Functional Dependencies in Fuzzy Databases
Brian Hartlieb
This concept was called Fuzziness and the theory was called
Fuzzy Set Theory.
ABSTRACT
Integrity constraints play a critical role in a logical database
design in which data dependencies are of more interest. One of
the most important data dependencies is the functional
dependency in relational databases, representing the dependency
relationships among attribute values in a relation. We will
examine the two most popular frameworks for extending classical
Functional Dependencies into Fuzzy Functional Dependencies.
We will examine some of the extensions made to the classical
relational data model that attempt to tackle the challenges of
incorporating fuzzy logic. While the frameworks’ foundational
premises are different, the similarities are so strong, that a
unification model may emerge in the future.
Keywords
functional dependency, fuzzy functional dependency, relational
database, fuzzy database.
While there are many types of uncertainty, Fuzzy set theory
attempts to address only one aspect of uncertainty. It would seem
beneficial to database systems, if it could incorporate fuzzy logic.
Databases that use crisp and non-crisp data can beneficial from
use of a query language that uses Fuzzy logic. A branch of fuzzy
set theory is the use of “hedging” adjectives. A query language
that provides the use of these adjectives will provide a more
natural language for the user, than traditional SQL. It could be
said that purpose of a fuzzy database is to provide an intelligent
interface to a relational database system by facilitating
approximate query handling and producing approximate answers
when exact data may not be unavailable. [3] As an example
consider a student record database system. We want to find bright
and young students in the whole batch. For a crisp system we
would specify the query as
PROJECT (Student_Name)
WHERE 19 ≤ AGE ≤ 23 and 3 ≤ GPA ≤ 4
1. INTRODUCTION
This paper deals with the application of fuzzy logic in a relational
database environment with the objective of showing how the
definition of classical functional dependency (FD) has been
extended in to a fuzzy functional dependency (FFD).
The paper is organized as follows. In Section 2, the paper deals
with some of the basic definitions and concepts of fuzzy logic. In
Sections 3, we introduce the two major classes of fuzzy relational
data models. In Section 4 and section 5, the two main approaches
of fuzzy functional dependency frameworks are described.
Finally, some concluding remarks.
2. BACKGROUND
Databases are one form of modeling the real world. A world that
is imprecise and vague. Database models can be either precise or
imprecise. Query languages are designed to express the user’s
retrieval requests in either a crisp manner or not. Most database
models today are crisp. A crisp database model is one that is
highly quantifiable - all relationships are fixed and all attributes
have one value. The classical approach to uncertainty in databases
is to reduce retrieval to 3-value logic. Each database object is
surely, maybe, or surely-not, in response to a query.
Several extensions have been brought to the relational data model
to capture the imprecise parts of the real world. In general there
are two approaches. The approaches differ mainly in the method
they use, while still being based on the work of a single man,
Lotfi A. Zadeh, father of fuzzy logic. The first approach is the
similarity-based approach, first introduced by Zadeh in1970. [1]
The second approach was later introduced by Zadeh in 1978 [2]
Zadeh explained how a possibility distribution can be used in
conjunction with his earlier work of fuzzy sets. Zadeh’s initial
work introduced a theory whose objects fuzzy sets are sets with
boundaries that are not precise and the membership in this fuzzy
set is not a matter of true or false, but rather a matter of degree.
But this system has a major flaw. Consider a student, Bob whose
age is 24 and has a good GPA of 4 out of 4. He should have been
selected but is not. It is because of the rigid boundary conditions
set by the crisp logic of the query. In fuzzy logic, we would
specify two fuzzy sets, YOUNG (fig.1a) and GPA (fig.1b), and
each student will have some membership grade associated with
the two sets. So according Bob will have a non–zero membership
grade although it will be less than other students in the age group
19-23. Hence Bob will now be included in the result set to be
considered. Bob now satisfies the query to some extent, which is
represented by his membership grade. [4]
1
0
1
18
19
23
24
0
3
3.5
4
Figure 1. (a) Young (b) GPA
2.1 Fuzzy Definitions
When A is a fuzzy set and x is a relevant object, the proposition
“x is a member of A” is not necessarily either true or false, as
required by the two-valued logic, but it may be true only to some
degree, the degree to which x is actually a member of A, is a real
number in the interval [0, 1]. Theoretically, if X is a collection of
objects denoted generically by x, then a fuzzy set F in X is a set
of ordered pairs,
F = {(x, µF(x)) | x ε X}
µF(x) is called the membership function (or grade of membership)
of x in F that maps X to the membership space M. The range of
the membership function is a subset of the nonnegative real
numbers whose supremum is finite.
