Rough Sets in Flux: Crispings and Change

Abstract

The paper aims to initiate the research upon a new conceptualisation of rough set theory (RST), which allows one to manage the change of the background knowledge. In the classical approach to RST, a rough set is defined as a pair $(\underline{A},\overline{A})$ of definable (crisp) subsets (of some universe U), called the lower and the upper approximation of A ⊆ U, respectively. Since both approximations are idempotent, a rough set is fixed (static). By mimicking the behaviour of a topological praclosure operator, we shall generalise the above ideas and introduce a notion of a crisping concept, which will set the stage for rough sets in flux. For each crisping concept we distinguish its generalised rough set, which can be modified by means of dynamic operators such as expansion, revision or contraction (but within the scope of the crisping concept). Since this approach can cope with the change of knowledge, it may be regarded as a step towards approximating processes (i.e., sequences of changes).