Inference in fuzzy rule bases with conflicting evidence

Inference based on fuzzy 'If ... then' rules has played a very important role since when Zadeh proposed the Compositional Rule of Inference and, especially, since the first successful application presented by Mamdani. From the mid-1980's when the 'fuzzy boom' started in Japan, numerous industrial applications appeared, all using simplified techniques because of the high levels of computational complexity. Another feature is that antecedents in the rules are distributed densely in the input space, so the conclusion can be calculated by some weighted combination of the consequents of the matching (fired) rules. The CRI works in the following way: If R is a rule and A* is an observation, the conclusion is computed by B* = R o A* (o stands for the max-min composition). Algorithms implementing this idea directly have an exponential time complexity (maybe the problem is NP-hard) as the rules are relations in X x Y, a k1 x k2 dimensional space, if X is k1, Y is k2 dimensional. The simplified techniques usually decompose the relation into k1 projections in X(sub i) and measure in some way the degree of similarity between observation and antecedent by some parameter of the overlapping. These parameters are aggregated to a single value in (0,1) which is applied as a resulting weight for the given rule. The projections of rules in dimensions Y(sub i) are weighted by these aggregated values and then they are combined in order to obtain a resulting conclusion separately in every dimension. This method is unapplicable with sparse bases as there is no guarantee that an arbitrary observation matches with any of the antecedents. Then, the degree of similarity is 0 and all consequents are weighted by 0. Some considerations for such a situation are summarized in the next sections.