They are used to query to fuzzy databases that are enhanced
from relational databases in a way that fuzzy sets are allowed in
both attribute values and truth values. A fuzzy calculus query
language is constructed based on the relational calculus and
a fuzzy algebra query language is also constructed based on
the relational algebra. In addition, this paper proves a fuzzy
relational completeness theorem such that both the languages
have equivalent expressive power to each other.
Index Terms-Fuzzy database, query languages, relational algebra,
relational calculus, relational completeness.
I. INTRODUCTION
DA TABASE technology has been advanced up to the relational database stage with the purpose that user
interfaces with databases may approach a level of human
interfaces. It is recognized that the fuzzy theory is suitably
applied to some human-oriented engineering fields, one
FUZZY database
..._________________------.---
Select Name where Name I Age I Tuple truth
Age i "vepy young", iiiiiiiiii=j*iiiiiiiiii..=iiii
__....--___> __To. _I_ Youns _I_ t_r u e_ ...----
(---.-...-----------...----- Mary I 30 I 0.7
Tom: A r e = " ~ o u n g ' . Bob I mBddle I nearly 0 . 5
John Inearlr 401 quite true
Ron I old I 0.3
..____________________________
Fig. 1. A query to a fuzzy database.
11. A FUZZY DATABASE MODEL
A fuzzy database is defined as an enhanced relational
database that allows fuzzy attribute values and fuzzy truth
values; both of these are expressed as fuzzy sets. An example
of the fuzzy database is shown in Fig. 1.
of which is information processing, in particular database
retrieval. In fact, fuzzy database models that allow fuzzy
attribute values and fuzzy truth values in enhanced relational
databases have been studied in [3] and [4]. However, these
studies are restricted to just some particular applications and
not grounded on theories of fuzzy database query languages.
Thus fuzzy database systems would not be systematically
developed on the basis of these studies; it is due to Codd's
relational database theory that relational database systems have
been systematically developed. It is desirable that theoretical
foundations of fuzzy databases be established in order to
systematically develop fuzzy database systems.
fuzzy database theory; it develops a theoretical foundation for
the fuzzy functional dependencies of fuzzy databases [I]. The
work encourages further research for the rest of theoretical
foundations of fuzzy databases. This paper thus aims to
databases. It proposes two fuzzy database query languages:
a fuzzy calculus query language and a fuzzy algebra query
language. In addition, it proves a relational completeness theorem
such that both the languages are equivalent in expressive
power to each other. With these theoretical foundations, fuzzy Truth Of any tup1es are either (= true) Or (=
database query systems will be developed systematically.
A. Data
A fuzzy database consists of relations: a relation is a relation
R(tl, ,tn) in a Cartesian product PI x PZ x ... x P, of
domains Pi; each P, is a set of fuzzy sets t, over an attribute
domain D, (1 5 i 5 n). It is assumed that key attributes
take ordinary nonfuzzy values. For the notational convenience,
fuzzy sets are identified with their representative membership
functions; for example, t; also denotes a membership function.
B. Fuzzy Attribute Values
Attribute values such as age have nonfuzzy values such as
fuzzy predicates such as ccyoung9a9n d <Gabout forty" in the
fuzzy database. For example, a fuzzy attribute value of "age
of Dr. x is is expressed as a possibility distribution
p (age of x) = YOUNG; YOUNG denotes a fuzzy set that
are identified with fuzzy sets such as YOUNG.
c. Fuzzy
in the relational database; truth values of any tuples are defined
as fuzzy predicates such as "0.7" and "completely true" in the
fuzzy database. Consider, for example, a tuple t that asserts
a fuzzy proposition: "It is completely true that Dr. x iS Very
much older than twenty." The truth value of t is expressed as
a possibility distribution P[T(t)]= N; T( t )d enotes a truth
value o f t and N denotes a fuzzy set that represents the fuzzy
an work has been done in the Of the 20 in the relational database; attribute values are defined as
a foundation of query languages to represents the fuzzy predicate "young." Thus attribute values
Manuscript received November 21, 1989; revised September 6, 1991 and
The author is with NlT Network Information Systems Laboratories,
IEEE Log Number 9205829.
May 29, 1992.
Kanagawa, Japan.
