioned using equivalence relations. First, we may partition documents. Here the equivalence relation would place documents that are similar to each other in the same class. This is the approach advocated by [21]. In their work they partition documents into classes such that similar documents are placed in the same class based on a subjective evaluation of similarity. However, document similarity may also be computed using any of a variety of similarity measures based on the indexing information. Or the strategy may be to group together documents that are relevant to the same set of queries. In addition, citations and other such linkages may be used to derive classes of similar documents. The second set of objects that may be partitioned is the query set. Here the equivalence relation may be selected such that each equivalence class contains queries that are similar as determined by the terms in common or the similarity may be determined as a function of the relevant documents in common. A third strategy may be to partition the indexing (search) vocabulary of the database. The idea here is to group terms that are similar to each other. This provides maximum versatility with respect the types of objects that can later be compared to each other. For example, if we choose to represent queries and documents by groups of terms then such a partition enables comparisons between documents, between queries and between documents and queries. In comparison, equivalence classes of documents, such as approached by [21], or equivalence classes of queries are limited in their utility. Let D be a collection of documents, which has an indexing vocabulary V. Let R1, be an equivalence relation, which divides V into elementary sets of terms such that the terms within a set are similar to each other. Now, given a particular document D i represented by the set of index terms belonging to V,the theory suggests that D i is either definable in the approximation space A = (V, R 1 ) or it may be approximated by A(X i ) and A(X i ). (It should be noted that X i may represent a query instead of a document.) In this context A (X i ) and A (X i ) represent composed sets in the approximation space VI. COMPARATIVE VIEW WITH APPLICATION The Fuzzy model arose as one-way of providing both Boolean logic as well as ranking of documents [22,23] in information retrieval systems. Documents are assigned membership values with respect to index terms during indexing and are assigned membership values with respect to queries during retrieval. However there are a number of problems here. First it is not clear how to define the membership functions, which assign the numerical values between documents and terms. Further there are well known problems with regards to the way in which the membership values are combined in response to a Boolean query to compute the document s retrieval status or rank [24]. The model allows the weighting of Boolean operators as well as complete sub expressions of the search specification [26]. Weighting of the AND and OR operators indicate (at best)) that the user is not entirely comfortable with the operator(s) chosen or the strictness with which it is implemented. This suggests that other intermediate strategies should be provided such as those within the rough set model and [25]. Advantages accorded to the fuzzy model are the inclusion of Boolean ISSN: ISBN:

6 logic with the simultaneous ability to weight terms and operators. VII. CONCLUSION Membership relation is the most primary relation in classical set theory. From the generalized view, we have analyzed the difference and identity of fuzzy set and rough set theory on the basis o vagueness and uncertainty. The identity of rough sets and fuzzy sets is that they are both the generalized membership relation of classical set theory, and the difference of rough sets and fuzzy sets is they are the generalized membership relation from different facets. From the comparisons we can get an initially unified view for the two theories. On this basis we can generalize other contents of classical set theory to make our work further. On the other hand we also can these two theories as the fuzzy rough set theory to extend the study and give better results. Many attempts have been made to combine theories of fuzzy sets and rough sets in order to have algebra, which is both an extension and a deviation of classical set algebra. One may introduce additional set-theoretic operators in the theory of fuzzy sets, or use graded binary relations in the theory of rough sets. Our comparisons made here of fuzzy sets and rough sets lead to further interesting results, which can be applied in user centric systems. REFERENCES [15] Wu Xuemou, "The pansystems view of the world", Press of Chinese People University, Beijing, pp , [16] Yao. Y.Y., "A comparative study of fuzzy sets and rough sets", in Information Sciences, Vol. 109, No. 1-4, 1998, pp [17] Tsau Young Lin, "Context Free Fuzzy Sets", The Fourth Annual International Conference on Fuzzy Theory and Technology, Proceedings of Second Annual Joint Conference on Information Science, Wrightsville Beach, North Carolina, Sept. 28-Oct. 1, 1995, pp [18] L.A. Zadeh, Toward a restructuring of the foundations of fuzzy logic (FL), Abstract of BISC Seminar, Computer Science Division, EECS, University of California, Berkeley,1997. [19] Yongli Li, Zhilin Li, Yong-qi Chen, Xiaoxia Li and Xuemou Wu, "Pansystems relativity of observation", in Int. J. Advances in Systems Science and Application, No2, pp , [20] Cohen, M. E., Hudson, D. L., Yazici, A., George, R., Buckles, B. P., Petry, F.E. (1992). A survey of conceptual logical models for uncertainty management, Fuzzy Logic for Management of Uncertainty, John Wiley and Sons Inc., pp [21] Wong, S. K. M. et al. A Machine Learning Approach to Information Retrieval. Proceedings 1986 ACM SIGIR conference, Italy, [22] Bookstein, A. Probabilistic and Fuzzy-Set Applications to Information Retrieval. Annual Review of Information Science and Technology, Vol. 20: 117. [23] Radecki, T. Fuzzy Set Theoretical Approach to Document Retrieval. Information Processing and Management, 15(5): , [24] Bookstein, A. On the Perils of Merging Boolean and Weighted Retrieval Systems. Journal of the ASIS, Vol. 29: , [25] Salton, G et al. Automatic Assignment of Soft Boolean Operators. Proceedings 1985 ACM SIGIR conference, Montreal, [26] BuelI, D. A. A General Model of Query Processing in Information Retrieval Systems. Information Processing and Management, Vol.17 (5): ,1981. [1] Z. Pawlak, Rough sets, International Journal of Computer and Information Sciences, 11: (1982). [2] Z. Pawlak, Rough sets and fuzzy sets, Fuzzy Sets and Systems, 17: (1985). [3] L.A. Zadeh, Fuzzy sets, Information and Control, 8: (1965). [4] S. Chanas and D. Kuchta, Further remarks on the relation between rough and fuzzy sets, Fuzzy Sets and Systems, 47: (1992). [5] L.A. Zadeh, Forward, in: E. Orlowska (Ed.), Incomplete Information: Rough Set Analysis, Physica-Verlag, Heidelberg, pp. v-vi, 1998 [6] D. Dubois and H. Prade, Rough fuzzy sets and fuzzy rough sets, International Journal of General Systems, 17: (1990). [7] M. Wygralak, Rough sets and fuzzy sets some remarks on interrelations, Fuzzy Sets and Systems, 29: (1989). [8] G.J. Klir, Multivalued logics versus modal logics: alternative frameworks for uncertainty modelling, in: P.P. Wang (Ed.), Advances in Fuzzy Theory and Technology, Department of Electrical Engineering, Duke University, Durham, North Carolina, pp. 3-47, 1994 [9] Z. Pawlak, Rough Sets: Theoretical Aspects of Reasoning about Data, Kluwer Academic publishers, Boston, 1991 [10] Z. Pawlak and A. Skowron, Rough membership functions, in: L.A. Zadeh and J.Kacprzyk (Eds.), Fuzzy Logic for the Management of Uncertainty, John Wiley & Sons, New York, pp , 1994 [11] N. Rescher, Many-valued Logic, McGraw-Hill, New York, 1969 [12] E.J. Lemmon, Algebraic semantics for modal logics I, II, Journal of Symbolic Logic, 31:46-65, (1966) [13] T.Y. Lin, Topological and fuzzy rough sets, in: R. Slowinski (Ed.), Intelligent Decision Support: Handbook of Applications and Advances of the Rough Sets Theory, Kluwer Academic Publishers, Boston, pp , [14] R.E. Moore, Interval Analysis, Englewood Cliffs, New Jersey, Prentice-Hall, 1966 ISSN: ISBN: