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INTERNATIONAL JOURNAL OF COMPUTATIONAL COGNITION (HTTP://WWW.IJCC.US), VOL. 9, NO. 2, JUNE 2011
A Multivalued Integrity Constraint in Fuzzy Relational Database Jaydev Mishra and Sharmistha Ghosh
Abstract— This paper introduces a new definition of fuzzy multivalued dependency, called α-fmvd, on the basis of the αequality of tuples as defined in [1]. Next the definition is shown to be consistent, i.e., it reduces to that of classical multivalued dependency (mvd) when the choice parameter takes the value one. Finally, a set of sound and complete inference axioms have been c 2011 Yang’s designed and proved for the α-fmvd. Copyright ° Scientific Research Institute, LLC. All rights reserved. Index Terms— Fuzzy relational database, Integrity constraint, α-equality of tuples, Fuzzy functional dependency, Fuzzy multivalued dependency.
I. I NTRODUCTION It is widely recognized that many real world relations are inherently fuzzy in nature. To incorporate such imprecise or fuzzy data, the classical relational data model introduced by Codd [5] in 1970 has been extended by several authors on the mathematical framework of fuzzy set theory proposed by Zadeh [15], [16]. Such an extended database model is called a fuzzy relational database model which may be able to represent and manipulate uncertain or vague information. Integrity constraints play an important role in any logical database design as well as database manipulation. In fact, in classical relational database literature, integrity constraints and the associated inference rules constitute a major area of research. Different types of integrity constraints such as functional dependency, multivalued dependency, join dependency etc. have been identified and studied widely in literature [6], [7] which are considered to be crucial for the purpose of designing a database. Thus, as one extends the classical relational database into a fuzzy relational database, it would also be necessary to study the integrity constraints in the light of fuzzy concepts and several authors [1]–[4], [8]–[14] have contributed in this direction also. In this context, in the year 2006, Al-Hamouz and Biswas [1] have introduced a new definition of fuzzy functional dependency (ffd), called α-ffd based on equivalence relation. The α-ffd defined in [1] with the concepts of (α)R -nearer elements and (α)R -equality of tuples made the development of the Manuscript received March 21, 2010; revised May 27, 2010. Jaydev Mishra, Assistant Professor, Department of Computer Science and Information Technology, College of Engineering and Management, Kolaghat, West Bengal, India. Pin: 721 171. Sharmistha Ghosh, Associate Professor, School of Advanced Sciences, Vellore Institute of Technology University, Vellore, Tamilnadu, India. Pin: 632 014. Emails:
[email protected] (Jaydev Mishra),
[email protected] (Sharmistha Ghosh). Publisher Item Identifier S 1542-5908(11)10411-X/$20.00 c Copyright °2011 Yang’s Scientific Research Institute, LLC. All rights reserved. The online version posted on June 08, 2011 at http://www.YangSky.com/ijcc/ijcc92.htm
theory of fuzzy database easier compared to the definition of ffd given by other authors. In the existing concepts of ffd, the comparison of two data in a domain is done with the help of fuzzy equality relations, which are not equivalence relations. Thus, the exploitation of the transitivity property is missed by them. A detailed study of fuzzy functional dependency based on the definition of α-ffd together with verification of fArmstrong’s axioms may also be found in the work of Mishra and Ghosh [8]. In the present paper, the authors have made an attempt to extend the idea of α-equality of tuples based on [1] to define fuzzy multivalued dependency (fmvd). Multivalued dependency is as such another very important integrity constraint that may arise in many applications that occur in dayto-day life. For example, “a customer can stay in two different addresses throughout the year” or “a teacher can access more than one text book for a particular subject”. Now, if a teacher wants to find books for Distributed Database he can access the list of books which are specific for Distributed Database by using a classical relational database. But the teacher may also read books on Database or Advanced Database to learn the concepts of Distributed Database since few chapters on Distributed Database are also included in these books. So, basically the books whose contents have resemblance should be fetched. Our traditional database cannot tackle this idea. Thus, in order to design a fuzzy relational database that can treat similar problems, one should study fuzzy multivalued dependency (fmvd). In this paper, we present a new definition of fuzzy multivalued dependency (fmvd), called α-fmvd, on the basis of the α-equality of tuples as defined in [1]. The new definition provides a very easy and straightforward way of extending mvd to fmvd for a fuzzy relational database model and thus it differs from the existing definitions in literature. Then a set of sound and complete inference axioms have been proposed and proved for the α-fmvd. 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 (αffd) as in reference [1]. In Section III, we have proposed the new definition of fuzzy multivalued dependency (α-fmvd) which have been shown to be consistent. Then the inference rules for the α-fmvd have also been designed and proved. Finally, the concluding remarks have been given in Section IV. II. BASIC D EFINITIONS In this section, we first review some basic definitions from fuzzy set theory that will be useful throughout the paper and then define fuzzy functional dependency (ffd) as introduced in [1] and revisit the basic propositions related to α-ffd.
