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S.429-431
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.
This In this paper, we introduce a concept of granular fuzzy rule-based system, offer a motivation behind its emergence and elaborate on ensuing algorithm developments. It is shown that the granularity of the fuzzy rules is directly associated with a reduction (compression) process in which the number of rules becomes reduced in order to enhance the readability (transparency) of the resulting rule base. The retained rules are made more abstract (general) by admitting a granular form of the fuzzy sets forming their antecedents. In other words, while the original rules read as "if A <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> then B <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> " their reduced subset comes in the form "if G(A <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> ) the B <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> " with G(.) denoting a certain granular extension of the original fuzzy set (which can be realized e.g. in the form of interval fuzzy sets, fuzzy sets of type-2 or rough fuzzy-sets). It is shown that the optimization of the reduced set of rules is realized through an optimal distribution of information granularity among fuzzy sets forming the conditions of the reduced rules. In particular, it is shown that the distribution of information granularity, being regarded as an important design asset, is realized through a minimization of a certain objective function quantifying how well the granular fuzzy set formed by reduced rules set represents (covers) all rules. In this study, we use a technique of particle swarm optimization (PSO) as a vehicle of forming a subset of rules and the optimal allocation of information granulation to construct a granular fuzzy rule-based system. In the sequel, we introduce and idea of a granular representation of results of inferences realized in fuzzy rule-based system.