Granular Symmetric Implicational Method

Abstract

As a means of describing realistic problems, fuzzy sets can be included into the category of information granules from a broader perspective. Then the interval-valued fuzzy set itself is an expression of information granule which has more elaborate and stronger characterization abilities than generic fuzzy sets. In this study, facing up with modeling situations involving the use of interval-valued fuzzy sets, we come up with the granular symmetric implicational (GSI) method of fuzzy inference in view of the symmetric implicational idea and granular computing, which includes the basic GSI method and the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\zeta (w, z)$</tex-math></inline-formula> -GSI method. First, complete residuated lattices are employed as the structures of truth-values for interval-valued fuzzy sets. Second, unified expressions of optimal solutions to two GSI methods are gained for R-implications and (S, N)-implications. Lastly, it is shown through examples that the GSI method is superior over corresponding interval-valued fully implicational method. The originality of this work is three-fold. To begin with, the interval-valued fuzzy operators are introduced to the symmetric implicational mechanism, and novel symmetric implicational principles are presented which ameliorate the previous ones. Moreover, we offer a new construction method for interval-valued implications and corresponding adjoint couples, and on the strength of it we validate the reversibility and continuous properties of the GSI method. Finally, the hierarchical granular inference strategy is established for the GSI method in allusion to the circumstance of multiple rules.