Remarks on a Fuzzy Approach to Flexible Database Querying, Its Extension and Relation to Data Mining and Summarization - Advanced Database Query Systems - page 127

tive, the OR connective is replaced by the corresponding max operator of fuzzy logic. Again, it may be
further replaced by a linguistic quantifier and importance coefficients may be assigned to subconditions.
Representation and Definitions of Linguistic Terms
Basically, all linguistic terms are represented as fuzzy sets. Obviously, the universe used to define
particular fuzzy set depends on the type of their corresponding linguistic term. Namely, in order to rep-
resent terms used along with the numeric fields (attributes) we use trapezoidal fuzzy numbers. It seems
obvious, that these fuzzy numbers are defined over the domains of corresponding fields. Namely, if the
price of the house ranges from, say, 10,000 up to 100,000 USD, then the fuzzy value low , to be used
in the atomic condition “price is low”, should be defined as a fuzzy number over the interval [10,000,
100,000]. However, then if we strictly stick to this scheme, we have to define low separately for each
field which we are going to use together with this linguistic term. In order to avoid this apparent incon-
venience, we propose to use, in parallel, “context-free” definitions of numeric linguistic terms. Namely,
the corresponding fuzzy numbers are defined over the universal interval, e.g., [-10, 10]. Additionally, for
each relevant field, its range of variability, the [LowerLimit, UpperLimit] interval, has to be provided
before a given linguistic term may be used along with this field in an atomic condition. Then, during
the calculation of a matching degree the definitions of relevant fuzzy numbers are translated into the
domains of corresponding fields.
The same approach is adopted for dealing with fuzzy relations . Basically, a fuzzy relation is a fuzzy
set defined in the universe of the Cartesian product of two domains. In the context considered here,
these are the domains of one or two numeric attributes involved. Obviously, it would be rather tedious
to define a fuzzy relation directly on this Cartesian product. Thus, in our approach as the universe for
the definition of a fuzzy relation we use the set of possible values of the difference of values of the two
attributes involved. More precisely, the same universal interval [-10,10] is used for the definition which
is then translated into the real interval of possible differences during the matching degree calculation.
The definitions again take the form of trapezoidal fuzzy numbers.
The idea of a universal domain is also applied for the definition of a linguistic quantifier. In fact, such
a universal domain in the form of the [0,1] interval is present in the original Zadeh's (1983) approach to
modeling a special class of so-called proportional linguistic quantifiers , which is most relevant for our
purposes. Particular numbers from the interval [0,1] are interpreted as expressing proportion of elements
possessing a property - in our case, this is a proportion of satisfied conditions.
In FQUERY for Access, a linguistic quantifier has to be first defined in the sense of Zadeh (1983).
Then, for the matching degree calculation two options exist. First, the quantifier's original form may be
employed directly. Second, an original definition may be automatically translated in order to obtain a
corresponding Yager's OWA operator (cf. Yager, 1988; Yager & Kacprzyk, 1997).
The linguistic terms meant for the use along with scalar attributes, i.e., fuzzy set constants , have to be
defined directly in the appropriate universe of discourse corresponding to the domain of a given scalar
attribute. More precisely, the set of given attribute values is taken as the universe of discourse. Thus,
these constants are, by definition, context dependent. Their discrete membership functions are defined
as arrays of numbers. Fuzzy set constants may be defined for any field of character (text) type. Fuzzy
set constants may also be defined for - and used together with - multi-valued attributes . Obviously, these
attributes have to be separately defined, according to the explanation given in the previous section. The
definition consists in indicating the list of fields which may then be treated jointly as a virtual attribute.