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2.2 Fuzzy Set Operators and Fuzzy Logic
1
For crisp sets, the basic operations are, namely,
Union, OR
Intersection, AND
Complement, NOT
0
Fuzzy sets have defined fuzzy operators that allow the
manipulation of the fuzzy sets. There are fuzzy complements,
intersection and union operators but they are not uniquely
defined. However there is an important distinction between
traditional set logic and fuzzy set theory. In traditional set theory
there is a distinction between the union operation of sets and OR
operator, was well as in the case of intersection and AND
operator. But in fuzzy theory there is no such distinction between
the logical and set operators;
Fuzzy union ≡ Fuzzy OR
Fuzzy intersection ≡ Fuzzy AND
Fuzzy complement ≡ Fuzzy NOT
d
There are four parameters associated with a linguistic term as a,
b, c and d as shown in the Fig. 3. For the range [b,c] the
membership value is 1.0, while for the range [a, b] and [c, d] the
membership value remains between [0.0, 1.0].
This example, introduces fuzzy sets and linguistic terms on the
attribute domains and linguistic variables (e.g. on the attribute
domain AGE we may define fuzzy sets as YOUNG, MIDDLE
and OLD). [3]
YOUNG
0 a Y bY
MIDDLE
OLD
c Y , a M dY , bM
c M , a O bO , dM
cO
dO
Figure 4. Age
Data can be classified as crisp, when there is no vagueness in the
information (e.g., X = 13). With Fuzzy data, there is vagueness in
the information and this can be further divided into two types as:
(1) Approximate Value: The information data is not totally
vague and there is some approximate value, which is known
and the data, lies near that value (e.g., 10 < X < 15). These
are considered have a triangular shaped possibility
distribution as shown below
1
X
c
1
Fuzzy Complement,
~A(x) = 1 - A(x)
Fuzzy Union,
(A∪B)(x) = max[A(x), B(x)].
Fuzzy Intersection,
(A∩B)(x) = min[A(x), B(x)].
-d
b
Figure 3. Possibility Distribution for a Linguistic Term
SMALL for the Linguistic Variable HEIGHT
Some standard fuzzy operations are:
0
a
d
Figure 2. Possibility Distribution for an approximate
value. (Approximately X)
The parameter, d gives the range around which the
information value lies.
(2) Linguistic Variable: A linguistic variable is a variable
that apart from representing a fuzzy number also represents
linguistic concepts interpreted in a particular context. Each
linguistic variable is defined in terms of a variable which
either has a physical interpretation (speed, weight etc.) or
any other numerical variable (salary, absences, GPA etc.)
The information in this case is totally vague and we associate a
fuzzy set with the information. A linguistic term is the name
given to the fuzzy set (e.g., X is SMALL). These are considered
have a trapezoidal shaped possibility distribution as shown below
3. APPROCH AND METHODOLOGY
There have been many extensions developed for fuzzy relational
data model. These extensions can be classified into two
categories: The similarity-based and the possibility-based models.
In a similarity-based model, some similarity relationships are
specified for some attributes so that values of these attributes may
be grouped into similarity classes. Each similarity class contains
values that are similar to each other to, and above a given degree.
Thus they are indistinct, and form an uncertain representation of a
real-world value. In a possibility-based model, an ill-known data
is represented by a possibility distribution which describes the
possibility for each crisp attribute value to be the actual value of
the data. In both types of models, membership degrees may be
associated with tuples of a fuzzy relation.
