SOFSEM 2001 28th Annual Conference on Current Trends in Theory and Practice of Informatics November 24 - December 1, 2001 Piestany, Slovak Republic, Europe

Inference in Rule-Based Systems by Interpolation and Extrapolation Revisited

by Sandor Jenei

We deal with the problem of rule interpolation and rule extrapolation for fuzzy and possibilistic systems. Such systems are used for representing and processing vague linguistic If-Then-rules, and they have been increasingly applied in the field of control engineering, pattern recognition and expert systems. The methodology of rule interpolation is required for deducing plausible conclusions from sparse (incomplete) rule bases. For this purpose the well-known fuzzy inference mechanisms have to be extended or replaced by more general ones. The methods proposed so far in the literature for rule interpolation are mainly conceived for the application to fuzzy control and miss certain logical characteristics of an inference. This serves as a motivation for looking for a more flexible method which is superior to the proposed ones with respect to its general applicability to fuzzy as well as to possibilistic systems.

First, a set of axioms is proposed. With this, a definition is given for the notion of interpolation, extrapolation, linear interpolation and linear extrapolation of fuzzy rules. The axioms include all the conditions that have been of interest in the previous attempts and others which either have logical characteristics or try to capture the linearity of the interpolation. A new method for linear interpolation and extrapolation of compact fuzzy quantities of the real line is suggested and analyzed in the spirit of the given definition. The method is extended to non-linear interpolation and extrapolation. Finally, the method is extended to the general case, where the input space is n-dimensional, by using the concept of aggregation operators.