Advanced Concepts in Fuzzy Logic and Systems with Membership Uncertainty

Vagueness and uncertainty are intrinsic aspects of engineering design. Therefore, in this chapter, we introduce mathematical tools for modelling various types of vagueness and uncertainty, including fuzzy sets, interval-valued fuzzy sets, fuzzy-valued (type-2) fuzzy sets, rough sets, rough approximations of fuzzy sets, and two different definitions of fuzzy-rough sets. Finally, we aim to categorize different types of uncertainty regarding various sources of it.

In this chapter, new analytical formulae for membership functions of extended t-norms are derived. We consider the following cases: extended minimum, minimum-based extensions of continuous t-norms, extended continuous t-norms based on drastic-product, and extended Łukasiewicz t-norm based on continuous Archimedean t-norms. As a dual concept to extended t-norms, extended t-conorms and their formulae are considered. These cases cover almost all practical engineering situations when we implement type-2 fuzzy logic systems. In many cases, we get formulae that preserve shapes, which enable us to derive adaptive network fuzzy inference systems of type- 2. Otherwise, some approximations are needful, or more general notion of a triangular norm on fuzzy truth values (t-norm of type-2 for short) is needed, whose axiomatics we provide briefly. Finally, implications on fuzzy truth values, especially their family called simplicatoins, are derived in order to prepare foundations for structures of uncertain fuzzy logic systems.

In this chapter, basic constructions of fuzzy logic systems with uncertain membership functions are presented. We begin with historical approaches to reasoning with interval-valued fuzzy sets and known formulations of general type-2 fuzzy logic systems. Next we provide new formulations grounded in non-singleton fuzzification. In the context of ordinary fuzzy systems, we demonstrate that variously interpreted non-singleton fuzzification, for typical structures fuzzy logic systems, can be implemented by the classical singleton structures only using modified antecedent fuzzy sets. The first approach to fuzzification of premises is done by the interpretation in terms of possibility distributions of actual inputs. Consequently, the possibility and necessity measures of antecedent fuzzy sets create boundaries for the interval-valued antecedent membership function. The second approach applies rough approximations to antecedent fuzzy sets by non-singleton fuzzy premise sets considered as fuzzy-rough partitions. Two known definitions, the one of Dubois and Prade, and the second proposed by Nakamura, lead to different formulations of fuzzy logic systems. Employing fuzzy-rough sets of Dubois and Prade, we obtain the interval-valued fuzzy logic system. Then, it can be immediately proved that upper approximations in fuzzy-rough systems are concurrent to fuzzification in conjunction-type fuzzy systems. Unexpectedly, lower approximations in fuzzy-rough systems coincide with fuzzification in logical-type fuzzy systems. Therefore, the proposed methods can be viewed as extensions to the conventional non-singleton fuzzification method. Fuzzyrough sets in the sense of Nakamura result with a formulation of a general fuzzy-valued fuzzy logic system. For this purpose, three realizations of general fuzzy-valued fuzzy systems: triangular, trapezoidal and Gaussian, are presented in details.

In this chapter, against the background of existing methods, we provide several methods to generate membership uncertainty. In particular, we present an approach to multiperson decision making that generates triangular secondary memberships. Then, we make use of nonlinear fitting to expand interval (as well as triangular) secondary membership functions over data partitioned by the fuzzy C-means algorithm. We also introduce an incomplete and discrete information reasoning schema based on rough-fuzzy sets. Finally, we apply generalized fuzzification performed either via possibility and necessity measures or by fuzzy-rough sets.

This chapter provides a complete methodology for construction of uncertain fuzzy logic systems. The methodology comprises all techniques delivered by this book including: rough-fuzzy discretization of input domains (as well as imputation of missing inputs), possibilistic and fuzzy-rough fuzzification of inputs, fusion of multiple expert designs. Besides, this chapter answers the question whether it is worth to make use of fuzzy-valued fuzzy logic systems instead of ordinary crisp-valued fuzzy systems at the cost of the complexity. In response, two methods for the approximation of intervalvalued fuzzy systems by ordinary fuzzy logic systems are presented. In the first approximation the interval-valued fuzzy system is assumed to perform the extended minimum Cartesian product and conjunction reasoning, and to use uniform uncertainty of membership functions. In the latter approximation the interval system is assumed to perform the algebraic Cartesian product and employ lower membership functions proportional to their upper counterparts. The chapter is complemented by a comparative analysis of the interval and non-interval fuzzy systems and a brief discussion about generalization of this analysis to general fuzzy-valued fuzzy logic systems.