Abstract
In this paper, an attempt is made to study approximate reasoning based on a Type-2 fuzzy set theory. In the process, we have examined the underlying fuzzy logic structure on which the reasoning is formulated. We have seen that the partial/incomplete/imprecise truth-values of elements of a type-2 fuzzy set under consideration forms a lattice. We propose two new lattice operations which ultimately help us to define a residual and thereby making the structure of truth- values a residuated lattice. We have focused upon two typical rules of inference used mostly in ordinary approximate reasoning methodology based on Type-1 fuzzy set theory. Our proposal is illustrated with typical artificial examples.
Highlights
In 1965, the concept of a fuzzy set was introduced by Zadeh[16] and it has already established its usefulness through successful applications in different fields
In dealing with vagueness/impreciseness using fuzzy set theory we come across situations, where it is difficult to find satisfactorily the degree of membership of an element of the universal set in a particular fuzzy subset
In a research work[5], the authors established a small set of terms that allowed us communicate with Type-2 fuzzy sets and helped us define such sets rather precisely
Summary
In 1965, the concept of a fuzzy set was introduced by Zadeh[16] and it has already established its usefulness through successful applications in different fields. Mendel and Qilian Liang [4] introduced a Type-2 fuzzy logic system (FLS), which can handle rule uncertainties The implementation of this Type-2 FLS involves the operations of fuzzification, inference, and output processing which, consists of type reduction and defuzzification. In a research work[5], the authors established a small set of terms that allowed us communicate with Type-2 fuzzy sets and helped us define such sets rather precisely There they presented a new representation for Type-2 fuzzy sets, and used it to derive formulas for union, intersection and complement of Type-2 fuzzy sets without having the use of extension principle. This is followed by a list of references in the last section
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