Solvability of systems of partial fuzzy relational equations revisited – a short note

Fuzzy inference systems have been widely investigated from different perspectives, including their logical correctness. It is not surprising that the logical correctness led mostly to the questions on the preservation of modus ponens. Indeed, whenever such a system processes an input equivalent to one of the rule antecedents, it is natural to expect the modus ponens to be preserved and the inferred output to be identical to the respective rule consequent. This leads to the related systems of fuzzy relational equations where the antecedent and consequent fuzzy sets are known values, the inference is represented either by the direct product related to the compositional rule of inference or by the Bandler-Kohout subproduct, and the fuzzy relation that represents the fuzzy rule base is the unknown element in the equations. The most important question is whether such systems are solvable, i.e., whether there even exists a fuzzy relation that models the given fuzzy rule base in such a way that the modus ponens is preserved.

Partiality allows dealing with partially defined truth values which allows to deal with situations, where we cannot define the truth value for a given predicate. The partial logics have been recently extended to partial fuzzy logics and the partial fuzzy set theory has been developed. Partial fuzzy sets then may have undefined membership degrees for some values from the given universe.

This background leads naturally to the problem of solvability of partial fuzzy relational equations, which are equations with partial fuzzy sets in the role of antecedents and consequents and with partial fuzzy relation as the model of the given fuzzy rule base. Such a setting led to a recent publication that uncovers the solvability and even the shape of the solutions. In this contribution, we revisit the problem and consider a specific case that allows partiality only in the input. In other words, antecedents, as well as consequents expressing the knowledge, are fully defined. Thus, the model of the fuzzy rules is one of the standard and fully defined relations. We investigate what happens if the input differs from one of the antecedents only by a few undefined values. This mimics the situation when a description of an observed object misses a few values in its feature vector. We show that under specific conditions, we still may preserve the modified modus ponens, i.e., that the inferred output is identical with the fully defined consequent of the respective rule.