Integrity constraints play a critical role in a logical database
design. Among these constraints, data dependencies are of most
interest. Various types of data dependencies such as functional
and multivalued dependencies are used for the design of classical
relational schema that are conceptually meaningful and free of
certain anomalies. For example, if one attribute determines
another, we say that there exists a functional dependency between
these attributes. Functional dependencies in databases relate the
values of one set of attributes to the values of another set. Fuzzy
functional dependencies can represent the dependency
relationships among attribute values in fuzzy relations, such as
“the salary almost depends on the job position and experience.”
[5]
We will review the two main approaches used at extending
relational functional dependencies into fuzzy functional
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dependencies. These two approaches are the similarity-based
approach, and possibility-based approach.
4. THE POSSIBLIITY-BASED APPROACH
In a relational data model that can support imprecise information,
it is necessary to accommodate two types of impreciseness, the
impreciseness in data values and impreciseness in the association
among data values. As an example of impreciseness in data
values, consider the Employee(Name, Salary) database, where
Salary of an employee, John, may be known to the extent that it
lies in the range $60,000-80,000, or may be known that John has a
“high salary.” Similarly, as an example of impreciseness in the
association among data values, let Likes(Student, Course)
represent how much a student likes a particular course. Here the
data values may be precisely known, but the degree to which a
student, John, likes the course DBMS, is imprecise. It is not
difficult to envision examples where both ambiguity in data
values as well as impreciseness in the association among them are
both present. [6]
Fuzzy data is represented by possibility distributions and a grade
of membership is used to represent the association between
values. Also this grade of membership may itself be a possibility
distribution. Fuzzy similarity relations facilitate the estimation of
the extent to which possible values of an attribute can be regarded
as being interchangeable. By introducing an extra element, e, for
the situations where a nonzero possibility can mean the nonapplicability of an attribute. The traditional null value no longer
has to mean that an attribute is completely unknown. It has also
been proposed that possibility distributions be used to represent
fuzzy values as well as uncertainty, when concerning the value of
an attribute. [5]
Depending on the complexity of dom(Ai), i = 1, . . . , n, we
classify fuzzy relations into two categories. In type-l, fuzzy
relations, dom(Ai) can only be a fuzzy set (or a classical set). A
type-l fuzzy relation may be considered as a first-level extension
of classical relations, where we will be able to capture the
impreciseness in the association among entities. The type-2 fuzzy
relations provide further generalization by allowing dom(Ai) to be
even a set of fuzzy sets (or possibility distributions). By enlarging
dom(Ai), type-2 relations enable us to represent a wider type of
impreciseness in data values. Such relations can be considered as
a second-level generalization of classical relations. For example,
Type-2 allows a domain that hold both numerical and linguistic
values; we may define the domain AGE as positive integers, and
functions for YOUNG, MIDDLE and OLD. The three functions
will convert the values of YOUNG, MIDDLE and OLD to
numerical values. [6]
4.1 Fuzzy Integrity Constraints
The integrity constraints in relational database systems can be
broadly classified into two groups:
(1) Domain dependency: Domain dependency restricts
admissible domain values of the attributes, e.g., “age of an
employee is less than 65 years,” or “no one is 10 feet tall.”
(2) Data dependency: Data dependency requires that if
some tuples in the database fulfill certain equalities, then
either some other tuples must also exist in the database, or
some values of the given tuples must be equal. [6]
As we generalize relational database systems to deal with fuzzy, it
will be necessary to consider integrity constraints that involve
fuzzy constructs. Thus in a relation PLAYERS(Name, Age,
Height, Sport, Income), an integrity constraint may be stated as,
“Most basketball players are tall,” or “Many baseball players have
high income.” These integrity constraints impose restrictions on
the admissible values of height or income of the basketball or
tennis players, respectively. Similarly, as an example of a fuzzy
data
dependency,
consider
the
relation
scheme
EMPLOYEE(Name, Department, Job, Experience, Salary), where
an integrity constraint may be stated as “in any department
employees having similar jobs and experience must have almost
equal salary.” [6]
4.2 Fuzzy Functional Dependencies
In the fuzzy domain, equality of domain values defines a fuzzy
proposition and may even be specified as “approximately equal,”
“more or less equal,” etc. For instance, a fuzzy data dependency
in the relation EMPLOYEE(Name, Job, Experience, Salary) can
be stated as “Job and Experience more or less determines Salary.”