1041-4347/93$03.00 0 1993 IEEE
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TAKAHASHI: FUZZY DATABASE QUERY LANGUAGES 123
predicate "completely true." Thus the truth values T(t) are
identified with fuzzy sets such as N over z E [0,1]; the value
z E [0,1] has the following meaning.
1) z = 0 means that the tuple t is completely false.
2) 0 < z < 1 means that the tuple t is true to the degree
3) z = 1 means that the tuple t is completely true.
In particular, each tuple t of the relation R(t1, . . . , tn) is
given a unique truth value T(t) by system designers at system
generation time. In this case, T(t) determines a mapping
T :P I x PZ x . . . x Pn + P([O1, 1) where P([O,1 1) is a set
of fuzzy sets over z E [O, 11.
expressed by the real number z.
111. QUERY BY TUPLE FUZZY CALCULUS
A. Tuple Fuzzy Calculus
A tuple fuzzy calculus (query language) is constructed as
an enhancement of the tuple relational calculus. Formulas in
the tuple fuzzy calculus are of the form ( t l f ( t ) )t: i s a fuzzy
tuple variable each ith component ti, which is a fuzzy set in
P;; f is a tuple fuzzy well-formed formula (WFF).
Tuple fuzzy WFF's are enhanced from those of the tuple
relational calculus as follows.
1) Atomic Tuple Fuzzy WFF's: An atomic tuple fuzzy
WFF consists of fuzzy sets and a fuzzy comparison operator *.
The fuzzy comparison operator * is one of the operators: equal;
not equal; proper inclusion; inclusion. The fuzzy comparison
operator * is an enhancement from the arithmetic comparison
operator (=, #, <, >, 5,z) in the relational calculus. Then
the atomic tuple fuzzy WFF's are either of the following two
types:
1) (t;)* ( s j ) ; here, it is assumed that t and s are fuzzy tuple
variables such that D; = Dj (1
2 ) (ti)* (c), (c) * (ti);h ere, it is assumed that c is a fuzzy
set over D;.
2) Logical Connectives and Quantifiers: Logical connectives
("AND," "OR," and "NOT") are used for tuple fuzzy
WFF's.
Also, quantifiers ("for all" and "there exists") are used for
tuple fuzzy WFF's.
3) Others: Other definitions concerning tuple fuzzy WFF's
are the same as in the tuple relational calculus.
Thus tuples in any relation R(tl,. . . , tn) that satisfy the
formula { t l f ( t ) }fo rm a set of Cartesian products of fuzzy
sets.
It should be considered further whether or not to include
fuzzy comparison operators * expressed by fuzzy relations
such as "much greater than," "is close to," "is similar to," and
"is relevant to."
i , j 5 n).
B. Query Evaluation
Queries expressed in the tuple fuzzy calculus are evaluated
by two steps as follows.
(Step 1) Selecting resultant tuples: Consider that the query
{tlf(t)} is issued to the relation R(t1,. . . , tn). Resultant
tuples are those r E R(t1, + . . , tn) each of which satisfies
the formula f(r).
(Step 2) Calculating truth values of resultant tuples: Let
any resultant tuple r be expressed as rkl . . . T k j ' . . rk, and
r be a projection of t E R(t1,. . . , tn) onto the components
ICl,...,ICj,...,km (1 5 m 5 n , l 5 k l , . . . , k j , . . . , k , 5
n). Then the truth value T(r) is defined as a projection of T(t)
onto the components k l , . . . , k j , . . . IC,: T(r) = Max .T(t),
where the maximum is taken over those components tk (1 5
Duplicate removal schemes are out of the scope of this paper
and left for future work: if two tuples T I , r2 having different
truth values T( r l ) ,T (r2)a re found to be duplicated, it is left
up to fuzzy database designers which one will be selected. The
fuzzy database designers will also choose which tuples from
the resultant tuples r should be returned to the users:
1) full sets or appropriate subsets of resultant tuples r should
be returned;
2) tuples r that contain truth values T(r) should be returned;
or when users need not make use of truth values T ( r ) ,tu ples r ,
from which truth values T(r) are removed, should be returned.
k 5 n), such that tk # tkj.
