The approximation of piecewise linear membership functions and Łukasiewicz operators

In this paper we propose an approximation of piecewise linear membership functions with the help of sigmoid functions and certain arithmetic operations. The gradient-based tuning of piecewise linear membership functions can be achieved with the proposed efficient approximation because it has simple continuous derivatives. With this construction we can even approximate the Łukasiewicz operator family which plays an important role in fuzzy logic, first of all from the theoretical point of view, although in practice in optimization and learning it is rarely used because the lack of good analytical properties, e.g. a continuous gradient. The proposed approximation enlarges the applicability of fuzzy methods to the operators and membership functions where the differentiability is desirable.