[6]
There are two families of approaches where possibility-based
attribute values are compared for “equality”. The classification is
based on the nature of comparison between two ill-known values.
In one approach the comparison is made in terms of
representations (i.e., the result is a degree to which the two
underlying fuzzy sets are equal). One interpretation of this FFD
is: "when two tuples have the same value (or representation) on
X, they should have the same value (or representation) on Y".
This FFD may contain tuples whose X-representations share some
more or less possible values while they do not share a single value
in the Y-representations.
In the other approach the comparison is made in terms of values
(i.e., its result is degree of possibility of the equality between two
ill-known values.) The interpretation of this FFD is: "when two
tuples have the same value (or representation) on X, they should
have the same value (or representation) on Y”. This FFD uses a
critical threshold value, which is undecided and remains arbitrary
to the design. [7]
4.3 Inference Rules
An important concept related to data dependencies is the concept
of inference rules. Given a set of dependencies, inference rules
introduce other dependencies that are logical consequences of the
given dependencies. These rules are dependency generators and
so they are closely related to the definition and semantics of the
dependencies. Given a set of data dependencies that hold on a
database, it is often possible to derive other data dependencies
that also hold on the same database. An important point to make
for the inference rules is that they can only be useful if the
dependencies they generate form a sound and complete set. By
sound, we mean that the generated dependency is valid in all
relation instances provided the given set of inferences is also
valid. By complete, we mean that all of the valid dependencies
can be generated using only these rules. Thus, when defining the
dependencies and their inference rules, it is crucial that the
dependencies are well defined in terms of definition and
semantics, and their inference rules are sound and complete.
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In particular, we focus on the establishment of inference rules for
the following reasons:
(1) By the inference rules, we sometimes get simpler
functional dependencies than the original ones. It is
convenient to use simpler functional dependencies whenever
possible in order to infer actual values of unknown values.
(2) We can obtain minimal sets of functional dependency by
using inference rules. In integrity checking, if we can use the
minimal sets of functional dependencies, we are free from
excessive evaluations.
(3) When we obtain functional dependencies by data mining
or knowledge discovery, we can get another functional
dependency by using the inference rules.
The inference rules for classical FDs are Armstrong’s 3-value
logic axioms which are sound and complete. [8]
Armstrong’s inference rules [9]
Al. Reflexivity
5. THE SIMILARITY-BASED APPROACH
The similarity-based fuzzy relational model is not an extension to
the original relational model, but actually a generalization of it. It
allows a set of values for an attribute rather than only atomic
values, and replaces the identity concept with a similarity
concept.
The similarity-based relational model allows a set of values for a
single attribute providing that all the values are from the same
domain. The model, allows multiple values, while keeping the
property of a strongly typed attribute value present in the classical
relational model. This property is useful for query processing and
Update operations. If the attribute value is precise and crisp, then
the value is atomic, if it is imprecise and inexact, then a set of
values that are similar to this value are stated in place of it. The
level of similarity among the values is defined by the explicitly
defined similarity relation for the domain of the attribute values.
Similarity relations are useful for describing how similar two
elements from the same domain are. A similarity relation, s(x,y),
for a given domain D, is a mapping of every pair of elements in
the domain onto the unit interval [0,1]. The identity relation used
in non-fuzzy relational databases Identity relation is a special case
of this similarity relation. [5]
If Y ⊆ X, then X → Y.
A2. Augmentation
If X → Y, then X → XY.
