Scalar cardinalities for divisors of a fuzzy cardinality

The cardinality of a finite fuzzy set can be defined as a scalar or a fuzzy quantity. The fuzzy cardinalities are represented by means the generalized natural numbers, where it is possible to define arithmetical operations, in particular the division by a natural number. The main result obtained in this paper is that, if determined conditions are assured, the scalar cardinality of a finite fuzzy set, B, whose fuzzy cardinality is a rational part of the fuzzy cardinality of another fuzzy set, A, is obtained by the same division of the scalar cardinality of A

Bell inequalities in cardinality-based similarity measurement

In this thesis a parametric family of cardinality-based similarity measures for ordinary sets (on a finite universe) harbouring numerous well-known similarity measures is introduced. The Lukasiewicz- and product-transitive members of this family are characterized. Their importance derives from the one-to-one correspondence with pseudo-metrics. Also a parametric family of cardinality-based inclusion measures for ordinary sets (on a finite universe) is introduced, and the Lukasiewicz- and product-transitivity properties are also studied. Fuzzification schemes based on a commutative quasi-copula are then used to transform these similarity and inclusion measures for ordinary sets into similarity and inclusion measures for fuzzy sets on a finite universe, rendering them applicable on graded feature set representations of objects. One of the main results of this thesis is that transitivity, and hence the corresponding dual metrical interpretation (for similarity measures only), is preserved along this fuzzification process. It is remarkable that one stumbles across the same inequalities that should be fulfilled when checking these transitivity properties. The inequalities are known as the Bell inequalities. All Bell-type inequalities regarding at most four random events of which not more than two are intersected at the same time are presented in this work and are reformulated in the context of fuzzy scalar cardinalities leading to related inequalities on commutative conjunctors. It is proven that some of these inequalities are fulfilled for commutative (quasi-)copulas and for the most important families of Archimedean t-norms and each of the inequalities, the parameter values such that the corresponding t-norms satisfy the inequality considered, are identified. Meta-theorems, stating general conditions ensuring that certain inequalities for cardinalities of ordinary sets are preserved under fuzzification, when adopting a scalar approach to fuzzy set cardinality, are presented. The conditions pertain to a commutative conjunctor used for modeling fuzzy set intersection. In particular, this conjunctor should fulfill a number of Bell-type inequalities. The advantage of these meta-theorems is that repetitious calculations (for example, when checking the transitivity properties of fuzzy similarity measures) can be avoided

Type-2 Fuzzy Alpha-cuts

Systems that utilise type-2 fuzzy sets to handle uncertainty have not been implemented in real world applications unlike the astonishing number of applications involving standard fuzzy sets. The main reason behind this is the complex mathematical nature of type-2 fuzzy sets which is the source of two major problems. On one hand, it is difficult to mathematically manipulate type-2 fuzzy sets, and on the other, the computational cost of processing and performing operations using these sets is very high. Most of the current research carried out on type-2 fuzzy logic concentrates on finding mathematical means to overcome these obstacles. One way of accomplishing the first task is to develop a meaningful mathematical representation of type-2 fuzzy sets that allows functions and operations to be extended from well known mathematical forms to type-2 fuzzy sets. To this end, this thesis presents a novel alpha-cut representation theorem to be this meaningful mathematical representation. It is the decomposition of a type-2 fuzzy set in to a number of classical sets. The alpha-cut representation theorem is the main contribution of this thesis. This dissertation also presents a methodology to allow functions and operations to be extended directly from classical sets to type-2 fuzzy sets. A novel alpha-cut extension principle is presented in this thesis and used to define uncertainty measures and arithmetic operations for type-2 fuzzy sets. Throughout this investigation, a plethora of concepts and definitions have been developed for the first time in order to make the manipulation of type-2 fuzzy sets a simple and straight forward task. Worked examples are used to demonstrate the usefulness of these theorems and methods. Finally, the crisp alpha-cuts of this fundamental decomposition theorem are by definition independent of each other. This dissertation shows that operations on type-2 fuzzy sets using the alpha-cut extension principle can be processed in parallel. This feature is found to be extremely powerful, especially if performing computation on the massively parallel graphical processing units. This thesis explores this capability and shows through different experiments the achievement of significant reduction in processing time.The National Training Directorate, Republic of Suda