MISHRA & GHOSH, A MULTIVALUED INTEGRITY CONSTRAINT IN FUZZY RELATIONAL DATABASE
A. Basic Preliminaries on Fuzzy Set Theory 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 U , then B is de[ the fuzzy union of A and [ B and is defined as A B = noted by A f uzzy
{(x, max{µA (x), µB (x)}) : x ∈ U }.
f uzzy
Definition II.3. If A and B are two fuzzy sets of the universe U , then the fuzzy intersection of A and B is f uzzy f uzzy \ \ denoted by A B and is defined as A B = {(x, min{µA (x), µB (x)}) : x ∈ U }. Definition II.4. 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.5. 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. B. Definition of Fuzzy Functional Dependency To introduce the new notion of α-ffd as defined in [1], we now give the following definitions and terminologies. Let X be a universal set and R be a fuzzy tolerance relation on X. Consider a choice parameter α ∈ [0, 1] to be predefined by the database designer. Definition II.6. (α)R -nearer or α-nearer elements: Two elements x1 , x2 ∈ X are said to be (α)R -nearer (or α-nearer) if µR (x1 , x2 ) ≥ α. We denote this by the notation x1 N (α)R x2 . Definition II.7. (α)R -equality or α-equality: Two elements x1 , x2 ∈ X are said to be (α)R -equal (or α-equal) if 1) either x1 N (α)R x2 or 2) ∃y1 , y2 , y3 , . . . , yr−1 , yr ∈ X such that {x1 N (α)R y1 , y1 N (α)R y2 , y2 N (α)R y3 , . . . , yr−1 N (α)R yr and yr N (α)R x2 }. This equality is denoted by the notation x1 E(α)R x2 . Now, consider a relation r(R) of a relation schema R(A1 , A2 , . . . , An ). Let us consider a fuzzy tolerance relation Ri on the domain dom(Ai ), ∀i = 1, 2, . . . , n. Let R denote the set {Ri , R2 , . . . , Rn } of fuzzy tolerance relations. Let X = {x1 , x2 , . . . , xk } ⊆ R, here we consider attributes set as x1 , x2 , . . . , xk because X is any k-combination of A1 , A2 , . . . , An . We now define (α)R -equality of two tuples t1 [X] and t2 [X] in a relational database design as follows: Definition II.8. (α)R -equality of t1 [X] and t2 [X]: For any two tuples t1 and t2 , t1 [X] and t2 [X] are said to be (α)R equal if t1 [xi ]E(α)Ri t2 [xi ], ∀i = 1, 2, . . . , k. The equal-
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ity notation is denoted as t1 [X]ε(α)R t2 [X] or simply by t1 [X]εα t2 [X]. The following definition is then straightforward. Definition II.9. [Fuzzy Functional Dependency (α-ffd)] Let X, Y ⊂ R(A1 , A2 , . . . , An ). Choose a parameter α ∈ [0, 1] and propose a fuzzy tolerance relation R. A fuzzy functional dependency (ffd), denoted by X −−−→ Y or simply by X − → (α)R
α
Y , is said to exist if whenever t1 [X]ε(α)R t2 [X], it also implies that t1 [Y ]ε(α)R t2 [Y ]. This ffd can be read as “X fuzzy functionally determines Y at α-level of choice” or “Y fuzzy functionally depends on X at α-level of choice” and is called an α-ffd. Clearly, by definition of α-ffd, it follows that for any subset X of R and for any α ∈ [0, 1], X − → X. Further, from the references [1] α and [8], we have the following propositions: Proposition II.1. If 0 ≤ α2 ≤ α1 ≤ 1, then t1 [X]εα1 t2 [X] ⇒ t1 [X]εα2 t2 [X]. Proposition II.2. If Y ⊆ X, then for any two tuples t1 and t2 in R and for any 0 ≤ α ≤ 1, t1 [X]εα t2 [X] ⇒ t1 [Y ]εα t2 [Y ]. Proposition II.3. If 0 ≤ α2 ≤ α1 ≤ 1, then X −→ Y ⇒ α1 X −→ Y . α2
