On the non-differentiability of fuzzy logic systems

Tuning the parameters of fuzzy logic systems has become an important issue for their efficient development and utilization. Many techniques, mainly based on the application of gradient descent, have been applied to this task. The class of fuzzy logic systems using piecewise-linear membership functions (e.g., triangular or trapezoidal) and/or minimum or maximum operators possesses an error function that is non-differentiable (i.e., at any point in the search space) with respect to some of its parameters. Therefore the gradient does not always exist, and thus any convergence proof (even if only at a zero gradient point) no longer holds. The paper discusses the problem and shows some of the issues it raises