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A Vague Relational Model and Algebra 1 http://www.paper.edu.cn A Vague Relational Model and Algebra 1 Faxin Zhao, Z.M. Ma*, and Li Yan School of Information Science and Engineering, Northeastern University Shenyang, P.R. China 110004 [email protected] Abstract Imprecision and uncertainty in data values are pervasive in real-world environments and have received much attention in the literature. Several methods have been proposed for incorporating uncertain data into relational databases. However, the current approaches have many shortcomings and have not established an acceptable extension of the relational model. In this paper, we propose a consistent extension of the relational model to represent and deal with fuzzy information by means of vague sets. We present a revised relational structure and extend the relational algebra. The extended algebra is shown to be closed and reducible to the fuzzy relational algebra and further to the conventional relational algebra. Keywords: Vague set, Fuzzy set, Relational data model, Algebra 1 Introduction Over the last 30 years, relational databases have gained widespread popularity and acceptance in information systems. Unfortunately, the relational model does not have a comprehensive way to handle imprecise and uncertain data. Such data, however, exist everywhere in the real world. Consequently, the relational model cannot represent the inherently imprecise and uncertain nature of the data. The need for an extension of the relational model so that imprecise and uncertain data can be supported has been identified in the literature. Fuzzy set theory has been introduced by Zadeh [1] to handle vaguely specified data values by generalizing the notion of membership in a set. And fuzzy information has been extensively investigated in the context of the relational model. Based on various fuzzy relational database models, many studies have been done for data integrity constraints [2, 3]. Also there have been research studies on fuzzy query languages [4] and fuzzy relational algebra [5]. In [4], an existing query language, namely SQL, for fuzzy queries was extended and some fuzzy aggregation operators were developed. In fuzzy set theory, for a given fuzzy set F, each object u∈U is assigned a single value, called the grade of membership, between zero and one (here U is a universe of discourse). In [6], the concept of vague sets was introduced, which is a generalized version of fuzzy sets. Gau and Buehrer pointed out that the drawback of using the single membership value in fuzzy set theory is that the evidence for u∈U and the evidence against u∈U are in fact mined together. They also pointed out that the single number reveals nothing about its accuracy. In order to tackle this problem, they proposed the notion of vague set. A vague set, as well as an Intuitionistic Fuzzy Set (IFS) [7], is a further generalization of a fuzzy set. But vague set is not isomorphic to the IFS, there are some interesting features for handling vague data that are unique to vague sets, such as vague sets allow for a more intuitive graphical representation of vague data, which facilitates significantly better analysis in data relationships, incompleteness, and similarity 1 Support by the National Research Foundation for the Doctoral Program of Higher Education of China under Grant No.20050145024. 1 http://www.paper.edu.cn measures [8]. In [8], the notion of vague sets was initially incorporated into relations. In this paper, we propose an extension of the relational model−the vague relational database (VRDB) model. Based on the proposed model and the semantic measure of vague sets, we define vague relational operations in the relational algebra, which can be used to query and update vague relational databases. The remainder of this paper is organized as follows. Section 2 presents some basic knowledge about the vague set theory. Based on the theory, we propose a vague relation model in Section 3. Vague data redundancies and removal are investigated in Section 4. Section 5 defines vague relational operations. Section 6 concludes the paper. 