Abstract. This paper concerns to fuzzy dependencies of attributes on fuzzy
object class. Base on the possibility distribution approach, we proposes the
formal definition of fuzzy functional and multivalued dependencies on object
class. The dependencies in this study allows a sound and complete set of
inference rules.

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JOURNAL OF SCIENCE OF HNUE
Natural Sci., 2011, Vol. 56, No. 7, pp. 23-31
FUZZY FUNCTION DEPENDENCIES
IN FUZZY OBJECT-ORIENTED DATABASES
Ho Cam Ha(∗)
Hanoi National University of Education
Vu Duc Quang
Quang Nam University
(∗)E-mail: hahc@hnue.edu.vn
Abstract. This paper concerns to fuzzy dependencies of attributes on fuzzy
object class. Base on the possibility distribution approach, we proposes the
formal definition of fuzzy functional and multivalued dependencies on object
class. The dependencies in this study allows a sound and complete set of
inference rules.
Keywords: object-oriented database, fuzzy dependencies, fuzzy ob-
ject, possibility distribution
1. Introduction
Not all the elements (data) that occur in the real world are fully known or
defined in a perfect way. Classical relational database models and its extension of
imprecision and uncertainty do not satisfy the need of modelling complex objects
with imprecision and uncertainty. So many researchers have concentrated on the
development of object-oriented database models to deal with uncertain data [9]. The
fuzzy object-oriented approach have recently received increasing attention. Integrity
constraints play a crucial role as in traditional databases, and also in fuzzy object-
orientation. These constraints are used to define the semantics and guidelines for
database design.
Based on the possibility distribution approach and resemblance relationship
we introduce a measurement of semantic equivalence of fuzzy data. In this paper, we
propose formal definitions of dependencies in the fuzzy class. The formal definition
fuzzy functional and multivalued dependencies in this study allows a sound and
complete set of inference rules. It is also shown that, such extended dependencies,
the classical dependency is a special case.
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Ho Cam Ha and Vu Duc Quang
The remainder of the paper is organized as follows: In the next section we
recall basic notions from fuzzy set and resemblance relationships. Then we define
fuzzy functional dependency, fuzzy multi-valued dependency on object class. These
formal definitions of dependencies allows a sound and complete set of inference rules.
Finally, we present some concluding remarks.
2. Content
2.1. Representations Fuzzy set and Possibility Distribution
As pointed out [10], fuzzy data is originally described as fuzzy set. Let U be a
universe of discourse, a fuzzy set F is defined in U by a membership function. The
membership function µF : U → [0, 1] assigns to each element u of U a number, in
the close interval [0,1], that characterizes the degree of membership of u in F . A
fuzzy set F can be written as:
F = {µF (u1)/u1, µF (u2)/u2, ..., µF (un)/un} (U is a finite set)
or F =
∫
u∈U
µF (u)
u
(U is a infinite set).
When µF (u) is viewed to be a measure of the possibility that a variable X has
the value u in this approach, where X takes values in U , a fuzzy value is described
by a possibility distribution πX .
πX = {πX(u1)/u1, πX(u2)/u2, ..., πX(un)/un},
where πX(ui), ui ∈ U denotes the possibility that ui is true. Let πX , F be the
possibility distribution representation and the fuzzy set representation for a fuzzy
value, respectively. It is apparent that πX = F is true.
2.2. Semantic relationship between fuzzy data
Let U = u1, u2, , un be the universe of discourse, A is a variable, a fuzzy data
of A on U based on possibility distribution is denoted as follows:
πA = {πA(u1)/u1, πA(u2)/u2, ..., πA(un)/un}, ui ∈ U, πA(ui) ∈ [0, 1].
Definition 2.1 (11). For fuzzy data, its semantics correspond to an area in space
(U× [0, 1]), where the universe of discourse is its X-axis and possibility is its Y -axis.
The semantics of a fuzzy data πA expressed by possibility distribution corre-
sponds to an area in space, the so called semantic space, denoted by SS(πA) (see
Figure 1).
