A mathematical model for concept granular computing systems

Abstract

Extent-intent and intent-extent operators are introduced between two complete lattices in this paper and a mathematical model for concept granular computing system is established. We proved that the set of all concepts in this system is a lattice with the greatest element and the least element. This framework includes formal concept lattices from formal contexts, L fuzzy concept lattices from L fuzzy formal contexts and three kinds of variable threshold concept lattices, i.e. the extension and the intension of a concept are a crisp set and a crisp set, a crisp set and a fuzzy set, a fuzzy set and a crisp set, respectively. Finally, some iterative algorithms for constructing concepts are proposed and they are proved to be optimal concepts under some conditions in this system. CitationQiu G F, Ma J M, Yang H Z, et al. A mathematical model for concept granular computing systems.

Concept learning via granular computing: A cognitive viewpoint

“…The existing literature mainly focused on concept learning via concept systems (e.g., [18,19,31,45]) which were established by axiomatic ways, regardless of the analysis on cognitive mechanism. However, the current study has carefully analyzed the cognitive mechanism of forming concepts based on the principles from philosophy and cognitive psychology, and then naturally found the constraints for cognitive operators to better simulate intelligence behaviors of the brain including perception, attention and learning.…”

“…However, the current study has carefully analyzed the cognitive mechanism of forming concepts based on the principles from philosophy and cognitive psychology, and then naturally found the constraints for cognitive operators to better simulate intelligence behaviors of the brain including perception, attention and learning. The existing work put forward cognitive computing systems (e.g., [18,19,31,45]) by means of different axiomatic ways, which cannot integrate past experiences into itself to deal with e.g., dynamic data. However, the current study has proposed such a cognitive computing system that is composed of a series of cognitive computing states and is able to integrate past experiences into itself through recursive thinking.…”

“…In addition, granular computing has also been integrated into the proposed cognitive computing system to decrease the computation time sharply. The existing researches on cognitive process mainly investigated the problem of learning concept(s) from a given pair of object and attribute sets by means of iterative algorithms [31,45], regardless of discussing whether the pair of object and attribute sets is concept-inducible or not, let alone the learning accuracy. However, the current paper has put forward the notion of a concept-inducible pair of object and attribute sets and a simple way of learning concept(s) from the concept-inducible pair via set approximations.…”

“…For example, information granules have been studied in rough set theory [17,23,29,30,35,44,46,47] extensively which is considered as one of the approaches of granular computing, and applied in formal concept analysis [10], evidence analysis [33], etc. Recently, studies on combination of granular computing with formal concept analysis have been made by several researchers [7,12,19,31,41,43,45,56]. And what is particularly worth mentioning is that information granules in formal concept analysis mean granular concepts [43] which are the basic concepts used to deduce others.…”

“…To the best of our knowledge, axiomatic systems of concept learning were often called concept systems instead. In recent years, there have been several concept systems proposed for certain concept learning such as cognitive system [45], concept granular computing system [31], generalized concept system [19] and generalized dual concept system [18]. At the same time, it should be pointed out that these concept systems can also be used to learn the certain concepts which were obtained by using the constructive methods in [9,11,16,40,42,48,49].…”