A GEOMETRICAL FUZZY PARTITIONS APPROACH TO FUZZY QUERY AND FUZZY DATABASE 
RETRIE VAL. 
E. G. Mtalo and E. Derenyi 
Department of Surveying Engineering 
University of New Brunswick 
Fredericton, N.B. CANADA 
COMMISSION III 
ABSTRACT 
Real world knowledge is usually vague and ambiguous and human beings generally think and communicate in fuzzy non-precise 
terms. The fuzzy sets theory introduced by Zadeh in 1965 forms the mathematical and practical basis for the representation and 
manipulation of such fuzzy information. 
Conventional databases, however, must contain precisely defined facts or data because database query languages such as the SQL 
impose strict formats for data entry and query. Conventional query languages do not permit ambiguous or non-precise queries . 
This paper presents a new method for the representation of fuzzy numerical quantities in a way favorable to the storage and retrieval of 
fuzzy values or vague expressions in a database. 
Key Words: Fuzzy Sets, Fuzzy Retrieval, Geometric Partitions, GIS, Database. 
1. INTRODUCTION. 
Conventional databases must contain precisely defined facts and 
numeric values because database query languages such as the 
SQL impose strict formats for data entry and query. 
Conventional query languages do not permit ambiguous or 
non-precise queries. Real world data and knowledge is usually 
vague and ambiguous and human beings generally think 
vaguely and communicate in fuzzy non-precise terms (Zadeh et 
al, 1975; Zadeh, 1989). However vague or fuzzy real world 
knowledge does not lend itself to easy manipulation by 
conventional database management systems. 
The fuzzy sets theory introduced by Zadeh in 1965 (Zadeh et al. 
1975; Kandel, 1986) forms the mathematical and practical basis 
for the representation and manipulation of fuzzy information. 
1.1 Representation of Vague Information by Fuzzy 
Sets. 
Consider the statement that the distance from the Earth to the 
Sun is "very great", or the statement "100 is much greater than 
5". Using Zadeh's fuzzy sets theory, the terms "great" and 
"greater" may be regarded as fuzzy sets and can therefore be 
defined in terms of fuzzy membership functions. Once these 
fuzzy sets have been defined the modifiers, "very" and "much", 
can be applied to transform them into the corresponding fuzzy 
sets very great and much greater respectively(see Zadeh et al. 
1975; Kaufmann and Gupta, 1988; Shmucker, 1984; Kandel, 
1986; Zadeh, 1989). 
Typically the fuzzy set great will be represented by a fuzzy 
membership function Mgrea(X): X-2[0,1] and greater can be 
represented by the membership function 
Hgreater than(X): X->{0,1]. Generally MAX) denotes the 
membership function of the elements of the universe X in the 
fuzzy set A such that the elements, x, take value in the 
universe X. The membership values, HA(X) on the other hand 
take value in the evaluation space of the fuzzy set, generally 
considered to be the continuous interval [0,1] (Zadeh et al, 
1975; Dubois and Prade, 1980; Kandel, 1986). The 
membership value Ha(x), is a real number between O and 1 
inclusive, expressing the strength of the membership of an 
element, x, of the universe X in a the fuzzy set A. 
Practical specification of a membership function involves the 
assignment of the parameters of some standard membership 
function, such as the S-function and x-function (Dubois and 
Prade 1980; Kandel 1986; Klir and Folger 1988). Important 
characteristic points of standard membership functions include; 
the cut-out points (points with zero membership value), the 
peak point (where the membership function attains the 
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maximum value of 1) and the turnover points (points where the 
membership function has the value of 0.5). Dubois and Prade 
(1988, 1990) suggest that for most applications simpler piece- 
wise linear membership functions such as triangular functions 
and trapezoidal functions provide satisfactory results. 
The method proposed in this paper differs from the standard 
approach in that, fuzzy expressions such as "greater than five" 
are characterized by geometric partitions induced by them in the 
real numbers domain. The paper begins by defining the concept 
of fuzzy geometric partitions and then goes on to show how it 
can be applied to the problem of storing and querying fuzzy 
database objects. For the purposes of fuzzy database retrieval, 
the universe of discourse is some search space X containing 
crisp and fuzzy objects. To facilitate database retrieval each 
fuzzy query and fuzzy database object can be associated with a: 
fuzzy partition defined over the search space. The condition for 
a successful search is obtained when the partition induced by 
the query object contains the partition generated by the fuzzy 
database object. 
The paper also looks at the potential use of the fuzzy 
geometrical partitions method in the construction of membership 
functions, which may then be used to represent fuzzy numerical 
sets in the usual way(Zadeh ,1975, 1989). 
2. THE CONCEPT OF FUZZY GEOMETRIC 
PARTITIONS. 
The concept of fuzzy partitions is not new, what is new is, 
however, the manner in which this concept is used to 
characterize and represent fuzzy valued expressions or fuzzy 
numbers. An earlier use of the term radial partition to 
characterize fuzzy sets can be found in Kaufmann (1975). 
Kaufmann shows that a fuzzy set, M, induced by the binary 
relation y >> x , for y = kx, k > 1, in the two dimensional real 
space constitutes a radial partition of the real numbers space. 
In Dowsing et al.(1986) the concept of the diagonal set of the 
universe of discourse, is used to define and characterize the 
equality operator. This research extends and generalizes the 
concept of the diagonal set generated by the equality operator, 
defined in Dowsing et al.(1986), and uses it to define general 
fuzzy comparison operators or fuzzy binary relations, R(x,y), 
in terms of the radial sets or partitions induced by them in a two 
dimensional real space X. 
Intuitively the operators "equal", "greater", and "less" define 
basic geometric partitions of the two dimensional space (Figure 
1). The partition generated by the equality operator, =, is called 
the diagonal set (Dowsing et al, 1986) or diagonal partition. By 
extension the partition corresponding to the operator greater (>) 
is called the upper diagonal partition, while the partition of the