An Argument in Favor of Piecewise-Constant Membership Functions

Abstract

Theoretically, we can have membership functions of arbitrary shape. However, in practice, at any given moment of time, we can only represent finitely many parameters in a computer. As a result, we usually restrict ourselves to finite-parametric families of membership functions. The most widely used families are piecewise linear ones, e.g., triangular and trapezoid membership functions. The problem with these families is that if we know a nonlinear relation $y=f(x)$ between quantities, the corresponding relation between membership functions is only approximate—since for piecewise linear membership functions for x, the resulting membership function for y is not piecewise linear. In this paper, we show that the only way to preserve, in the fuzzy representation, all relations between quantities is to limit ourselves to piecewise constant membership functions, i.e., in effect, to use a finite set of certainty degrees instead of the whole interval [0, 1].