Abstract: |
Fuzzy inference systems, i.e. systems consisting, in principle, of a fuzzy rule base and an inference mechanism, 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 that is 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.
The solvability of systems of fuzzy relational is a classical topic with lots of deep and inspiring results however, only recently, it has not been combined with partiality. Partiality allows dealing with partially defined truth values, which is again a classical logical topic being studied since 1920's. Indeed, in distinct situations, we cannot define the truth value for a given predicate. We may encounter e.g. missing values in databases, various N/A values in questionnaires, meaningless undefined values from predicates with undenoted terms, e.g., a statement about the french king is neither true nor false as France has no king and so, the truth-value of such a statement is simply undefined. The partial logics have been recently extended to partial fuzzy logics and consequently, the partial fuzzy set theory has been developed based on numerous extensions of partial algebras of truth values. Partial fuzzy sets then may have undefined membership degrees for some values from the given universe and the partial algebras have to extend operations from the residuated lattice in such a way that they may incorporate a dummy value representing the undefinedness.
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 antecedent and consequents as well as 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 for most of the known algebras. 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 also the model of the fuzzy rules is one of the standard and fully defined relations. And we investigate what happens if the input differs from one of the antecedents only by a few undefined values which 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. |