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