Envelope of plausibility defined by a new fuzzy implication operator

Abstract

A new implication operator is proposed, based on fuzzy entailment. Using a kernel and new measures of distance between fuzzy sets, it generates a set of fuzzy conclusions of various degrees of plausibility, rather than a single fuzzy conclusion. This set of plausible conclusions generates an envelope of plausibility, which is also a fuzzy set. The kernel can be chosen to make the envelope assume any reasonable shape surrounding the rule output, e.g., a Gaussian curve, or a set with a finite support, or the output of the Zadeh implication operator. As operator input becomes increasingly dissimilar to rule input, the envelope spreads further outside the rule output. The kernel can also be chosen to encode assumptions about system response, such as the degree of predictability/nonlinearity, the fractal dimension and the maximum possible rate of change of the output with respect to the input. The new distance functions permit speculative conclusions even when operator input does not intersect rule input. Since the operator's output is speculative, an analogous assumption-free belief envelope is also defined for comparison.