Extending the Concepts of Type-2 Fuzzy Logic and Systems

The work presented in this dissertation is a contribution to the field of fuzzy sets and fuzzy logic systems theory. Firstly, we will approach the controversial discussion that has been effusively debated among scholars of fuzzy logic for many years. Some authors argue that the ability of type-2 fuzzy logic systems to perform better than their type-1 counterparts relies on the higher number of parameters they need to be defined. On the other hand, other authors pose the argument that this ability is due to how those parameters are used, and how type-2 fuzzy sets model uncertainty in a more suitable way. Although other previous works have tackled this discussion, we propose a new approach based on a function approximation framework, using type-1 fuzzy logic systems with a varying number of parameters. This part of the work aims to support the previous findings related to this topic, and justify the further research on type-2 fuzzy sets and fuzzy logic systems in the rest of the dissertation. Secondly, after shedding some light on the previous discussion, we will focus on the development of the theory about type- 2 fuzzy sets and fuzzy logic systems. Traditionally, although type-2 fuzzy logic has proven to perform better than type-1, its use has been somehow limited. One of those reasons has been the limitation to operate with those sets; although the operations of intersection and union on these sets were defined at the same time that type-2 fuzzy sets themselves, the operations were computationally intensive, and closed formulas were only available for type-2 fuzzy sets having normal and convex secondary grades. The main contribution of this work to the fuzzy sets theory is to provide two new theorems for the intersection and union operations, regardless of the specific shape of the sets’ secondary grades. Those new theorems, which allow us to operate on type-2 fuzzy sets having non-convex secondary grades, are the keystone to further developing the theory of interval type-2 fuzzy logic systems. Interval type-2 fuzzy sets have been recently shown to be more general than interval-valued fuzzy sets, and can actually have non-convex secondary grades. Hence, a whole new theory needs to be developed in order to provide those fuzzy logic systems with the appropriate theoretical framework; we aim to do so in the last part of this dissertation.