The semantic relationship between two fuzzy data can be described by the
relationship between their semantic spaces. Let SS(πA) and SS(πB) be semantic
24
Fuzzy function dependencies in fuzzy object-oriented databases
Figure 1. Semantic space of fuzzy data.
spaces of two fuzzy data πA and πB, respectively. If SS(πA) ⊇ SS(πB), πA seman-
tically includes πB or πB is semantically included by πA. If SS(πA) ⊇ SS(πB) and
SS(πA) ⊆ SS(πB), πA, πB are semantically equivalent to each other. We employ
semantic inclusion degree to measure the semantic relationship of fuzzy data.
Definition 2.2 (11). Let πA and πB be two fuzzy data, and their semantic spaces
be SS(πA) and SS(πB), respectively. Let SID (πA, πB) denotes the degree that πA
semantically includes πB. Then
SID(πA, πB) = (SS(πA) ∩ SS(πB))/SS(πB).
For two fuzzy data πA and πB, the meaning of SID(πA, πB) is the percentage
of the semantic space of πB which is wholly included in the semantic space of πA.
Definition 2.3 (11). Let πA and πB be two fuzzy data, SE(πA, πB) denote the
degree that πA and πB are equivalent to each other, then
SE(πA, πB) = min(SID(πA, πB), SID(πB, πA)).
2.3. Evaluation of Semantic Measures
Definition 2.4 (11). Let U = {u1, u2, , un} be the universe of discourse. Let πA
v πB be two fuzzy datas on U based on possibility distribution and πA(ui), ui ∈ U
denote the possibility that ui is true. SID(πA, πA) is then defined
SID(πA, πB) =
n∑
i=1
min
ui∈U
(πB(ui), πA(ui))/
n∑
i=1
πB(ui).
Definition 2.5 (11). Let U = u1, u2, , un be the universe of discourse. Let πA v πB
be two fuzzy data on U based on possibility distribution and πA(ui), ui ∈ U denote
25
Ho Cam Ha and Vu Duc Quang
the possibility that ui is true. Let Res be a resemblance relation on domain U , α for
0 ≤ α ≤ 1 be a threshold corresponding to Res. SIDα(πA, πB) is then defined by
SIDα(πA, πB) =
n∑
i=1
min
ui,uj∈U ;Res(ui,uj)≥α
(πB(ui), πA(uj))/
n∑
i=1
πB(ui).
In there, resemblance relationship Res on domain U is mapping U×U → [0, 1]
which satisfies two terms as follows:
(1) ∀u ∈ U,Res(u, u) = 1;
(2) ∀ui, uj ∈ U,Res(ui, uj) = Res(uj, ui).
2.4. Fuzzy functional dependencies and fuzzy multi-valued
dependencies
Assumption:
OID is a set of object-identities.
C is a set of classes.
A is a set of names of attributes.
nil is used to denote a special constant, that presents uncertainty value.
dom includes all crisp atom values and fuzzy values.
Fuzzy Object: A fuzzy object is a pair of (id, v), in there, id is the object
identifier taken from the OID, v is a fuzzy (or clearly) value on the OID. The OID
values is signed as val(OID) and defined as follows [1]:
(a) an element of dom, an element of IOD and the nil values on OID
(b) if v1, v2, , vn are values and A1, A2, ..., An are names of attributes, the tuple
value is (A1 : v1, A2 : v2, ..., An : vn) and the set value is {v1, v2, , vn} on OID.
Fuzzy class [11]: The objects having the same properties are gathered into
classes. A class is fuzzy because of the following several reasons. First, some objects
are fuzzy ones, which have similar properties. A class defined by these objects may
be fuzzy. These objects belong to the class with membership degree of [0,1]. Second,
when a class is defined, the domain of an attribute may be fuzzy and a fuzzy class
is formed. Third, the subclass produced by a fuzzy class by means of specialization
and the superclass produced by some classes (in which there is at least one class
which is fuzzy) by means of generalization are also fuzzy.
