Interpolative reasoning with insufficient evidence in sparse fuzzy rule bases

Rule based fuzzy approximate reasoning uses various techniques of modified modus ponens. The observation is in most cases not identical with any of the antecedents in the rules. However, a conclusion still can be computed by using some combination of all consequents where an overlapping of observation and antecedent is present. If the rule base is sparse, i.e., it contains insufficient information on the total state space, it might occur that an observation has absolutely no overlapping with any of the antecedents and so not even a single rule is fired, i.e., no conclusion can be computed on the basis of modus ponens. In such a case, interpolative reasoning in the strict sense can be applied: some kind of (weighted) average of the flanking rules is calculated. This technique can be extended to a form of extrapolation, when the observation is not flanked from both sides. Linear interpolation and extrapolation is presented, and then the idea is extended to arbitrary approximation.