On the satisfaction of Moser-Navara axioms for fuzzy inference systems

In [1], B. Moser and M. Navara defined three “interpolation” axioms for fuzzy inference systems and showed, that usually neither very popular Mamdani-Assilian nor logically motivated implicative fuzzy systems, do satisfy them all simultaneously. The investigated inference mechanism was the compositional rule of inference (abb. CRI). Therefore, the authors introduced so-called conditionally firing rules and proved, that under very mild and practically very feasible conditions, all three axioms may be satisfied simultaneously. Note, that the conditionally firing rules were stemming from the Mamdani-Assilian rule interpretation and from the CRI. In [2], M. Štěpnička and S. Mandal showed that if the original axioms are supposed to express the intended meaning for the implicative interpretation of rules, they have to be re-defined. Furthermore, after such a re-definition, very similar results were obtained. Here, the so-called Bandler-Kohout subproduct (abb. BKS) was used as the inference mechanism. Therefore, analogously to [1], conditionally firing implicative rules were proposed and in order to satisfy the re-defined axioms simultaneously. However, as we know from the field of solvability of fuzzy relational equations, the advantageous composition/ image, which is directly related to the inference, is the direct image (related to CRI) when dealing with the implicative interpretation of fuzzy rules, and the subdirect image (related to BKS) when dealing with Mamdani-Assilian interpretation of fuzzy rules, not vice-versa. This article aims directly at this point and investigates the satisfaction of Moser-Navara axioms by the preferable combinations of rule interpretations and inference mechanisms.