Types(C) is defined as follows:
(1) Base type: integer, string, bool, float and the fuzzy type is generated base
on the base types.
26
Fuzzy function dependencies in fuzzy object-oriented databases
(2) Class name in C is fuzzy (clearly) type object type.
(3) If T is a fuzzy type, set(T) is fuzzy (clearly) set type.
(4) If T1, T2, , Tn are types and A1, A2, ..., An are names of attributes in A, the
tuple type is (A1 : T1, A2 : T2, ..., An : Tn).
2.4.1. Fuzzy Functional Dependencies
Definition 2.6. Let c be a class. Suppose that U is set of attributes of c and
X, Y ⊆ U . A fuzzy functional dependency X fc−→ Y is said to hold in class c if for
every pairs of objects O1, O2 of class c, we have SE(O1.X,O2.X) ≤ SE(O1.Y, O2.Y ).
Note that: The equivalent measurement of semantics is defined as follows:
(i) If X = A1, A2, ..., Ak, then
SE(O1.X,O2.X) =min(SE(O1.A1, O2.A1), SE(O1.A2, O2.A2), ..., SE(O1.Ak, O2.Ak)).
(ii) If Ai refers to class ci(Ui), type of Ai is object type, then
SE(O1.X,O2.X) = 1 when O1.X ≡ O2.X ,
E(O1.Ai, O2.Ai) = SE(O1.Ai.Ui, O2.Ai.Ui); otherwise.
(iii) If type of Ai is set type, O1.Ai = {s1, s2, s3, ..., si, ..., sn}, O2.A =
{s1, s2, s3, ..., si, ..., sm}, then
SE(O1.Ai, O2.Ai) = min(min(maxi(SID(sj, si)
i=1..n,j=1..m
),min(maxj(SID(si, sj)
j=1..m,i=1..n
)).
(iv) If type of A is tuple type, O1.Ai = (v1, v2, v3, ..., vi, ..., vn),
O2.Ai = {w1, w2, w3, ..., wj, ..., wn}, then SE(O1.Ai, O2.Ai) = min(SE(vi, wj
i,j=1..n
)).
The equivalent measurement of semantic on Y and on X is defined similarly.
Example 1: Given c(U), where c is class c and U is set of attributes of c.X, Y ⊆
U , Res(X) and Res(Y ) are similarity related on X and Y , with α1 = 0.9, α2 = 0.95
are threshold of Res(X) and Res(Y ) respectively.
Res(X) a b c d e
a 1.0 0.2 0.3 0.2 0.4
b 1.0 0.92 0.4 0.1
c 1.0 0.1 0.3
d 1.0 0.2
e 1.0
Res(Y) f g h i j
f 1.0 0.3 0.2 0.96 0.2
g 1.0 0.4 0.2 0.3
h 1.0 0.3 0.1
i 1.0 0.4
j 1.0
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Ho Cam Ha and Vu Duc Quang
Instances of object class is shown as follows:
Id X Y ...
Id1 {0.7/a, 0.4/b, 0.5/d} {0.9/f, 0.6/g, 1.0/h} ...
Id2 {0.5/a, 0.4/c, 0.8/d} {0.6/g, 0.9/h, 0.9/i} ...
Id3 {0.3/d, 0.8/e} {0.6/h, 0.4/i, 0.1/j} ...
For every objects O1, O2 of c, we have:
SE(O1.X,O2.X) = min(SID(O1.X,O2.X), SID(O2.X,O1.X))
= min(0.824, 0.875) = 0.824 v
SE(O1.Y, O2.Y ) = min(SID(O1.Y, O2.Y ), SID(O2.Y, O1.Y ))
= min(1.0, 0.96) = 0.96,
It is easy to realize that SE(O1.Y, O2.Y ) > SE(O1.X,O2.X) from the definition of
fuzzy functional dependency X
fc−→ Y .
Lemma 2.1. Let c(U) is a class with set of attributes U . Suppose X ⊆ U,A ∈ U . If
X
fc−→ A holds and A: tuple(A1 : T1, A2 : T2, ..., Ak : Tk) then X fc−→ Ai, i = 1, 2, ..., k.
