A large number of papers have been devoted to point out which combinations of the several fuzzy implication, composition and aggregation operators are necessary to satisfy a natural requirement for meaningful reasoning, that is, given a system observation which matches the antecedent of a rule, the conclusion of the inference process matches the consequent of that rule. Nevertheless, only few of these papers have analysed, once fixed a combination, which constraints on the reciprocal position of input and output fuzzy sets should be satisfied. In this paper, we consider fuzzy implication operators which are extensions of the two-valued logic implication operator and are non-decreasing with respect to their second argument, generic Sup-T composition operators, and minimum as aggregation operator. First, we analyse some features (namely, uniform level of indetermination, perfect matching with the consequent of a rule, and complete indetermination) of the conclusions inferred by those fuzzy implication operators with regard to fuzzy reasoning with one rule. Then, as regards approximate reasoning with multiple rules, we demonstrate that, if the fundamental requirement for fuzzy reasoning is satisfied, then the fuzzy sets which partition the input and output universes have to meet appropriate constraints. Finally, we provide a sufficient condition defined on input fuzzy sets to obtain a reasonable inference result.
Some considerations on input and output partitions to produce meaningful conclusions in fuzzy inference
LAZZERINI, BEATRICE;MARCELLONI, FRANCESCO
2000-01-01
Abstract
A large number of papers have been devoted to point out which combinations of the several fuzzy implication, composition and aggregation operators are necessary to satisfy a natural requirement for meaningful reasoning, that is, given a system observation which matches the antecedent of a rule, the conclusion of the inference process matches the consequent of that rule. Nevertheless, only few of these papers have analysed, once fixed a combination, which constraints on the reciprocal position of input and output fuzzy sets should be satisfied. In this paper, we consider fuzzy implication operators which are extensions of the two-valued logic implication operator and are non-decreasing with respect to their second argument, generic Sup-T composition operators, and minimum as aggregation operator. First, we analyse some features (namely, uniform level of indetermination, perfect matching with the consequent of a rule, and complete indetermination) of the conclusions inferred by those fuzzy implication operators with regard to fuzzy reasoning with one rule. Then, as regards approximate reasoning with multiple rules, we demonstrate that, if the fundamental requirement for fuzzy reasoning is satisfied, then the fuzzy sets which partition the input and output universes have to meet appropriate constraints. Finally, we provide a sufficient condition defined on input fuzzy sets to obtain a reasonable inference result.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
