  Concise granule descriptions for definable granules and approaching
descriptions for indefinable granules are challenging and important issues in
granular computing. The concept with only common attributes has been
intensively studied. To investigate the granules with some special needs, we
propose a novel type of compound concepts in this paper, i.e.,
common-and-necessary concept. Based on the definitions of concept-forming
operations, the logical formulas are derived for each of the following types of
concepts: formal concept, object-induced three-way concept, object oriented
concept and common-and-necessary concept. Furthermore, by utilizing the logical
relationship among various concepts, we have derived concise and unified
equivalent conditions for definable granules and approaching descriptions for
indefinable granules for all four kinds of concepts.
