Fuzzy inference systems preserving Moser-Navara axioms

Satisfiability of three naturally proposed axioms, which are desirable for fuzzy inference system, was discussed by Moser and Navara in [36]. The fuzzy rule base models considered in this study were mainly the Mamdani-Assilian ones and partly also logically motivated implicative fuzzy systems. The investigation leads to the fact that often none of these fuzzy systems do satisfy the given axioms simultaneously. Therefore, Moser and Navara proposed so-called conditionally firing rules and proved that the simultaneous satisfaction of the three axioms is feasible under very mild and practically non-restrictive conditions. Note that the inference mechanism, tailored to the fuzzy rules in the investigation, was the compositional rule of inference (abb. CRI) and similarly, the concept of conditionally firing rules were stemming from the CRI and the Mamdani-Assilian model of a fuzzy rule base. For the implicative interpretation of fuzzy rules, the original axioms proposed by Moser and Navara were later on re-defined by Štěpnička and Mandal with the aim to express the meaning of the original axioms initially proposed for the Mamdani-Assilian rules. Furthermore, conditionally firing implicative rules were proposed based on the so-called Bandler-Kohout subproduct (abb. BKS) inference mechanism and a study similar to the one by Moser and Navara was undertaken for them. The study encompasses the satisfaction of the modified axioms for the use of implicative rules and the BKS. Furthermore, some additional results confirming the applicability of the conditionally firing rules satisfying the Moser-Navara axioms are also provided, including the universal approximation ability. However, the solvability criterions for fuzzy relational equations give a strong preference to the usage of the other combinations, namely, to the combination of CRI with the implicative interpretation of fuzzy rules and to the combination of BKS with the Mamdani-Assilian interpretation of fuzzy rules. Therefore, also these combinations of fuzzy inference mechanisms and interpretations of fuzzy rules should be investigated with respect to the satisfaction of Moser-Navara axioms. The main purpose of this article, which extends the short conference contribution [47] by Štěpnička, is to provide such an investigation jointly with the review on conditionally firing rules and supplementary results.