"Possible Definitions of an ’A Priori’ Granule\ud in General Rough Set Theory" by A. Mani

We introduce an abstract framework for general rough set theory from a mereological perspective and consider possible concepts of ’a priori’ granules and granulation in the same. The framework is ideal for relaxing many of the\ud relatively superfluous set-theoretic axioms and for improving the semantics of many relation based, cover-based and dialectical rough set theories. This is a\ud relatively simplified presentation of a section in three different recent research papers by the present author.\u

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