Conservation of relative fuzziness: Retrospective and triangular extension

Fuzzy rule interpolation is one of the tools for reducing computational complexity of fuzzy systems, and can be used when there are gaps in the knowledge base. These gaps can be natural, due to cost, or due to rule base reduction. The fuzzy interpolation methods are all descendent techniques of Kóczy and Hirota's linear interpolation. In this paper we provide a retrospective on the development of these techniques, and then focus on an early technique of conservation of fuzziness which has advantages in interpolation in hierarchical fuzzy systems as only near flank information is meant to be used and this allows the interpolation between different levels in the fuzzy rule base hierarchy. We point out an error and rectify it using a triangular extension which restores the intuitive, philosophical and practical nature of the approach.