Parameterized R-implications

Authors:
Thomas Whalen
Affiliations:
Professor of Decision Sciences, J. Mack Robinson College of Business, Georgia State University, Atlanta, Georgia
Venue:
Fuzzy Sets and Systems - Implication operators
Year:
2003

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Abstract

A residuated implication operator (R-implications) is a fuzzy material implication operator defined in terms of its corresponding T-norm. This paper is concerned with a family of R-implications derived from the Schweizer-Sklar family of parameterized T-norms. Analysis of fuzzy and defuzzified outputs from two- and three-rule systems includes proper and improper fuzzy set outputs and exact defuzzified solutions for important special cases such as Gödel-Brouwer, Goguen, and Lukasiewicz implications.A rule can be characterized as strong or weak in its interaction with a neighboring rule. If two rules both interact with one another weakly, there exists a range of y values all of which are 100% compatible with both rules given any x in their domain of interaction. The paper concludes with a discussion of the joint effect of the strength of rule interaction and the value of the Sehweizer-Sklar parameter.