Solvability of Fuzzy Relational Equations Employing Undefined Values

Abstract

The paper aims at studying the solvability of fuzzy relational equations where the input fuzzy sets may have undefined values. The appropriate operations dealing undefined values are taken from the algebras built in partial fuzzy set theory, in particular, the Bochvar operations, the Sobociński operations, the Kleene operations and the lower estimation operations are used for the investigation. We first elaborate new models using distinct operations handling undefined values and mirroring the shapes of the implicative model and Mamdani-Assilian model. Then the solvability criterions for such new systems of fuzzy relational equations are set up, and consequently, we show a way of checking whether the systems are solvable or not.

Sufficient Solvability Conditions for Systems of Partial Fuzzy Relational Equations

“…has been shown to be a solution of the system (10) under the assumption of its solvability [4]. And the solvability can ensured by the boundary condition, see Theorem 9.…”

“…Recently, investigations of the systems of fuzzy relational equations allowing the appearance of undefined values in the involved fuzzy sets were initiated [4,7]. Partial fuzzy logic which is considered as a generalization of the threevalued logic, and the related partial fuzzy set theory has been established [1][2][3]11].…”

“…Recently, further algebras for partial fuzzy logics motivated by dealing with missing values were designed [6,18]. In [4], the initial investigation on the solvability of the systems of partial fuzzy relational equations was provided. The study was restricted on the equations with fully defined (non-partial) consequents.…”