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.
