Similarity rules and gradual rules for analogical and interpolative reasoning with imprecise data

Similarity rules and gradual rules in the form of functional mappings of imprecise data are proposed to implement analogical and interpolative reasoning. The rules are deduced from the property of the horizontal graduality of fuzzy sets, and the horizontal graduality is represented by introducing a horizontal difference function as an alternative to the proposed extensions of distance between fuzzy concepts. A thorough discussion of the main characteristics of fuzzy sets and horizontal difference function is given. An analogical reasoning algorithm based on these characteristics and a set of intuitively sound axioms are proposed. The algorithm is consistent with the intuitive concepts of analogical reasoning expressed by the axioms. It guarantees a convex and normal output regardless of the shape of the precondition and action patterns and input data. When some axiomatic conditions are not satisfied, the algorithm relaxes the less important ones and produces an approximate solution. The fuzziness, cardinality and shape of an output fuzzy set are easily controlled. Some steps of the algorithm can be carried out in parallel. Numerical examples confirm the theoretical analysis of the properties of the algorithm. Some extensions for interpolative reasoning are discussed. The proposed rules are particularly well suited for systems which support qualitative reasoning and qualitative modelling.