III. F UZZY M ULTIVALUED D EPENDENCY Before we proceed to define fuzzy multivalued dependency (fmvd), let us first see what multivalued dependency (mvd) is. A. Multivalued Dependency (mvd) Multivalued dependency (mvd), as introduced by Fagin [7], is an important data dependency which helps the database designer to remove redundancy from the database. Informally, mvd in databases relates a value of an attribute (or a set of attributes) to a set of values associated with a set of attributes, independent of the other attributes in the relation. Formally, the definition of mvd may be written as follows. Definition III.1. Let r be any relation instance on scheme R(A1 , A2 , . . . , An ) and let X, Y ⊂ R. Then X →→ Y (read as X multidetermines Y ) if for any two tuples t1 and t2 in r with t1 [X] = t2 [X], there exists a third tuple t3 in r with t3 [X] = t1 [X], t3 [Y ] = t1 [Y ] and t3 [R − X − Y ] = t2 [R − X − Y ]. As with functional dependency, multivalued dependency also faces the problem of the inability to handle imprecise attribute values. Thus, we need to extend the definition of mvd to deal with impreciseness. The extended version of mvd is called fuzzy multivalued dependency (fmvd) for which we present here a new definition using the idea of α-equality of tuples as defined in [1] and we call it α-fmvd. This extension to α-fmvd is straightforward where we choose the same methodology as α-ffd and soften the strict equality in mvd by α-equality as follows.
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B. Fuzzy Multivalued Dependency (α-fmvd) Let r be any relation instance on scheme R(A1 , A2 , . . . , An ). Let X and Y be two subsets of R. Now, the relation r(R) satisfies X →→ Y at the level α (called X fuzzy multidetermines Y at the level α) is denoted by X →→α Y and is defined as: For any two tuples t1 and t2 in r, if t1 [X]εα t2 [X], then there must exist a third tuple t3 in r with t1 [X]εα t2 [X]εα t3 [X] t1 [Y ]εα t3 [Y ] t2 [R − X − Y ]εα t3 [R − X − Y ]. Next, let us show the consistency of this new definition of α-fmvd. In order to do this, we prove that this definition of α-fmvd reduces to that of classical mvd when α = 1, i.e. the fmvd X →→α=1 Y ⇒ mvd X →→ Y .
•
(A) t1 [X]εα=1 t2 [X]εα=1 t3 [X] (B) t1 [Y ]εα=1 t3 [Y ] (C) t2 [R − X − Y ]εα=1 t3 [R − X − Y ]. Now, from Definition II.8, t1 [X]εα=1 t2 [X] ⇒t1 [xi ]Eα=1 t2 [xi ], ∀i = 1, 2, . . . , k ⇒µ(t1 [xi ], t2 [xi ]) ≥ α = 1, ∀i = 1, 2, . . . , k ⇒µ(t1 [xi ], t2 [xi ]) = 1, ∀i = 1, 2, . . . , k [∵ µ(t1 [xi ], t2 [xi ]) ∈ [0, 1]] ⇒t1 [xi ] = t2 [xi ] ∀i = 1, 2, . . . , k ⇒t1 [X] = t2 [X]. Using this in the above relations (A), (B) and (C), we see that for any two tuples t1 and t2 in r, if t1 [X] = t2 [X], then there must exist a tuple t3 in r with t1 [X] = t2 [X] = t3 [X], t1 [Y ] = t3 [Y ], t2 [R − X − Y ] = t3 [R − X − Y ], which implies X →→ Y . Thus, our definition of α-fmvd with α = 1 is equivalent to the classical definition of mvd and hence the definition is consistent. C. Inference rules for α-fmvd Next, we propose and prove the following inference rules for α-fmvd. • Replication rule for α-fmvd (α-fmvd-replication): If X − → Y , then X →→α Y . α
rule
for
α-fmvd
(α-fmvd-
If X →→α Y , then X →→α R − X − Y . •
Inclusion rule for α-fmvd (α-fmvd-inclusion): If X →→α1 Y and α1 ≥ α2 , then X →→α2 Y .
•
Additivity or Union rule for α-fmvd (α-fmvd-union): If X →→α1 Y and X →→α2 Z, then X →→min(α1 ,α2 ) Y Z.
