Interpretable Fuzzy Systems: Analysis of T-norm interpretability

The issues of Interpretable Fuzzy Systems (IFS) spin from the fundamental definitions of the concept of IFS to practical design of such systems. This paper addresses the current issues of formalization of the concept of interpretability, its dimensions, evaluations and design of interpretable Fuzzy Systems, including fuzzy control systems. T-norms and T-conorms are in the core of Fuzzy Systems, therefore we consider the following questions about them. Where is the adequate sphere for T-norms in IFS to be interpretable operations? How to modify the T-norms to satisfy the IFS requirements? What are the alternatives to T-norms and t-conorms in IFS? This analysis of T-norms is done in the context of their interpretation as measurements. We show that the approach used in the representative measurement theory is a source of the interpretability definition, which is in line with Tarsky´s definition of interpretability. This approach is used to define a concept of the interpretable fuzzy operations. Next, it is shown that an adequate scope for scalar T-norms is to be compact approximations of 2-D lattice operations. Such lattice operations are an alternative to T-norms and T-conorms in IFS because they have better interpretability. It is shown that such popular T-norms as the minimum and the product operations are similar as approximations of the Pareto optimal set, but quite different in representation of the lattice structure, where the product is preferable because it distorts the lattice structure less.