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

Layered map reasoning

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Rough relation algebras revisited

Rough relation algebras arise from Pawlak’s information systems by considering as object ordered pairs on a fixed set X. Thus, the subsets to be approximated are binary relations over X, and hence, we have at our disposal not only the set theoretic operations, but also the relational operators;,˘, and the identity relation 1′. In the present paper, which is a continuation of [6], we further investigate the structure of abstract rough relation algebras