•
Reflexivity rule for α-fmvd (α-fmvd-reflexivity): If Y ⊆ X, then X →→α Y .
•
Projectivity or Decomposition rule for α-fmvd (α-fmvdprojectivity): If X →→α1 Y and X →→α2 Z, then X →→min(α1 ,α2 ) Z − Y , X →→min(α1 ,α2 ) Y − Z and X →→min(α1 ,α2 ) Z ∩ Y .
Lemma III.1. The definition of the α-fmvd is consistent. Proof. Let us assume that X →→α=1 Y . Then, from the definition of α-fmvd it follows that for any two tuples t1 and t2 in r, if t1 [X]εα=1 t2 [X], then there must exist a third tuple t3 in r with
Complementation complementation):
•
Augmentation rule for α-fmvd (α-fmvd-augmentation): If X →→α Y and W ⊆ U , then U X →→α Y W .
•
Transitivity rule for α-fmvd (α-fmvd-transitivity): If X →→α1 Y and Y →→α2 Z, then X →→min(α1 ,α2 ) Z − Y .
The above inference axioms are similar to that of classical mvd and also they have been shown to be sound and complete in the fuzzy paradigm [2]. In the following sub-sections, we present a formal proof for each of the axioms proposed for our α-fmvd. 1) α-fmvd-replication rule: If X − → Y , then X →→α Y : α
Proof. We prove the replication rule by contradiction. Suppose some relation instance r of R satisfies X − → Y but violates α X →→α Y . Then, for tuples t1 and t2 in r with t1 [X]εα t2 [X], there exists a third tuple t3 in r for which t1 [X]εα t2 [X]εα t3 [X], but t1 [Y ]εα t3 [Y ] and t2 [R − X − Y ]εα t3 [R − X − Y ] does not hold. Now since X − → Y holds, then by Definition II.9, we have α for any two tuples t1 and t3 in r, whenever t1 [X]εα t3 [X], it also implies t1 [Y ]εα t3 [Y ] which contradicts our assumption. Hence X →→α Y holds. 2) α-fmvd-replication rule: If X →→α Y , then X →→α R−X −Y: Proof. Given X →→α Y . Then, from the definition of α-fmvd it follows that for any two tuples t1 and t2 with t1 [X]εα t2 [X], there exists a tuple t3 such that t1 [X]εα t2 [X]εα t3 [X], t1 [Y ]εα t3 [Y ], t2 [R − X − Y ]εα t3 [R − X − Y ].
(1) (2) (3)
MISHRA & GHOSH, A MULTIVALUED INTEGRITY CONSTRAINT IN FUZZY RELATIONAL DATABASE
Since R − (R − X − Y ) − X ⊆ Y and t1 [Y ]εα t3 [Y ] by (2), then from Proposition II.2 we can say that
Then, Y 0 ⊆ R − X − Z, so that from (6) using Proposition II.2, we get (9)
t1 [R − (R − X − Y ) − X]εα t3 [R − (R − X − Y ) − X]. (4) Then the α-fmvd-complementation rule follows from the relations (1), (3) and (4). 3) α-fmvd-inclusion rule: If X →→α1 Y and α1 ≥ α2 , then X →→α2 Y : This rule follows directly from Proposition II.1. 4) α-fmvd-union rule: If X →→α1 Y and X →→α2 Z, then X →→min(α1 ,α2 ) Y Z: Proof. Case-I: α1 ≥ α2 Given X →→α1 Y and X →→α2 Z. Now X →→α1 Y ⇒ X →→α2 Y by α-fmvd-inclusion rule. Further, from the definition of X →→α2 Y , we have (1) (2) (3)
t1 [X]εα2 t2 [X]εα2 t3 [X], t1 [Y ]εα2 t3 [Y ], t2 [R − X − Y ]εα2 t3 [R − X − Y ],
and then combining the relations (9) and (10), we get (11)
Then, from the relations (1) and (4) we have (A)
t1 [X]εα2 t2 [X]εα2 t3 [X]εα2 t4 [X].
Next, if X, Y and Z are disjoint sets, then Y ⊆ R − X − Z, so that from (6) using Proposition II.2 we may write (7)
t1 [X]εα2 t4 [X] and t1 [Z]εα2 t4 [Z] so that combining the above two relations, we may write (12)
t1 [XZ]εα2 t4 [XZ].