A3. Transitivity
If X → Y and Y → Z, then X → Z.
As augmentation, “If X →Y, X Z →YZ” is sometimes used, which
is deduced from A1-A3. Some useful inference rules deduced
from Armstrong’s inference rules
D1. Union
If X →Y and X → Z, then X → Y Z.
Similarity relations are useful for describing how similar two
elements from the same domain are, as the name implies. Given
two elements, the similarity relation maps these two elements into
an element in the interval [0, 1]. The more similar two elements
are, the higher the value of the mapped element. If the two
elements are the same, that is, if we compare an element with
itself, the mapped element is 1, the highest possible value. An
ordinary relation is considered to be a similarity relation when it
satisfies the three conditions stated below.
Definition 5.1. A similarity relation is a mapping s: D X D → [0,
1] such that for x, y, z∈D,
D2. Decomposition
If X → Y Z, then X → Y and X → Z.
Fuzzy inference rules [6]
By extending Armstrong’s rules to a multivalued logic system.
FFl. Reflexivity
s(x, x) =1 (reflexivity)
s(x, y) = s(y, x) (symmetry)
s(x, z) ≥ max y∈D (min s(x, y) , s(y, z))) (max-min
transitivity)
Example 5.1 For a domain D, we have D= s{a, b, c, d}. We
define a relationship s for domain D, such that;
If Y ⊆ X, then X →FY.
FF2. Augmentation
If X →F Y, then XZ →F YZ.
FF3. Transitivity
If X →F Y and Y →F Z, then X →F Z.
FF4. Union
If X →FY and X →F Z, then X →F Y Z.
s
a
b
c
d
a
1
0
8
0
b
0
8
1
0
c
0
0
1
0.7
d
0
0
0.7
1
Relation s satisfies the three conditions stated in Definition 5.1.
Thus, it is a similarity relation.
FF5. Decomposition
If X →F Y Z, then X →F Y and X →F Z.
FF6. Generalized augmentation
If X →F Y and X ⊆ U and V ⊆ X Y, then U →F V.
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Example 5.2. The equivalence classes induced by s in Example
2.1 are;
[1, 0.8):
[0.8, 0.7):
[0.7, 0):
0:
{a}, {b}, {c}, {d}.
{a, b}, {c}, {d}.
{a, b}, {c, d}.
{a, b, c, d}
Example 5.3. This is the instance of fuzzy relation car and the
similarity relations of its attribute domains. [5]
Type
Color
price
t1
{sportscar}
{blue, green}
{expensive}
t2
{wagon}
{red}
{modest, affordable}
t3
{truck}
{blue}
{modest, affordable}
t4
{wagon}
{green}
{expensive}
TYPE
S
W
T
Sportscar (S)
1
0
0
Wagon (W)
0
1
0
Truck (T)
0
0
1
COLOR
B
G
R
Blue (B)
1
0.7
0
Green(G)
0.7
1
0
Red (R)
0
0
1
PRICE
C
A
M
A
Cheap (C)
1
0.3
0.3
0
Modest (M)
0.3
1
0.8
0
Affordable(A)
0.3
0.8
1
0
Expensive (E)
0
0
0
1
5.1 Fuzzy Functional Dependencies
Fuzzy functional dependencies reflect some kind of semantic
knowledge about attribute subsets of the real world. FFDs are
used to design similarity-based fuzzy databases where data
redundancy and update anomalies are reduced. In a fuzzy
relational data model, the degree of “X determines Y” may not
necessarily be 1 as in the crisp case. Naturally, a value ranging
over the interval [0, 1] may be accepted. Then the definition of
FFD turns into “similar Y values correspond to similar X values.”
[5]
FFDs are functional constraints that are specified among the
attributes of a fuzzy relation schema. The similarity-based
relational model compares two attributes by measuring the
closeness of the values in terms of the explicitly declared
similarity relation of the attribute domain. The degree of
closeness between two tuples in a fuzzy relation instance is called
the conformance of them. The conformance is defined both on a
single attribute and on a set of attributes.
