INTERNATIONAL JOURNAL OF COMPUTATIONAL COGNITION (HTTP://WWW.IJCC.US), VOL. 6, NO. 4, DECEMBER 2008
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A Study of Fuzzy Relational Database Jaydev Mishra and Sharmistha Debnath Ghosh
Abstract— The present work deals with detail study of functional dependency in fuzzy relational databases. The paper is an extension of the work done by Al-Hamouz and Biswas [1]. Here, the f-Armstrong’s axioms of [1] have been modified with verification and the well-known concepts of partial ffd, fuzzy key, fuzzy closure of attribute(s) and redundant ffd have been studied with the new notion of α-ffd as defined in [1]. Finally fuzzy key and fuzzy closure of attribute(s) have been tested with c 2008 Yang’s Scientific Research a real life example. Copyright ° Institute, LLC. All rights reserved. Index Terms— Fuzzy set, fuzzy union, fuzzy functional dependency(ffd), f-Armstrong’s axioms, partial ffd, fuzzy key, redundant ffd.
I. I NTRODUCTION
I
N recent years, a lot of attention has been attracted to fuzzy relational databases that generalize the classical relational data model introduced by Codd [5] by allowing uncertain and imprecise information to be represented and manipulated. Data is often partially known, vague or ambiguous in many real world applications. Fuzziness is introduced in the classical model to deal with such imprecise information and several extensions of the model are available in literature [1][3][4][6][7]. Data dependency plays a crucial role in logical database design and functional dependency of one set of attributes upon another is one of the most important concepts in relational databases. When data is of fuzzy nature, the concept of functional dependency has been extended by several authors [1][2][4][7][8] to define fuzzy functional dependency(ffd) using the concept of fuzzy logic. The comparison of two data of a domain is done with the help of fuzzy equality relations in these existing concepts of ffd. Recently, in 2006, a new notion of ffd based on equivalence relation has been introduced by Al-Hamouz and Biswas [1] and the f-Armstrong’s axioms have been studied and verified with that ffd. The definition of ffd defined in [1] with the concept of (α< )-nearer and (α< )equality made easy for developing the theory of fuzzy database compare to the definition of ffd defined by other authors. The present work is based on the ffd defined by Al-Hamouz and Biswas which introduces a choice parameter α ∈ [0, 1] that may be set by the database designer and is called αffd. In this paper, the f-Transitive, f-Union, f-Pseudotransitive Manuscript received May 13, 2008; revised July 09, 2008. Jaydev Mishra, Department of CSE & IT, College of Engineering and Management, Kolaghat-721171, West Bengal, India. Email:
[email protected]. Sharmistha Debnath Ghosh, Department of Mathematics, College of Engineering and Management, Kolaghat-721171, West Bengal, India. Email:
[email protected]. Acknowledgment: The author would like to thank Prof. Ranjit Biswas for their contribution to the field of fuzzy databases. Publisher Item Identifier S 1542-5908(08)10403-1/$20.00 c Copyright °2008 Yang’s Scientific Research Institute, LLC. All rights reserved. The online version posted on February 17, 2009 at http://www.YangSky.com/ijcc/ijcc64.htm
and f-Decomposition rules of [1] have been modified with proper verification. I have also extended the work to define partial fuzzy functional dependency(partial ffd), fuzzy key, fuzzy closure of an attribute or a set of attributes, redundant fuzzy functional dependency(redundant ffd). The paper is organized as follows: In section II, we have revisited some basic definitions of fuzzy set theory and then defined fuzzy functional dependency(ffd) as in [1]. In section III, the f-Armstrong’s axioms have been studied and modified with verification. Next, in section IV, we have proceeded to define partial fuzzy functional dependency(partial ffd). The definition of fuzzy key at α-level of choice has been introduced and discussed with an example in section V. The idea of fuzzy closure of an attribute or a set of attributes has been studied in section VI together with an algorithm and an example. An example of real life application area has been taken in section VII to show how fuzzy key and fuzzy closure of attribute(s) can be computed. Redundant fuzzy functional dependency(redundant ffd) has been defined in section VIII with example. Finally, the concluding remarks have been given in the last section IX. II. F UZZY F UNCTIONAL D EPENDENCY ( FFD ) In this section, I first review some basic definitions of fuzzy set theory that will be useful throughout the paper and then define the Fuzzy Functional Dependency ffd) as introduced in [1]. Let U = {u1 , u2 , . . . , un } be a universe of discourse. Definition II.1 A fuzzy set A in the universe of discourse U is characterized by the membership function µA given by µA : U → [0, 1] and A is defined as the set of ordered pairs A = {(u, µA (u)) : u ∈ U }, where µA (u) is the grade of membership of element u in the set A. Definition II.2 If A and B are two fuzzy sets of the universe S U , then the fuzzy union of A and B is denoted by A B f uzzy S and is defined as A B = {(x, max{µA (x), µB (x)}) : x ∈ U }.
f uzzy
Definition II.3 Let X and Y be two sets. A fuzzy relation R from X to Y is a fuzzy set on X × Y and is denoted by R(X → Y ). Definition II.4 A fuzzy relation R(X → X) is said to be 1) reflexive: iff ∀x ∈ X, µR (x, x) = 1; 2) symmetric: iff ∀x1 , x2 ∈ X, µR (x1 , x2 ) = µR (x2 , x1 ). A fuzzy relation is said to be a fuzzy tolerance relation if it is reflexive and symmetric.
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INTERNATIONAL JOURNAL OF COMPUTATIONAL COGNITION (HTTP://WWW.IJCC.US), VOL. 6, NO. 4, DECEMBER 2008
Next, to introduce the new notion of ffd as defined in [1], we give the following definitions and terminologies. Let X be a universal set and < be a fuzzy tolerance relation on X. Consider a choice parameter α ∈ [0, 1] to be predefined by the database designer.
Proposition II.3 If 0 ≤ α2 ≤ α1 ≤ 1, then X −→ Y ⇒ α1 X −→ Y .
Definition II.5 (α)< -nearer or α-nearer elements. Two elements x1 , x2 ∈ X are said to be (α)< -nearer (or α-nearer), if µ< (x1 , x2 ) ≥ α, we denote this by the notation x1 N(α)< x2 .
t1 [X]εα1 t2 [X] ⇒ t1 [Y ]εα1 t2 [Y ]
Definition II.6 (α)< -equality orα-equality elements. Two elements x1 , x2 ∈ X are said to be (α)< -equal(or α-equal) if 1) either x1 N(α) < x2 or 2) ∃y1 , y2 , y3 , . . . , yr−1 , yr ∈ X such that {x1 N(α)< y1 , y1 N(α)< y2 , . . . , yr−1 N(α)< yr , yr N(α)< x2 }. This equality is denoted by the notation x1 E(α)< x2 . Definition II.7 δ(α)< relation on X. The crisp relation δ(α)< on X is defined as: For x1 , x2 ∈ X, x1 δ(α)< x2 if x1 E(α)< x2 . Proposition II.1 The relation δ(α)< defined on X is an equivalence relation (See ref. [1]) Now, consider a relation r(R) of a relation schema R(A1 , A2 , . . . , An ). Let us assume a fuzzy tolerance relation