Now, since Y −Y 0 ⊆ XZ, so from (12) using Proposition II.2, we have (13) t1 [Y − Y 0 ]εα2 t4 [Y − Y 0 ]. Then combining the relations (11) and (13), we again get t1 [Y ]εα2 t4 [Y ]
as also obtained for disjoint sets. Then, from equations (5) and (8), we have (B)
t1 [Y Z]εα2 t4 [Y Z].
Again, since R−X −Y −Z ⊆ R−X −Y and R−X −Y −Z ⊆ R − X − Z, from (3) and (6) using Proposition II.2, we get respectively t2 [R − X − Y − Z]εα2 t3 [R − X − Y − Z] and t3 [R − X − Y − Z]εα2 t4 [R − X − Y − Z] so that we may write
t3 [Y ]εα2 t4 [Y ].
Now, combining the relations (2) and (7), we get (8)
t1 [Y 0 ]εα2 t4 [Y 0 ].
Next, from the relations (4) and (5), we have
(8)
t1 [X]εα2 t3 [X]εα2 t4 [X], t1 [Z]εα2 t4 [Z], t3 [R − X − Z]εα2 t4 [R − X − Z].
t3 [Y 0 ]εα2 t4 [Y 0 ].
Again, since Y 0 ⊆ Y , so from (2) using Proposition II.2, we have (10) t1 [Y 0 ]εα2 t3 [Y 0 ]
and from X →→α2 Z, we have (4) (5) (6)
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t1 [Y ]εα2 t4 [Y ].
However, if the sets X, Y and Z are non-disjoint, let us consider the set Y 0 as shown in Fig. 1 such that Y 0 = Y − [Y ∩ X] − [Y ∩ Z]. R
(C)
t2 [R − X − Y − Z]εα2 t4 [R − X − Y − Z].
Hence, using the relations (A), (B) and (C), we have for any two tuples t1 and t2 if t1 [X]εα2 t2 [X] then ∃ a tuple t4 for which t1 [X]εα2 t2 [X]εα2 t4 [X] t1 [Y Z]εα2 t4 [Y Z] t2 [R − X − Y − Z]εα2 t4 [R − X − Y − Z]. which implies X →→α2 Y Z. Case-II: Similarly, for the case α2 ≥ α1 , one can show that
X
if X →→α1 Y and X →→α2 Z, then X →→α1 Y Z. Hence, from the above two cases we may conclude that if X →→α1 Y and X →→α2 Z, then X →→min(α1 ,α2 ) Y Z. Y/ Y
Z
Fig. 1: Venn Diagram showing Y 0 when X, Y and Z are non-disjoint sets.
5) α-fmvd-reflexivity rule: If Y ⊆ X, then X →→α Y : Proof. Given Y ⊆ X. Then, from the f -Reflexive rule for α-ffd [1], [8], it follows that X − → Y . Again, from α-fmvdα replication rule, if X − → Y , then X →→α Y . Hence, the α α-fmvd-reflexivity rule is proved.
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6) α-fmvd-projectivity rule: If X →→α1 Y and X →→α2 Z, then X →→min(α1 ,α2 ) (Y −Z), X →→min(α1 ,α2 ) (Z −Y ) and X →→min(α1 ,α2 ) Z ∩ Y : Proof. Case-I: α1 ≥ α2 Let X →→α1 Y . By α-fmvd-inclusion rule this implies X →→α2 Y . Now, from the definition of X →→α2 Y , we have (1) (2) (3)
t1 [X]εα2 t2 [X]εα2 t3 [X], t1 [Y ]εα2 t3 [Y ], t2 [R − X − Y ]εα2 t3 [R − X − Y ].
Also, from X →→α2 Z, we have (4) (5) (6)
t1 [X]εα2 t3 [X]εα2 t4 [X], t1 [Z]εα2 t4 [Z], t3 [R − X − Z]εα2 t4 [R − X − Z].
Again, from α-fmvd-reflexivity rule we have X →→α2 X, which implies (7) (8)
t1 [X]εα2 t2 [X]εα2 t3 [X], t2 [R − X]εα2 t3 [R − X],
and (9) t1 [X]εα2 t3 [X]εα2 t4 [X], (10) t3 [R − X]εα2 t4 [R − X]. Now, from (1), we have (A1 )
t1 [X]εα2 t2 [X]εα2 t3 [X].
