Sources, Measurements and Models of Type 2 Fuzziness in the New Millennium

Abstract

There are two main sources of Type 2 fuzziness: (1) the acquisition of membership functions for linguistic values of linguistic variables, and (2) the combination of linguistic values with linguistic operators. First, it has been shown that acquisition of membership functions, whether (1) they are obtained by subjective measurement experiments, such as direct or reverse rating procedures or else (2) they are obtained with the application of fuzzy clustering methods, they all reveal a scatter plot, which ought to be captured with Type 2 membership functions. Type 1 membership functions are a result of reducing the information content and uncertainty embedded within scatter points via curve fitting or averaging techniques, which ignore the spread and hence, the content and uncertainty embedded within scatter points. Type 2 fuzziness can be represented either with interval-valued Type 2 or with “full” Type 2 membership functions, which specify gradations between the upper and lower bounds of the interval of the spread. Secondly, it has been shown that the combination of linguistic values with linguistic operators, “AND”, “OR”, “IMP”, etc., as opposed to crisp connectives that are known as t-norms and t-conorms and standard negation, lead to the generation of Fuzzy Disjunctive and Conjunctive Canonical Forms, FDCF and FCCF, respectively. It is noted that in current literature most authors start with Type 1 membership functions and end up with Type 1 membership functions in knowledge representation and approximate reasoning. This approach naturally ignores the spread of gradation both in knowledge representation and in approximate reasoning. Some researchers start out with Type 1 membership representation, but uses FDCF and FCCF and end up with interval-valued Type 2 reasoning results. More recently, a few researchers finally began to capture Type 2 knowledge representation and develop reasoning schemas that determine Type 2 consequences. It is to be forecasted that in the new millennium more and more researchers will attempt to capture Type 2 representation and end up with Type 2 conclusions that reveal the information content available in information granules, as well as expose the risk associated with the gradation between the lower and the upper membership degrees. In turn, this will entail more realistic system model developments, which will help explore computing with perceptions.