General Type-2 Fuzzy Inference Systems: Analysis, Design and Computational Aspects Luis Alberto Lucas, Tania Mezzadri Centeno and Myriam Regattieri Delgado Abstract- The aim of this work is to handle Non-interval Type-2 Fuzzy Logic Systems (NIT2 FLS) in a simple manner. We retrieve an alternative representation of Type-2 Fuzzy Sets (12 FS) that we call General Footprint of Uncertainty. Such representation, not only lets us easily visualize T2 FS in two-dimensions but also makes the understanding of basic operations and the inference procedure easier. We introduce the concept of supremum of a T2 FS and translation and cylindric extension for vertical slices to support the adopted inference mechanism, based on the scaled inference mechanism of Type-I FIS. Finally, we propose a new defuzzification method, the Vertical Slice Centroid Type Reduction, which requires low computational effort. Some calculations are presented to illustrate that the theory and simplifications proposed in this paper make NIT2 FLS, referred here as General Type-2 Fuzzy Inference Systems, much more accessible to FIS designers. I. INTRODUCTION Although being well defined, Non-interval Type-2 Fuzzy Logic Systems (NIT2 FLS) [1] [2], that we call General Type-2 Fuzzy Inference Systems (GT2 FIS), still lack prac- tical applications [3]. The main reasons for this can be: 1) The use of Type-2 Fuzzy Sets (T2 FS) is not widespread yet. Nevertheless such sets were introduced by Zadeh a long time ago [4], only recently Mendel established a set of terms to simplify their communication [5]; 2) The number of necessary operations in a GT2 FLS is large; and 3) Some op- erations on T2 FS (e.g. defuzzification) are computationally intensive. Interval Type-2 Fuzzy Inference Systems (IT2 FIS) were proposed [6] [7] as an alternative to simplify GT2 FLS. Additionally, some operations on T2 FS like centroid cal- culation [8] [9] and meet [10] were simplified to make their processing simpler. Although IT2 FIS are applicable to a great variety of situations, they can still be considered a particular case of GT2 FLS. In this paper, we strive to overcome some difficulties associated with GT2 FIS by: 1) retrieving an alternative representation of uncertainty of T2 FS that we call Gen- eral Footprint of Uncertainty (GEFOU); 2) introducing the concepts of supremum of a T2 FS, translation and cylindric extension for vertical slices; 3) defining a general inference mechanism to derive the GT2 FIS outputs; and 4) proposing an alternative defuzzification method that we call Vertical Slice Centroid Type Reduction (VSCTR). Lufs A. Lucas, Tania M. Centeno and Myriam R. Delgado are with the Graduate School of Electrical Engineering and Applied Computer Science, Federal University of Technology of Parand. Av. Sete de Setembro, Curitiba - PR, 80230-901, Brazil (phone: +51 41 33104688); {lalucas, mezzadri, myriamdelg} @utfpr.edu.br II. TYPE-2 FuzzY SETS: BASIC DEFINITIONS In this Section, we present some basic concepts of T2 FS which will be extensively used in the remainder of this paper. A. Type-2 fuzzy set Type-2 fuzzy sets were first described by Zadeh as a development for his fuzzy set theory [4]. According to [11] type-2 fuzzy sets are "sets whose membership grades are themselves type-I fuzzy sets". A type-2 fuzzy set, denoted by A, can be defined at the universe X as A = J J X,A(x u)/(X, V), exX tE Jx (1) where J, C [0,1] is the set of primary membership grades of x e X, with u e Jx, Vx e X, and PAA(x, u) is the type-2 membership function [5] [12]. Since type-2 membership functions are defined in X3 [13] [14] (see Figure 1) there is a lot of obstacles for drawing, handling and understanding them [5]. B. Vertical slice According to [14], for a particular x' C becomes JCJVJC H, IIA( X U)/(x = exX GJX/ Pj / (x, u)/(xl, u). Equation (2) can be rewritten as IEJ,/ X, Equation (1) x', U) (2) PA(X, u)/(x', V) =J I E~~~~~~E JXI where PA (x u ) PAWS) =J fx, (u) / u 4 is called the vertical slice of PuA(x, u) [14] [15], Jx' C [0,11 is the set of primary membership grades of x' and fx' (u), o < fx' (u) < 1, is a function f of the primary membership grade a, that identifies the secondary membership grades of x' in A. Vertical slices are also known as secondary membership functions or secondary sets [5]. As discussed by Mendel and John [5], a type-2 fuzzy set can be represented by the union of vertical slices in the way presented by Equation (5). 1-4244-1210-2/07/$25.00 C 2007 IEEE. fx, (u) / u / x', (3)