This and the next special issues of Reliable Computing contain a rich set of articles
that attest to the synergism between interval analysis and fuzzy set theory. The
relationships between interval analysis and fuzzy set theory are many. The most
obvious relationship is arithmetic since fuzzy arithmetic is interval arithmetic on
alpha-cuts (see , ). The second more basic relationship, since it is how interval
and fuzzy arithmetic can be derived as well as how functions in the interval and fuzzy
analysis setting (and many mathematical relationships) are deﬁned, is the extension
principle (see , , ). The extension principle in interval analysis is called
the united extension (see –). The third relationship is perhaps more explicit;
this is fuzzy interval analysis (see , ). Here, the use of interval analysis in
fuzzy set theory is direct.
Of course, from the beginning (, , ) an interval [a
b] was considered
as a number and this lead to interval analysis. Moveover, an interval is also a set.
As a set, an interval is also a fuzzy set with rectangular membership function
Therefore, from the point of view of intervals as sets, interval analysis can be
considered as a subset of fuzzy set theory, so the fuzzy membership representing
intervals can represent all that is known about the value of a variable. This leads to
the fourth type of relationship, an analysis of uncertainty. The interval analysis as
developed by R. E. Moore (see, for example, –) can be thought as arising from
a need to understand and model the uncertainty and error of the digital computer
as well as to develop tools for automatic error analysis of numerical solutions of
continuous problems. There was a recognition in the late nineteen-forties, after the
application of the computer in solving linear problems, that the error associated
with solving problems of continuous mathematics on the computer presented a new
challenge (see ). The uncertainty associated with the digital computers is one
domain of interval analysis. On the other hand, fuzzy set theory may be thought
as arising from the need of a more complete and inclusive mathematical model of
uncertainty (see , ).
The special issues have a wide range of articles that include all of the relation-
ships mentioned above. There are eight research papers presented in two volumes.
In the ﬁrst volume, we have three articles. The ﬁrst article presents a new way of
capturing uncertainty that extends some ideas of fuzzy sets and logic. The second