Rough Granular Computing in Modal Settings: Generalised Approximation Spaces

Granular computing (GrC) is a methodology or – as ambitious at it sounds – a paradigm in computer science the aim of which is to define and solve computational problems in terms of information granules. These granules are conceived as clumps of objects drawn together due to some criteria such as, e.g., similarity, indiscernibility or functionality. The present paper is concerned with the rough granular computing methodology (RGrC) which is GrC expressed and developed within rough set theory. In this theory the elementary information granules usually are: indiscernibility classes (the classical model), tolerance classes (the tolerance rough set model), or minimal open neighbourhood (the topological model). The main idea which stays behind all these rough set models is to express all pieces of information about a given domain U as some binary relation E between objects belonging to U – e.g.: an equivalence relation, a tolerance relation, and a preorder, respectively. Each relation induces in turn rough approximations of any subset of the universe U . It allows one to express the approximations of sets/concepts in terms of modal operators of some modal logic – e.g.: S5, B, and S4, respectively. Then, given a set of objects |α| which satisfy the formula α, | α| is the interior of |α| (in the corresponding topology or pretopology) and |♦α| is the closure. However, the specific feature of modal logic is the locality: everything is computed starting from a single designated world (called the actual world). Thus a formula α is satisfied by a single world; in other words, formulae may say something only about single objects whereas in GrC one deals with information granules rather than single elements. So, the standard modal approach lacks the expressive power to “touch” information granules. Z. Pawlak and A. Skowron considered a generalisation of approximation spaces defined as a triple (U, I, v), where U is the universe of objects, I is an uncertainty function I : U → PPU (or, in simplified version, I : U → PU ), and v is a rough inclusion function v : PU × PU → [0, 1] telling to what an extent one set in included in another one. Of course, each function from U to PPU may be redefined as a relation R between objects and sets of objects R ⊆ U × PU ; one can also consider its inverse R−1 ⊆ PU × U . In contrast to the standard modal settings, here the granulation of the universe is given explicitly; moreover, both R and R−1 may be given GrC based interpretations as organisation and decomposition.