Implicative rules

Implicative rule aggregation is conjunctive, because each rule imposes a constraint on the output. The handling of this set of constraints is done using an intersection operator (t-norm). If no rule is active for a given input vector, the output is the universal fuzzy set, meaning that, in case of total ignorance, all output values are equally possible.

Three implication operators are available:

The Resher-Gaines operator is not fuzzy: the possibility distribution corresponds to the kernel of the other two, Goguen and Gödel.

The output partitions must be adapted to the aggregation mode specific to implicative rules. We propose the concept of Quasi standard partition (QSP) as a compromise between interpretability and coherence. Indeed the QSP is derived from a SFP partition, which is well known for its advantages in terms of interpretability. The extra fuzzy sets allow to have a non empty intersection when several rules are simultaneously activated. Figure 11 displays a SFP partition and the equivalent SFP one.

Figure 11: A SFP partition and the QSP equivalent one
Image qsp

Implicative rules offer several advantages compared to conjunctive ones: