Formal granular calculi based on rough inclusions

Rough inclusions are predicates of the form "an object x is a part of an object y to degree at least r", in symbolic form: μ(x, y, r). The partial containment, parallel and analogous to the partial membership of the fuzzy set theory, is not any surface phenomenon: formalized predicates μ induce in their universes fuzzy similarity relations in the sense of Zadeh. In this work, we address the problems of granular computing, the branch of approximate (or, soft) computing due to Zadeh whose underlying idea is to compute with granules of objects, i.e., "clumps of objects (...) which are drawn together by indistinguishability, similarity or functionality". As put by Lin:" granulation (...) appears (...) in different names, such as chunking, clustering, data compression, divide and conquer, information hiding, interval computations, and rough set theory, just to name a few". In addition to a formal granule calculus, we also introduce a certain form of granular information/decision systems derived from original given information/decision systems and we analyze by help of a simple example granular classifiers/decision algorithms induced from them as an approximation to classifiers/decision algorithms induced from original systems.