When dealing with fuzzy relational inference system, there are two
standard approaches to model a fuzzy rule base (by conjunction of
implications and by disjunction of conjunctions aka Mamdani-Assilian
models) and there are two appropriate images that can be used for
modeling the inference mechanism (Computational Rule of Inference and
Bandler-Kohout subproduct). The combination of the Mamdani-Assilian
model and Computational Rule of Inference (abbr. CRI) is the most
favourite one however, not the safest one from the interpolative point
of view. The combination of the conjuction of implications and the
Bandler-Kohout Subproduct is not safe from the interpolative point of
view either. From tje interpoolative point of view, one should give
preference to using conjunction of implications jointly with CRI or to
using the combination of Mamdani-Assilian model jointly with the
Bandler-Kohout subproduct. The other side of the coin is that the
unpreferred combinations give advatages in terms of computational costs,
robustness or other aspects.
However, in 2002, B. Moser and M. Navara showed, that in the case of the
frequently used combination of Mamdani-Assilian model and the CRI, the
interpolativity often cannot be assured at all. Thus, the apparent
advantages of low computational costs or robustness are heavily redeemed
by the price of broken modus ponens. Therefore, B. Moser and M. Navara
proposed a specific modification of the CRI inference mechanism, that
together with Mamdani-Assilian model easily assures interpolativity. The
proposal was entitled: Conditionally Firing Rules. In our presentation,
we focus on re-definition of the proposed axiomatic environment in such
a way, that the modified axioms express the same meaining for
implicative rules as the original axioms for the Mamdani-Assilian rules.
In this setting, we may introduce Conditionally Firing Implicative Rules
and show, that a modification of the Bandler-Kohout subproduct inference
similar to the one proposed by B. Moser and M. Navara jointly with the
implicative rules becomes a very safe combination from the
interpolativity point of view.