IV. QUERY BY DOMAIN FUZZY CALCULUS
A domain fuzzy calculus (query language) is obtained from
1) replacement of tuple variables t with domain variables,
2) replacement of the ith tuple component ti with a domain
the tuple fuzzy calculus through the following replacements:
211212.. . ,Un;
variable ui (1 5 i 5 n).
V. QUERY BY FUZZY ALGEBRA
A. Fuzzy Algebra
A fuzzy algebra (query language) is constructed as an
enhancement of the relational algebra. Fundamental fuzzy algebraic
operations are union, set difference, Cartesian product,
projection, and selection, which are defined as follows.
I ) Union: Let R and S denote any relations in the fuzzy
database. The union of R and S is a set of tuples that belongs
to R or S. The union is equal to that in set theory.
Any resultant tuple t by the union of R and S inherits the
truth value T(t) from its original tuple in R or S.
2) Set Difference: The difference R - S of R from S is a
set of tuples, each of which belongs to R and does not belong
to S. The difference is equal to that in set theory.
Any resultant tuple t by the set difference R - S inherits
the truth value T(t) from its original tuple in R.
3) Cartesian Product: The Cartesian product R x S of R
and S is a set of tuples, {(r, s ) l r : tuple in R, s: tuple in S}.
The Cartesian product is equal to that in set theory.
The truth value T(t) of the resultant tuple t = (rl s) by the
Cartesian product R x S is the minimum of T(r) and T(s)
where T(r) and T(s) are truth values of r and s, respectively.
4) Projection: The projection Proj(IC1, . . . , kj, . . . , k,)(R)
of R onto the lcjth attributes is a set of tuples of the lcjth
attribute values. The projection is equal to that in set theory.
Let r denote any resultant tuple of the projection
Proj(i1,i z,. . . , im) (R)o f t E R. Then the truth value T( r )i s
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-
124 IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING, VOL. 5, NO. 1, FEBRUARY 1993
the maximum of T(t) taken over those components tk, such
5) Selection: Let G denote a fuzzy WFF involving the
i) operands that are constant fuzzy sets and attribute item
ii) the fuzzy set comparison operators * (equal, not equal,
iii) logical connectives "OR," "AND," and "NOT."
The selection SelG(R) of the relation R is a set of tuples
t in R each of which satisfies the fuzzy WFF G when any
occurrences of the number i in G are replaced with the ith
component of T in R.
When any resultant tuple T is made by the selection
SelG(R)t, E R inherits the truth value T( t )f rom the original
tuple t in R: T(T) = T(t).
Some additional fuzzy algebraic operations such as intersection,
quotient, &join, and natural join are defined as
combinations of the fundamental fuzzy algebraic operations
defined previously in the same way as in the relational algebra.
For example, the &join and the natural join are defined as
follows.
6) 0-Join: The &join of R and 5' is defined as a combination
of two fundamental fuzzy algebraic operations: the
Cartesian product and the selection where 8 is enhanced
to a fuzzy comparison operator * (equal, not equal, proper
inclusion, inclusion). Truth values of resultant tuples by the
&join are calculated as those of combinations of the two
fundamental fuzzy algebraic operations.
7) Natural Join: The natural join of R and S is defined as a
combination of three fundamental fuzzy algebraic operations:
the Cartesian product, the selection, and the projection. Truth
values of resultant tuples by the natural join are calculated as
those of combinations of the three fundamental fuzzy algebraic
operations.
that tk # tkj.
following constituents:
numbers of the relation R;
proper inclusion, inclusion);
B. Que? Evaluation
Any query by the fuzzy algebra is expressed as a combination
of the fundamental fuzzy algebraic operations. Thus the
resultant tuples T and their truth values T ( T )by this query are
obtained as combinations of its constituent fundamental fuzzy
algebraic operations.
Duplicate removal schemes and return methods of resultant
tuples to users are the same as described in the fuzzy calculus.
VI. RELATIONACLO MPLETENESTSH EOREM
FOR FUZZY DATABASE QUERY LANGUAGES
The relational database theory establishes the relational
completeness theorem such that the relational calculus is
equivalent in expressive power to the relational algebra [2].
A similar theorem in the fuzzy database is given.
Theorem: The following three fuzzy database query languages
have the same expressive power:
1) tuple fuzzy calculus;
2) domain fuzzy calculus;
3) fuzzy algebra.
