International Journal of Computer Applications (0975 – 8887) Volume 39– No.8, February 2012 29 A New Functional Dependency in a Vague Relational Database Model Jaydev Mishra College of Engineering and Management, Kolaghat West Bengal, India. Sharmistha Ghosh Vellore Institute of Technology University, Vellore, Tamilnadu, India. ABSTRACT In order to model the real world with imprecise and uncertain information, various extensions of the classical relational data model have been studied in literature using fuzzy set theory. However, vague set, as a generalized fuzzy set, has more powerful ability to process fuzzy information than fuzzy set. In this paper, we have proposed a vague relational database model and have defined a new kind of vague functional dependency (called  -vfd) based on the notion of  -equality of tuples and the idea of similarity measure of vague sets. Next, we present a set of sound vague inference rules which are similar to Armstrong‟s axioms for the classical case. Finally, partial  - vfd and vague key have been studied with the new notion of  -vfd and also tested with examples. General Terms Vague Database Design Keywords Vague set, similarity measure of vague sets,  -vfd, partial  -vfd, vague key. 1. INTRODUCTION Information in the real world is very often imprecise or uncertain in nature. Fuzzy set theory, introduced by Zadeh in 1965 [1] has been widely used in literature [2, 3, 4, 5, 6, 7, 8, 9] to incorporate such imprecise data into classical relational databases. Extensive research has been carried out in this direction and several fuzzy relational database models have been proposed to model vague information in relational databases. Also, based on such fuzzy relational models, there have been many studies on different data integrity constraints [2, 3, 4, 8, 9], fuzzy relational algebra [5], fuzzy query languages [10] and so on. However, vague set theory was put forward by Gau and Buehrer [11] in 1993 as a more efficient tool to deal with imperfect or ambiguous data. A vague set, conceived as a generalization of the concept of fuzzy set, is a set of objects each of which has a grade of membership whose value is a continuous sub-interval of [0,1]. On the contrary, it is well known that a traditional fuzzy set F in the universe of discourse U is characterized by a single membership function F  that assigns to each object u  U a single membership value  u F  which is a real number lying between 0 and 1.  u F  is called the grade of membership of the element u in the set F. However, in real life it is hard to make sure of the precision degree that an element belongs to a fuzzy set. Further, it was pointed out by Gau et al. in [11] that the single membership value in the fuzzy set theory combines the evidence for u  U and the evidence against u  U without indicating how much there is of each. To resolve this problem, they had introduced the concept of vague set V which is characterized by a truth membership function V t and a false membership function V f . Thus, a vague set separates the positive and negative evidence for membership of an element in the set and provides lower and upper bounds on the grade of membership of an element in the set. These lower and upper bounds are used to create a sub-interval on [0, 1], namely,   ) ( 1 ), ( u f u t V V  , to generalize the membership function ) (u V  of fuzzy sets, where ) ( 1 ) ( ) ( u f u u t V V V     . Since vague sets have been introduced to deal with imprecise information in a more efficient manner than traditional fuzzy sets, classical relational databases may also be extended to represent and deal with uncertain data with the concept of vague set theory. The extended database model is then called a vague relational database model. However, compared to fuzzy relational databases, much less research has been reported so far in the area of vague relational database. This is, in particular, true for the study of vague functional dependency. But, it is well known that data dependencies play an important role in any database design and implementation of functional dependency of one set of attributes upon another is one of the most vital concepts in relational databases. Thus, similar to the theory of classical relational databases, vague functional dependencies can also be used as a guideline for the design of a vague relational schema. Zhao et al. [12] have proposed a vague relational model based on vague set theory and a new similarity measure (SE) between vague sets. They have also extended and studied the vague relational algebra in ref. [13]. In particular, Zhao et al. have focused on the issues of vague functional dependency and vague Armstrong‟s axioms in [12]. According to Zhao et al., a vague relation r on a relational schema R satisfies the vague functional dependency vfd: Y X vfd   , where XY  R if ])) [ ], [ ( ( ]) [ ], [ ( )( )( ( Y s Y t SE X s X t SE r s r t      . It may be clearly observed that the above definition of vfd fails if the ]) [ ], [ ( Y s Y t SE is slightly less than ]) [ ], [ ( X s X t SE , which should not be the case in reality. However, such situations have been taken care by Zhao et al. [12] in the satisfaction degree of vfds. Such vague functional dependencies have been utilized by Lu & Ng [14] to maintain the consistency of a vague database.