Proof. X
fc−→ A implies that SE(O1.A, O2.A) ≥ SE(O1.X,O2.X), for every O1, O2
in fuzzy object class. A1, A2, ..., Ak takes the place of A, so that SE(O1.A, O2.A) =
min(SE(O1.Ai, O2.Ai)), i = 1..k. Obviously SE(O1.Ai, O2.Ai) ≥ SE(O1.X,O2.X),
i = 1..k. Thus, according to definition of fuzzy functional dependency, we have
X
fc−→ Ai, i = 1, 2, ..., k.
Theorem 2.1. The functional dependency in relational database is a special case
of the fuzzy functional dependency in object oriented database (given above).
Proof. Suppose X → Y holds in relation r(U), X, Y ⊆ U . It means: for every pairs
of tuple t1, t2 ∈ r, if t1[X ] = t2[X ] then t1[Y ] = t2[Y ]. If consider U as the set of
attributes of class c, for every pairs object of c(O1, O2), we have SE(O1.X,O2.X) =
1, SE(O1.Y, O2.Y ) = 1. Thus, SE(O1.Y, O2.Y ) > SE(O1.X,O2.Y ) > 1. It means
X
fc−→ Y holds.
2.4.2. Decomposing tuple type attribute to fuzzy function dependencies
The group of some attributes in a class into the other one which has tuple
type. This can lead to binding semantics or fuzzy functional dependencies between
attributes are not preserved. In this case, we need to decompose these attributes to
preserve the semantic constraints.
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Fuzzy function dependencies in fuzzy object-oriented databases
Suppose A∗: tuple(Xi : Ti), i = 1, 2, 3, ..., k(k > 1). If Xi takes the place of A∗
and X
fc−→ Y holds, where X = k∪
i=1
X ′i, X
′
i ⊆ Xi, then A∗ can be decomposed into
attributes as follows:
Bi : tuple(Xi\X ′i : TXi\X′i), i = 1, 2, 3, ..., k − 1.
Bk : tuple(Xk\X ′k\Y : TXk\X′k\Y ), i = 1, 2, 3, ..., k − 1.
Ci : tuple(X
′
i : TX′i), i = 1, 2, 3, ..., k.
D : tuple(Y : TY ).
2.4.3. Fuzzy Multi-valued Dependencies
Definition 2.7. Let c be a class. Suppose that U is set of attributes of c. Suppose
also X, Y ⊆ U , Z = U\XY . A fuzzy multivalued dependency X fc→→Y is said to
be held in class c if whenever O1, O2 are two objects of class c, contains object O3 so
that SE(O3.X,O1.X) > SE(O1.X,O2.X) and SE(O3.Y, O1.Y ) > SE(O1.X,O2.X),
SE(O3.Z, O2.Z) > SE(O1.X,O2.X)
Example 2: Given class c(U), X, Y ⊆ U,Z = U\XY . Res(X), Res(Y ) and
Res(Z) are three similarity relations of X , Y and Z respectively, as in Example 1.
α1 = 0.9, α2 = 0.95, α3 = 0.90 are three thresholds of Res(X) and Res(Y ), Res(Z)
respectively.
