A new class of fuzzy logic systems
(FLS) --- type-2 fuzzy logic systems --- is introduced, one that
makes use of type-2 fuzzy sets for representing linguistic and/or
What are type-2 fuzzy sets? They are fuzzy sets having fuzzy
membership functions, i.e., the membership grade of each element
of such a set is an ordinary (type-1) fuzzy set. Type-2 sets
are useful in circumstances where it is difficult to define the
exact membership function for a fuzzy set, as in computing with
words, when words mean different things to different people.
The report includes the very basic operations of union, intersection
and complement of type-2 sets, and develops results that are
needed to implement a type-2 FLS, including: set theoretic and
algebraic operations for type-2 sets, properties of membership
grades of type-2 sets, and type-2 relations and compositions.
A new operation called type-reduction is introduced.
It is an extended version of type-1 defuzzification. Type-reduction
as well as defuzzification are examined in great detail. Results
are provided that greatly simplify the implementation of interval
and Gaussian type-2 FLS's. Whenever actual results are difficult
to implement or generalize, practical approximations are provided.
The report demonstrates the use of a type-2 FLS for two applications:
managing rules collected by means of a survey, and time-series
In the survey application, the report shows how linguistic
uncertainty about membership functions, as well as rule uncertainty
from multiple experts (each of whom may give different answers
to the same questions), can be handled in the type-2 framework.
Type-2 FLSs lets us combine expert opinions in a rational way.
In the time-series application, the report shows how information
about noise in the training data (i.e., unreliable training data)
can be incorporated into a type-2 FLS to obtain bounds on the
predictions as well as better predictions. The bounds are linguistic
2. Operations on Type-2 Sets
3. Properties of Membership Grades
4. Relations and Compositions
of Type-2 Fuzzy Logic
... Four Appendixes with detailed
...derivations and proofs