Knowledge-Based Systems 104 (2016) 62–73 Contents lists available at ScienceDirect Knowledge-Based Systems journal homepage: www.elsevier.com/locate/knosys Granule description based on formal concept analysis Huilai Zhi a , Jinhai Li b,∗ a School of Computer Science and Technology, Henan Polytechnic University, Jiaozuo 454000, Henan, PR China b Faculty of Science, Kunming University of Science and Technology, Kunming 650500, Yunnan, PR China a r t i c l e i n f o Article history: Received 31 December 2015 Revised 4 March 2016 Accepted 13 April 2016 Available online 15 April 2016 Keywords: Granule Granule description Granular computing Formal concept analysis Stability index a b s t r a c t Granule description is a fundamental problem in granular computing. Although the spirit of granular computing has been widely adopted in scientific researches, how to classify and describe granules in a concise and apt way is still an open, interesting and important problem. The main objective of our paper is to give a solution to this problem under the framework of granular computing. Firstly, by using stabil- ity index, we classify the granules into three categories: atomic granules, basic granules and composite granules. Secondly, in order to improve the conciseness and aptness of granules, we impose additional conditions on minimal generator to define a new term which is called the most apt minimal genera- tor. And then, based on the most apt minimal generator, we put forward methods for the description of atomic granules and basic granules. Moreover, for composite granules, we continue to divide them into three subcategories: ∧-definable granules, (∧, ¬)-definable granules and (∧, ∨)-definable granules, and their respective descriptions are provided as well. Finally, some discussions are also made on indefinable granules. © 2016 Elsevier B.V. All rights reserved. 1. Introduction Granular computing (GrC) is an emerging computing paradigm of information processing, which lies in the scope of cognitive sci- ence and cognitive informatics [2,43]. Granular computing studies information and knowledge processing in an abstract way, handles complex information entities in different granules, and allows us to view a phenomenon with different levels of granularity [1,60]. The spirit of granular computing has been adopted frequently in scien- tific researches, such as philosophy of structured thinking, struc- tured problem solving, and structured information processing. In this sense, all the methods which treat information in this perspec- tive will fall into the scope of granular computing [12,26,34,37]. To put it simply, information granules are collections of enti- ties which are arranged together due to their similarity, functional or physical adjacency, coherency, and so on [29,53,69]. At present, granular computing is not a coherent set of methods or principles but rather a theoretical perspective, which encourages researchers to deal with knowledge at different levels of abstraction or gen- eralization [9,40,52,66,68]. It often granulates the universe of dis- course into a family of disjoint or overlapping granules. Based on this idea, different views of the universe of discourse can be linked together, and a hierarchy of granulations can be established. Thus, ∗ Corresponding author. Fax: +86 871 65916703. E-mail addresses: zhihuilai@126.com (H. Zhi), jhlixjtu@163.com (J. Li). one of the main directions in the study of granular computing is to deal with the construction, interpretation, and representation of granules [50]. Rough set theory (RST), as an efficient tool of granular comput- ing, presented by Pawlak [31], has drawn many attentions from re- searchers over the past thirty-four years [14,19,33,48,54,67,70,71]. As is well known, the original idea of rough set theory is to par- tition the universe of discourse into disjoint subsets by a given equivalence relation, and then by using the obtained disjoint sub- sets, target sets are characterized by means of the so-called lower and upper approximations. Rough sets were used to describe a target set by the lower and upper approximations under one granulation, but multiple granulations are sometimes required to approximate a target set when dealing with multi-scale or multi-source data sets [35,36,51]. Under such a circumstance, pessimistic multigranulation rough sets and optimistic multigranulation rough sets were proposed for applying multi-source information fusion. These information fusion strategies were soon extended to cater the cases such as incomplete, neighborhood, covering and fuzzy environments [13,24,25,39,55,59]. Moreover, a byproduct is that “AND” and “OR” decision rules can be derived from decision systems with the pes- simistic and optimistic multigranulation rough sets [35,36], which was further exploited by Yang et al. [58] and Li et al. [23] in terms of local and global measurements of the “AND” and “OR” decision rules. http://dx.doi.org/10.1016/j.knosys.2016.04.011 0950-7051/© 2016 Elsevier B.V. All rights reserved.