Xor-Implications and E-Implications: Classes of Fuzzy Implications Based on Fuzzy Xor

Abstract

The main contribution of this paper is to introduce an autonomous definition of the connective "fuzzy exclusive or" (fuzzy Xor, for short), which is independent from others connectives. Also, two canonical definitions of the connective Xor are obtained from the composition of fuzzy connectives, and based on the commutative and associative properties related to the notions of triangular norms, triangular conorms and fuzzy negations. We show that the main properties of the classical connective Xor are preserved by the connective fuzzy Xor, and, therefore, this new definition of the connective fuzzy Xor extends the related classical approach. The definitions of fuzzy Xor-implications and fuzzy E-implications, induced by the fuzzy Xor connective, are also studied, and their main properties are analyzed. The relationships between the fuzzy Xor-implications and the fuzzy E-implications with automorphisms are explored.

Fuzzy XNOR connectives in fuzzy logic

“…Fuzzy implications have been widely studied, playing important roles in different domains. Recent papers studied different classes of fuzzy implications (Baczyński 2004;Baczyński andJayaram 2007, 2008;Balasubramaniam 2007;Bedregal et al 2009;Bustince et al 2003;Shi et al 2007;Yager 2004), such as R-implication (Baczyński 2004, Gottwald 2001, S-implication , QL-implication (Mas et al 2006;Shi et al 2007), D-implication (Mas et al 2006(Mas et al , 2007a, etc. They are usually derived from t-norms, t-conorms, and fuzzy negations.…”

“…Bedregal et al (2009) proposed the definition of the connective fuzzy Xor. Also, two kinds of fuzzy Xor connectives were presented.…”

“…Preposition 2 (Bedregal et al 2009) Let S, T and N be a t-conorm, a t-norm and a fuzzy negation, respectively. A fuzzy Xor connective can be given by the function E T : ½0; 1 2 !…”

“…In Bedregal et al (2009), an autonomous definition of the connective ''fuzzy exclusive or'' (fuzzy Xor, for short) and corresponding properties were introduced.…”

“…Definition 5 (Bedregal et al 2009) A function E: ½0; 1 2 ! ½0; 1 is a fuzzy Xor if it satisfies the properties:…”