Squashing Functions

Abstract

In fuzzy logic, the most commonly used membership functions are triangular and trapezoid ones. A crucial drawback of these functions is the lack of differentiability; a property that would be useful for learning systems. In this chapter, we introduce the so-called squashing functions; a differentiable parametrized family of functions that can not only be used for approximating piecewise linear membership functions but also Łukasiewicz-type logical operators. We show that the derivative of a squashing function is the difference of two sigmoid functions. This fact will come useful in gradient-based applications.