2 Basic Knowledge Let U be a universe of discourse, where an element of U is denoted by u. Definition 2.1. A vague set V in U is characterized by a truth-membership function tV and a false-membership function fV. Here tV (u) is a lower bound on the grade of membership of u derived from the evidence for u, and fV (u) is a lower bound on the negation of u derived from the evidence against u. tV (u) and fV (u) are both associated with a real number in the interval [0, 1] with each element in U, where tV (u) + fV (u) ≤ 1. That is tV: U → [0, 1] and fV: U → [0, 1] Suppose U = {u1, u2, …, un}. A vague set V of the universe of discourse U can be represented by n V = ∑[t v (ui ),1− fv (ui )]/ui ,∀ui ∈U , where tV (u) ≤ µV (u) ≤ 1 - fV (u) and 1≤ i ≤ n. i=1 This approach bounds the grade of membership of u to a subinterval [tV (u), 1 - fV (u)] of [0, 1]. In other words, the exact grade of membership µV (u) of u may be unknown, but is bounded by tV (u) ≤ µV (u) ≤ 1 - fV (u), where tV (u) + fV (u) ≤ 1. For a vague value [tV (u), 1 - fV (u)]/u, the vague value to the object u is the interval [tV (u), 1 - fV (u)]. For example, if [tV (u), 1 - fV (u)] = [0.5, 0.8], then we can say that tV (u) = 0.5, 1 − fV (u) = 0.8 and fV (u) = 0.2. It can be interpreted as “the degree that object u belongs to the vague set V is 0.5; the degree that the object u does not belong to the vague set V is 0.2”. In a voting process, the vague value [0.5,0.8] can be interpreted as “the vote for resolution is 5 in favor,2 against, and 3 neutral (abstentious).” The precision of the knowledge about u is characterized by the difference (1 - tV (u) - fV (u)). If this is small, the knowledge about u is relatively precise. But if it is large, we know correspondingly little. If tV (u) is equal to (1 - fV (u)), the knowledge about u is exact, and the vague set theory reverts back to fuzzy set theory. If tV (u) and (1 - fV (u)) are both equal to 1 or 0, depending on whether u does or does not belong to V, the knowledge about u is very exact and the theory reverts back to ordinary sets. In the following, we present several special vague sets and operations on vague sets. Definition 2.2. A vague set V is an empty vague set if and only if its truth-membership function tV = 0 and false-membership function fV = 1 for all u on U. We use ∅ to denote it. Definition 2.3. The complement of a vague set V, denoted V’, is defined by tV’ (u) = fV (u) and 1 - fV’ (u) = 1 - tV (u). Definition 2.4. A vague set A is contained in another vague set B, written as A ⊆ B, if and only if tA ≤ tB and 1 - fA ≤ 1 - fB. Definition 2.5. Two vague sets A and B are equal, written as A = B, if and only if A ⊆ B and B ⊆ A, namely, tA = tB, and 1 - fA = 1 - fB. Definition 2.6. (Union) The union of two vague sets A and B is a vague set C, written as A ∪ B, whose truth-membership and false-membership functions are related to those of A and B by tC = max (tA, tB) and 1 - fC = max (1 - fA, 1 - fB) = 1 – min (fA, fB). 2 http://www.paper.edu.cn Definition 2.7. (Intersection) The intersection of two vague sets A and B is a vague set C, written as C = A ∩ B, whose truth-membership and false-membership functions are related to those of A and B by tC = min (tA, tB) and 1 - fC = min (1 - fA, 1 - fB) = 1 – max (fA, fB). 3 Vague Relational Model In order to incorporate fuzzy information into relational databases, various attempts towards enhancing the relational database model by fuzzy extensions can be found in the literature [5, 9]. In this section, we describe an approach of enhancing the relational model by means of vague set theory, which results in the vague relational database (VRDB) model. To this purpose, we should extend some attribute domains as a set of vague sets and a vague relational instance is hereby defined as a subset of the Cartesian product of such attribute domains. Definition 3.1. Let Ai for i from 1 to n be attributes defined on the universes of discourse sets Ui, respectively. Then a vague relation r is defined on the relation schema R (A1, A2, …, An) as a subset of the Cartesian product of a collection of vague subsets: … r ⊆ V (U1) × V (U2) × × V (Un), where V (Ui) denotes the collection of all vague subsets on a universe of discourse Ui. Each tuple t of r consists of a Cartesian product of vague subsets on the respective Ui’s, i.e., t [Ai] = π (Ai) where π (Ai) is a vague subset of the attribute Ai defined on Ui for all i. It should be noticed that the vague relations can be considered as an extension of classical relations (all vague values are [1, 1]) and possibilistic relations (all vague values are possibility distributions, i.e. such that each degree is [a, a], 0 ≤ a ≤ 1), which can capture more information about vagueness.
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