Construction of fuzzy classification systems with the Lukasiewicz-t-norm

Fuzzy classification systems usually aim at describing a function from continuous domains-the given attributes-to a discrete domain that represents the classes, like ill and healthy in medical diagnosis or plastic, metal etc. in recycling tasks. It usually does not make sense to interpolate between the discrete classes. Therefore, one of the main issues for fuzzy classification systems is the question of whether different classes can be distinguished by such a fuzzy system. We discuss the question of how complex fuzzy classification rules have to be in order to distinguish classes that are separated by a number of (hyper-)planes. We restrict our investigations to the case of two classes. Nevertheless, our results can also be applied, when we are interested in a larger number of classes, since we can simply consider the separation of one class w.r.t. the union of all other classes. We concentrate on the two- and three-dimensional case. It turns out that even in the three-dimensional case the rules can become quite complicated