For precise FFDs, the similarity of Y values has to be greater than
or equal to the similarity of X values, where similarity is
measured in terms of conformance. For imprecise FFDs, the
impreciseness of the dependency is a threshold on the similarity
of Y values, thus weakening the dependency. [5] The FFDs should
also be checked whenever tuples are inserted into the fuzzy
relational database or they are modified, so that the integrity
constraints imposed by these FFDs are not violated. [8] The
definition of the FFD turns into: “if t[X] is similar to t’[X], t[Y] is
also similar to t’[Y]. The similarity between Y values is greater or
equal to the similarity between X values.” This dependency is
shown as X→F Y. A typical example of such a dependency is:
‘‘employees with similar experiences must have similar salaries.’’
In this case, while the values of attributes experience and salary
may be imprecise, the defined dependency is precise which can
be noticed from the ‘‘must have’’ clause in the example. This
definition of FFDs still has some missing point, which is the case
where the dependency itself is imprecise. An example to this kind
of FFD is: ‘‘the intelligence level of a person more or less
determines the degree of success,’’ where the ‘‘more or less’’
clause makes the dependency imprecise. Assume that there are
two people with identical intelligence levels, and the first person
is very successful. We cannot conclude that the second person
will be very successful, too, but we can state that the success level
of the second person will be ‘‘more or less similar’’ to the success
level of the first person, so a change in definition has to be made
in order to accommodate the imprecise FFDs in addition to
precise FFDs. One way to do this is to accept the linguistic
strength in the dependency as a threshold value for example, the
dependency ‘‘employees with similar experiences must have
similar salaries’’ has linguistic strength 1, and the dependency
‘‘the intelligence level of a person more or less determines the
degree of success’’ has linguistic strength (0.6). We choose this
method to describe imprecise FFDs as well as precise ones. This
threshold value naturally determines the strength of the
dependency. Thus, this value will be θ called the strength of the
FFD, shown as X→ θF Y. [6]
5.2 Inference Rules
For the similarity-based fuzzy model, we must examine inference
rules under two interpretations of functional dependencies. Under
the interpretation corresponding to using Godel’s multivalued
logic system, Armstrong’s inference rules are sound and complete
for any functional dependency with no weights, and the extended
inference rules of Armstrong’s ones are sound and complete for
any functional dependency with weights. On the other hand,
under the interpretation corresponding to using Diens’
multivalued logic system, Armstrong’s inference rules are sound
and complete for functional dependencies with identity relations
and no weights, and the extended inference rules of Armstrong’s
ones are sound and complete for functional dependencies with
identity relations and weights. However, Armstrong’s inference
rules and their extended inference rules are not sound for
functional dependencies with resemblance relations and no
weights and with resemblance relations and weights, respectively.
In these cases, another sound inference rules hold. (Armstrong,
Diens, and Godel, where mathematicians who worked separately
on 3-value and/or multivalued logic systems.) [9]
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6. CONCLUSION
Like the classical databases, the fuzzy databases not properly
designed suffer from the problems of data redundancy and update
anomalies. To provide a good fuzzy relational database design,
the concept of FFD is used to define the fuzzy normal forms and
dependency-preserving and lossless join properties. In this article,
we reviewed the two leading frameworks for Fuzzy relational
databases, the similarity-based approach, and possibility-based
approach. Zadeh’s fuzzy logic models are the bases for both
approaches. While the possibility-based model came later, work
has continued on both approaches by a number of researchers.
Both approaches have a number of parallels, and similarities. As
far as Fuzzy functional dependencies are defined in both
approaches, most of the problems have been addressed. There
seems to be no major advantage to using one approach over the
other. It is clear that, based on these basic fuzzy relational
models; there maybe one type of extended fuzzy relational model,
where possibility distributions and resemblance relations arise in
relational databases simultaneously. In the Future that is a good
possibility that a single universal Fuzzy relational database model
be proposed.
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