Proof: The fundamental idea of the proof of this theorem
is given by Ullman [2, pp. 114-1221; it presents the proof of
the relational completeness theorem for the relational database
query languages. Ullman's proof techniques consist of the
following three reduction techniques:
i) reduction of the relational algebra to the tuple relational
calculus;
ii) reduction of the tuple relational calculus to the domain
relational calculus;
iii) reduction of the domain relational calculus to the relational
algebra.
The reduction technique ii) is just the transformation between
variable expressions, and thus is not influenced by the
enhancements of the fuzzy database query languages. Therefore,
it should be proved here that the reduction techniques i)
and iii) can also be extended to cover the enhancements of the
fuzzy database query languages.
There are two essential enhancements in the fuzzy database
query languages from the relational database.
1) The fuzzy database allows fuzzy sets as attribute values;
the fuzzy comparison operators * (equal, not equal, proper
inclusion, inclusion) are used in the fuzzy database query languages
instead of the arithmetic comparison operators (=, #
, <, > ,z2,) used in the relational database query languages.
2) The fuzzy database allows fuzzy sets as truth values
T(t),t E R; truth values T(r) of resultant tuples T are
inherited from T(t) of original tuples t E R, or calculated
as combinations of Cartesian products or projections of T(t)
of original tuples t E R.
The enhancement 1) is easily incorporated into the reduction
techniques i) and iii) by replacing the arithmetic comparison
operators with the fuzzy comparison operators.
Next, consider the enhancement 2). Remember that the truth
value T(t) of the tuple t is defined just depending on the fuzzy
set and fuzzy set operations that have been established in fuzzy
theory; calculations of T(r) are made independently of any of
the definitions of the fuzzy database query languages. Thus the
reduction techniques i) and iii) can be extended to incorporate
the enhancement 2). This completes the proof. QED
VII. CONCLUDINGR EMARKS
This paper proposes two fuzzy database query languages
(fuzzy relational calculus and fuzzy relational algebra)
based on the relational database query languages. In
addition, it proves the relational completeness theorem
such that both the languages are equivalent in expressive
power to each other. As in the case of the relational
database, this relational completeness theorem in the fuzzy
database is expected to provide a criterion for the
minimum fuzzy database query capability that must be
implemented in any reasonable real fuzzy database query
languages.
There are interesting further theoretical studies still left.
More complicated fuzzy queries, including more general fuzzy
comparison operators such as "much greater than," and "is
close to", need to be studied. Such queries include, for
example, a statement "select several persons where their age is
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TAKAHASHI: FUZZY DATABASE QUERY LANGUAGES 125
a little over than that of v,o ung bovs." Other studies need to be 131 M. Umano. "Relational algebra in fuzzy database," IEICE Tech. Rep. U , . >
devoted to duplicate removal schemes and query optimization cih Japanese) v01r86, no. 192>-PP. 1986.
[4] M. Zemankova-Leech and A. Kandel, "Fuzzy relational databases-A techniques to improve execution efficiency of the fuzzy query key to expert systems," Verlag TUV Rheinland GmbH, 1984. I . -
languages; both of these are completely out of the scope of
this paper though these are essential to the fuzzy database.
Practically, there also should be an interesting further study
how to implement the fuzzy database query languages in this
paper by extending the existing real relational database query
languages, such as the international standard database language
SQL.
REFERENCES
[l] K. V. S. V. N. Raju and A. K. Majumdar, "Fuzzy functional dependencies
and lossless join decomposition of fuzzy relational database
systems," ACM Trans. Database Sysr., vol. 13, no. 2, pp. 129-166, June
1 OR!?
Yoshikane Takahashi received the M.Sc. degree in
mathematics from the University of Tokyo, Tokyo,
Japan in 1975.
He is currently with NlT Network Information
Systems Laboratories, Kanagawa, Japan. His
research fields include communications protocol,
fuzzy theory, neural networks, nonmonotonic logic,
genetic algorithms, and knowledge information theory.
Mr. Takahashi was awarded the Moto-oka Commemorative
Award in 1986. He is a member of the
[Z] J. D. Ullman, Principles of Database Systems. Rockville, MD: Com- Japanese Institute of Electronics, Information, and Communication Engineers,
puter Science, 1980. and the Information Processing Society of Japan.
Authorized
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