Some Considerations and a Benchmark Related to the CNF Property of the Koczy-Hirota Fuzzy Rule Interpolation

The goal of this paper is twofold. Once to highlight some basic problematic properties of the KH Fuzzy Rule Interpolation through examples, secondly to set up a brief Benchmark set of Examples, which is suitable for testing other Fuzzy Rule Interpolation (FRI) methods against these ill conditions. Fuzzy Rule Interpolation methods were originally proposed to handle the situation of missing fuzzy rules (sparse rule-bases) and to reduce the decision complexity. Fuzzy Rule Interpolation is an important technique for implementing inference with sparse fuzzy rule-bases. Even if a given observation has no overlap with the antecedent of any rule from the rule-base, FRI may still conclude a conclusion. The first FRI method was the Koczy and Hirota proposed "Linear Interpolation", which was later renamed to "KH Fuzzy Interpolation" by the followers. There are several conditions and criteria have been suggested for unifying the common requirements an FRI methods have to satisfy. One of the most common one is the demand for a convex and normal fuzzy (CNF) conclusion, if all the rule antecedents and consequents are CNF sets. The KH FRI is the one, which cannot fulfill this condition. This paper is focusing on the conditions, where the KH FRI fails the demand for the CNF conclusion. By setting up some CNF rule examples, the paper also defines a Benchmark, in which other FRI methods can be tested if they can produce CNF conclusion where the KH FRI fails.