An analysis of three types of partially-known formal concepts

Formal concept analysis with an incomplete context has received much attention recently, where an object is known to have one set of attributes and not have another set of attributes; for the rest of attributes, it is unknown if the object has or does not have them. This has led to a notion called partially-known formal concepts in a framework of three-way concept analysis with interval sets. The intent and/or extent of a partially-known concept may no longer be a set but an interval set. Depending on the set or interval set representation of the intent and extent, there are three different forms of partially-known formal concepts, namely SE-ISI (i.e., set extent and interval-set intent) formal concept, ISE-SI (i.e., interval-set extent and set intent) formal concept and ISE-ISI (i.e., interval-set extent and interval-set intent) formal concept. Although these three forms of partially-known formal concepts have been identified and proposed, their structures and relationships have not been fully investigated. The main objective of this paper is to provide such a study. We adopt a possible-world semantics of an incomplete formal context, i.e., an incomplete formal context is viewed as the family of all its possible completions. This enables us to systematically study the structures of the three different forms of partially-known formal concepts and their relationships. To be consistent with the possible-world semantics, we interpret a partially-known formal concept as the family of formal concepts in completions of an incomplete formal context. In addition to presenting theorems to summarize our results, we use an example to illustrate the main ideas.