Since Z ∩ Y ⊆ Z, so from (5) using Proposition II.3, we get (C2 )
Also R − X − (Z ∩ Y ) ⊆ R − X, so from (10) using Proposition II.2, we get (C3 )
t3 [R − X − (Z ∩ Y )]εα2 t4 [R − X − (Z ∩ Y )].
Using relations (C1 ), (C2 ), (C3 ), we get X →→α2 (Z ∩ Y ) Hence, given X →→α1 Y and X →→α2 Z, we have proved X →→α2 (Y −Z), X →→α2 (Z −Y ) and X →→α2 (Z ∩Y ) where α1 ≥ α2 . Case-II: α2 ≥ α1 Similarly, given X →→α1 Y and X →→α2 Z, we can prove X →→α1 (Y −Z), X →→α1 (Z−Y ) and X →→α1 (Z∩Y ). Thus, combining the above two cases we can say, if X →→α1 Y and X →→α2 Z, then X →→min(α1 ,α2 ) (Y − Z), X →→min(α1 ,α2 ) (Z − Y ), and X →→min(α1 ,α2 ) Z ∩ Y . 7) α-fmvd-augmentation rule: If X →→α Y and W ⊆ U , then U X →→α Y W : Proof. We prove the augmentation rule by contradiction. Assume that X →→α Y and W ⊆ U , but U X →→α Y W does not hold for some relation instance r of R. Since U X →→α Y W does not hold in r, then from the definition of α-fmvd we can say that for any two tuples t1 and t2 if t1 [U X]εα t2 [U X], then there exists a tuple t3 such that (A)
t1 [Y − Z]εα2 t3 [Y − Z].
Also, since R − X − (Y − Z) ⊆ R − X, so from (8) using Proposition II.2, we get (A3 )
t1 [R − X − (Y − Z)]εα2 t3 [R − X − (Y − Z)].
(B)
(B1 )
(C)
t2 [R − U X − Y W ]εα t3 [R − U X − Y W ],
does not hold. From Definition II.8 the relation (A) can split into two relations as follows:
(B2 )
t1 [Z − Y ]εα2 t4 [Z − Y ].
Also R − X − (Z − Y ) ⊆ R − X, so from (10) using Proposition II.3, we get (B3 )
t3 [R − X − (Z − Y )]εα2 t4 [R − X − (Z − Y )].
Using relations (B1 ), (B2 ), (B3 ), we get X →→α2 (Z − Y ) Next, from relation (9), we have (C1 )
t1 [X]εα2 t3 [X]εα2 t4 [X].
(D)
t1 [X]εα t2 [X]εα t3 [X]
(E)
t1 [U ]εα t2 [U ]εα t3 [U ].
and
t1 [X]εα2 t3 [X]εα2 t4 [X].
Since Z − Y ⊆ Z, so from (5) using Proposition II.2, we get
t1 [Y W ]εα t3 [Y W ]
and
Using relations (A1 ), (A2 ), (A3 ), we get X →→α2 (Y − Z) Again from (4), we have
t1 [U X]εα t2 [U X]εα t3 [U X],
but
Since Y − Z ⊆ Y , so from (2) using Proposition II.2, we get (A2 )
t1 [Z ∩ Y ]εα2 t4 [Z ∩ Y ].
Now given X →→α Y , so that from the definition of α-fmvd we have (D) (1) (2)
t1 [X]εα t2 [X]εα t3 [X], t1 [Y ]εα t3 [Y ], t2 [R − X − Y ]εα t3 [R − X − Y ].
Again, by α-fmvd-reflexivity rule, if W ⊆ U →→α W . Hence, from the definition of α-fmvd we have (E) (3) (4)
t1 [U ]εα t2 [U ]εα t3 [U ], t1 [W ]εα t3 [W ], t2 [R − U − W ]εα t3 [R − U − W ].
U , then
MISHRA & GHOSH, A MULTIVALUED INTEGRITY CONSTRAINT IN FUZZY RELATIONAL DATABASE
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Now from relations (1) and (3) using Definition II.8, we have t1 [Y W ]εα t3 [Y W ] which is contradictory to our assumption. Thus, U X →→α Y W holds and the augmentation rule is proved.