Instances of object class is shown as follows:
Id X Y Z
O1 Id1
{0.4/a, 0.6/b,
0.7/d} {0.6/g, 0.9/h, 0.8/i} {0.5/a, 0.7/c, 0.4/e}
O2 Id2 {0.4/c, 0.5/d, 0.2/e} {0.3/h, 0.6/i, 1.0/j} {0.2/b, 0.5/c, 0.9/e, 0.8/f}
O3 Id3
{0.4/a, 0.5/b,
0.6/d} {0.6/g, 0.7/h, 0.8/f} {0.6/d, 1.0/e, 0.7/f}
SE(O1.X,O2.X) = min(SID(O1.X,O2.X), SID(O2.X,O1.X))
= min(0.818, 0.529) = 0.529
SE(O1.X,O3.X) = min(SID(O1.X,O3.X), SID(O3.X,O1.X))
= min(1.0, 0.882) = 0.882 > SE(O1.X,O2.X)
SE(O1.Y, O3.Y ) = min(SID(O1.Y, O3.Y ), SID(O3.Y, O1.Y ))
= min(1.0, 0.913) = 0.913 > SE(O1.X,O2.X)
SE(O2.Z, O3.Z) = min(SID(O2.Z, O3.Z), SID(O3.Z, O2.Z))
= min(0.913, 0.875) = 0.875 > SE(O1.X,O2.X)
29
Ho Cam Ha and Vu Duc Quang
Thus, according to the definition of fuzzy multivalued dependency, X
fc→→Y holds
in fuzzy object class c.
Theorem 2.2. The multi-valued dependency in relational database is the special case
of the fuzzy multivalued dependency in object oriented database (as given above).
Proof. Suppose X →→ Y holds in relation r(U). It means ∀t1, t2, t1[X ] = t2[X ], ∃t3
such that t3[X ] = t1[X ], t3[Y ] = t1[Y ], t3[Z] = t2[Z], where Z = U −XY . Consider
U be as the set of attributes of class c. For every pair of objects of c(O1, O3), exits
object O3 ∈ c we have SE(O3.X,O1.X) = SE(O3.Y, O1.Y ) = 1, SE(O3.Z, O2.Z) =
SE(O1.X,O2.X) = 1.
Obviously, SE(O3.X,O1.X) > SE(O1.X,O2.X) > 1,
SE(O3.Y, O1.Y ) > SE(O1.X,O2.X) > 1
and SE(O3.Z, O2.Z) > SE(O1.X,O2.X) > 1. Thus, X
fc→→Y holds.
2.4.4. The inference rules of fuzzy dependencies
The inference rules of fuzzy functional dependencies:
Rule 1: Reflexivity. If X ⊇ Y then X fc→Y
Rule 2: Augmentation. If X
fc→Y then XZ fc→Y Z
Rule 3: Transitivity. If X
fc→ Y and Y fc→Z then X fc→Z
Rule 4: Union. If X
fc→Y , X fc→Z then X fc→Y Z
Rule 5: Pseudo transitivity. If X
fc→Y , Y W fc→Z then XW fc→Z
Rule 6: Decomposition. If X
fc→Y , Z ⊆ Y thenX fc→Z
The inference rules of fuzzy multivalued dependencies:
Rule 7: Reflexivity. If Y ⊆ X then X fc→→Y
Rule 8: Complementation. If X
fc→→Y then X fc→→U −XY
Rule 9: Augmentation. If X
fc→→Y and Z ⊆ W then XW fc→→Y Z
Rule 10: Transitivity. If X
fc→→Y and Y fc→→Z then X fc→→Z − Y
Rule 11: Union. If X
fc→→Y and X fc→→Z then X fc→→ZY
Rule 12: Pseudo transitivity. If X
fc→→Y and Y W fc→→Z then
XW
fc→→Z − Y W
Rule 13: If X
fc→→Y and X fc→→Z then X fc→→Z ∩ Y .
Rule 14: If X
fc→→Y and X fc→→Z then X fc→→Y −Z and X fc→→Z−Y .
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Fuzzy function dependencies in fuzzy object-oriented databases
Theorem 2.3. The inference rules (rule 1- rule 14) in fuzzy class are sound and
complete.
We suppose that procedure of proof for the soundness and completeness of
above inference rules is similar to the case of fuzzy relational database [7, 11].
3. Conclusion
When we design an object-oriented database schema, we need to eliminate data
redundancy. The reason is that as a consequence of the redundancy, the integrity
constraints are broken. It is the same in classical database, the fuzzy dependencies
are conceptually meaningful, which are integrity constrains in fuzzy object-oriented
database design. This paper describes an on going work. In order to continue, we
have already begun some studies of normalization in OODB design.
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