Approximations from Anywhere and General Rough Sets

Not all approximations arise from information systems. The problem of fitting approximations, subjected to some rules (and related data), to information systems in a rough scheme of things is known as the \emph{inverse problem}. The inverse problem is more general than the duality (or abstract representation) problems and was introduced by the present author in her earlier papers. From the practical perspective, a few (as opposed to one) theoretical frameworks may be suitable for formulating the problem itself. \emph{Granular operator spaces} have been recently introduced and investigated by the present author in her recent work in the context of antichain based and dialectical semantics for general rough sets. The nature of the inverse problem is examined from number-theoretic and combinatorial perspectives in a higher order variant of granular operator spaces and some necessary conditions are proved. The results and the novel approach would be useful in a number of unsupervised and semi supervised learning contexts and algorithms.Comment: 20 Pages. Scheduled to appear in IJCRS'2017 LNCS Proceedings, Springe

Algebraic Models for Qualified Aggregation in General Rough Sets, and Reasoning Bias Discovery

In the context of general rough sets, the act of combining two things to form another is not straightforward. The situation is similar for other theories that concern uncertainty and vagueness. Such acts can be endowed with additional meaning that go beyond structural conjunction and disjunction as in the theory of $*$-norms and associated implications over $L$-fuzzy sets. In the present research, algebraic models of acts of combining things in generalized rough sets over lattices with approximation operators (called rough convenience lattices) is invented. The investigation is strongly motivated by the desire to model skeptical or pessimistic, and optimistic or possibilistic aggregation in human reasoning, and the choice of operations is constrained by the perspective. Fundamental results on the weak negations and implications afforded by the minimal models are proved. In addition, the model is suitable for the study of discriminatory/toxic behavior in human reasoning, and of ML algorithms learning such behavior.Comment: 15 Pages. Accepted. IJCRS-202