Further, X−X 0 ⊆ Y Z, so that from (B) using Proposition II.2, we get
8) α-fmvd-transitivity rule: If X →→α1 Y →→α2 Z, then X →→min(α1 ,α2 ) Z − Y :
Then, from the above two relations (9) and (10), we have
Y
and
Proof. To prove the transitivity property, we first show that if X →→α1 Y and Y →→α2 Z, then X →→min(α1 ,α2 ) Y Z. Case-I: α1 ≥ α2 Given X →→α1 Y and Y →→α2 Z. Now X →→α1 Y ⇒ X →→α2 Y by α-fmvd-inclusion rule. Then, from the definition of X →→α2 Y , we have (1) (2) (3)
t1 [X]εα2 t2 [X]εα2 t3 [X], t1 [Y ]εα2 t3 [Y ], t2 [R − X − Y ]εα2 t3 [R − X − Y ].
and from Y →→α2 Z, we have (4) (5) (6)
t1 [Y ]εα2 t3 [Y ]εα2 t4 [Y ], t1 [Z]εα2 t4 [Z], t3 [R − Y − Z]εα2 t4 [R − Y − Z].
Now combining relations (4) and (5), we obtain (B)
t1 [Y Z]εα2 t4 [Y Z].
Next, if X, Y and Z are disjoint sets, then X ⊆ R − Y − Z, so that from (6) using Proposition II.2 we get (7)
t3 [X]εα2 t4 [X].
However, if the sets X, Y , Z are non-disjoint, let us consider a set X 0 as shown in Fig. 2 such that X 0 = X − [X ∩ Z] − [X ∩ Y ]. R Z
X/ X
Y
(10)
(11)
t1 [X − X 0 ]εα2 t4 [X − X 0 ]. t3 [X − X 0 ]εα2 t4 [X − X 0 ].
Finally, combining the relations (8) and (11), we may write (7)
t3 [X]εα2 t4 [X].
as already obtained for disjoint sets. Then, from (1) and (7), we have (A)
t1 [X]εα2 t2 [X]εα2 t3 [X]εα2 t4 [X].
Further, since R − X − Y − Z ⊆ R − X − Y as well as R − X − Y − Z ⊆ R − Y − Z, hence from (3) and (6) using Proposition II.2, we get respectively (8) and (9)
t2 [R − X − Y − Z]εα2 t3 [R − X − Y − Z] t3 [R − X − Y − Z]εα2 t4 [R − X − Y − Z].
Then, combining the above two relations we may write (C)
t2 [R − X − Y − Z]εα2 t4 [R − X − Y − Z].
Thus, using the relations (A), (B) and (C) we have for any two tuples t1 and t2 , if t1 [X]εα2 t2 [X], then there exists a tuple t4 for which t1 [X]εα2 t2 [X]εα2 t4 [X] t1 [Y Z]εα2 t4 [Y Z] t2 [R − X − Y − Z]εα2 t4 [R − X − Y − Z] which implies X →→α2 Y Z. Hence, X →→α1 Y and Y →→α2 Z implies X →→α2 Y Z where α1 ≥ α2 . Case-II: α2 ≥ α1 Similarly, for the case α2 ≥ α1 , one can show that if X →→α1 Y and Y →→α2 Z, then X →→α1 Y Z. Thus, combining the above two cases we may write, X →→α1 Y and Y →→α2 Z, then X →→min(α1 ,α2 ) Y Z. Next, applying the α-fmvd-projectivity rule we see that X →→α1 Y and X →→min(α1 ,α2 ) Y Z, then X →→min(α1 ,α2 ) Y Z − Y , i.e., X →→min(α1 ,α2 ) Z − Y [since, Z − Y = Y Z − Y ] Hence, the α-fmvd-transitivity rule follows. IV. C ONCLUSION
Fig. 2: Venn diagram showing X 0 when X, Y and Z are non-disjoint sets.
Then, X 0 ⊆ R − Y − Z, so that from (6) using Proposition II.2, we get (8)
t3 [X 0 ]εα2 t4 [X 0 ].
Again, since X − X 0 ⊆ X, so from (1) using Proposition II.2, we get (9)
t1 [X − X 0 ]εα2 t2 [X − X 0 ]εα2 t3 [X − X 0 ].
Integrity constraints play a vital role in the logical design of a relational database. Multivalued dependency (mvd) stands as a generalization of functional dependency and constitutes an important data integrity constraint. However, in many database designs it is necessary to consider integrity constraints of fuzzy nature. In this work, we have presented a new definition of fuzzy multivalued dependency (called α-fmvd) existing in fuzzy database relations with the idea of α-equality of tuples as introduced in [1]. It provides an easy and straightforward way of extension of mvd to fmvd and is different from the
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