Chapter 1
Ecological Applications of Fuzzy Logic
A. Sa1ski
1.1
Fuzzy Sets and Fuzzy Logic
The Fuzzy Set Theory developed by L. Zadeh (Zadeh 1965) as a possible way to
handle uncertainty is particularly useful for the representation of vague expert
knowledge and processing uncertain or imprecise information. The Fuzzy Set
Theory is based on an extension of the cIassical meaning of the term "set" and
formulates specific logical and arithmetical operations for processing information
defined in the form of fuzzy sets and fuzzy mies.
The theory of fuzzy sets deals with sub sets of a given uni verse, where the
transition between fuIl membership and no membership is gradual. Therefore the
boundaries of fuzzy sets are not sharp. An example of a fuzzy set is the set A of aIl
large carps as a sub set of aIl carps in Lake Belau (Salski and Kandzia 1996).
TraditionaIly, the grade of membership 1 is assigned to those objects of the
uni verse that fuIly belong to a set, while 0 is assigned to objects that do not belong
to the set. In traditional set theory, the sets considered are defined as coIIections of
objects having some property, for example the property "carp in Lake Belau". The
property "Iarge carp in Lake Belau" does not constitute a set in the usual sense, the
property does not offer a precisely defined criterion of membership. Intuitively, a
fuzzy set is a coIlection of objects that admits the possibility of partial
membership in it. Thus a fuzzy set A in a given uni verse is characterized by a
function J-lA (x) termed "the grade of membership of x in A". We shaIl assume
that the values of J-l A (x) are elements of the interval [0,1] , with the grades 1 and
o representing fuIl membership and non-membership, respectively. J-l A (x) is
caIled the membership function of A.
Fuzzy logic is based on the extension of the mies of conventional logic. This
extension enables us to process fuzzy mies in the "IP - THEN" form with fuzzy
sets in the premise and conclusion parts of these mies. These fuzzy sets represent
imprecise express ions used by experts to describe their knowledge. Therefore
fuzzy inference methods are particularly useful to work with such a vague
knowledge representation. The main difference to conventional methods is that the
Fuzzy Set Theory offers inference methods for the calculation of the conclusion
values of mies when the premises of these mies are not completely fulfiIled.
Ecological Applications of Fuzzy Logic
A. Sa1ski
1.1
Fuzzy Sets and Fuzzy Logic
The Fuzzy Set Theory developed by L. Zadeh (Zadeh 1965) as a possible way to
handle uncertainty is particularly useful for the representation of vague expert
knowledge and processing uncertain or imprecise information. The Fuzzy Set
Theory is based on an extension of the cIassical meaning of the term "set" and
formulates specific logical and arithmetical operations for processing information
defined in the form of fuzzy sets and fuzzy mies.
The theory of fuzzy sets deals with sub sets of a given uni verse, where the
transition between fuIl membership and no membership is gradual. Therefore the
boundaries of fuzzy sets are not sharp. An example of a fuzzy set is the set A of aIl
large carps as a sub set of aIl carps in Lake Belau (Salski and Kandzia 1996).
TraditionaIly, the grade of membership 1 is assigned to those objects of the
uni verse that fuIly belong to a set, while 0 is assigned to objects that do not belong
to the set. In traditional set theory, the sets considered are defined as coIIections of
objects having some property, for example the property "carp in Lake Belau". The
property "Iarge carp in Lake Belau" does not constitute a set in the usual sense, the
property does not offer a precisely defined criterion of membership. Intuitively, a
fuzzy set is a coIlection of objects that admits the possibility of partial
membership in it. Thus a fuzzy set A in a given uni verse is characterized by a
function J-lA (x) termed "the grade of membership of x in A". We shaIl assume
that the values of J-l A (x) are elements of the interval [0,1] , with the grades 1 and
o representing fuIl membership and non-membership, respectively. J-l A (x) is
caIled the membership function of A.
Fuzzy logic is based on the extension of the mies of conventional logic. This
extension enables us to process fuzzy mies in the "IP - THEN" form with fuzzy
sets in the premise and conclusion parts of these mies. These fuzzy sets represent
imprecise express ions used by experts to describe their knowledge. Therefore
fuzzy inference methods are particularly useful to work with such a vague
knowledge representation. The main difference to conventional methods is that the
Fuzzy Set Theory offers inference methods for the calculation of the conclusion
values of mies when the premises of these mies are not completely fulfiIled.