Set theory

A Test Space is a mathematical object which models the process of scientific inquiry. We examine the motivation of defining the Test Spaces and discuss connections between the Test Spaces and Metamathematics. A variant of Test Spaces called orthogonal partitions is introduced and we draw comparisons between the two. The combinatorial problems of counting finite Test Spaces and orthogonal partitions is highlighted. Some issues in manipulating infinite Test Spaces are discussed as well.

A methodology is presented to construct an approximate fuzzy-mapping algorithm that maps multiple inputs to single outputs given a finite training set of argument vectors functionally linked to corresponding scalar outputs. Its scope is limited to problems where the features are known in advance, or equivalently, where the expected functional representation is known to depend exclusively on the known selected variables. Programming and simulations to implement the methodology make use of Matlab Fuzzy and Neural toolboxes and a PC application of Prolog, and applications range from approximate representations of the direct kinematics of parallel manipulators to fuzzy controllers.