An analysis of three types of partially-known formal concepts
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ORIGINAL ARTICLE
An analysis of three types of partially‑known formal concepts Ruisi Ren1,2 · Ling Wei1 · Yiyu Yao2 Received: 10 April 2017 / Accepted: 6 November 2017 © Springer-Verlag GmbH Germany, part of Springer Nature 2017
Abstract 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. Keywords Incomplete formal context · Interval set · Partially-known formal concept · Formal concept analysis · Three-way decisions
1 Introduction Formal concept analysis (FCA), proposed by Wille [8, 9, 30], is formulated based on a formal context (OB, AT, 𝐈) consisting of a non-empty set of objects OB, a non-empty set of attributes AT, and a binary relation 𝐈 between the objects and the attributes. An object o is related to an attribute a under the binary relation, that is, o𝐈a, if the object has the attribute. A fundamental result of FCA is the representation * Ling Wei [email protected] Ruisi Ren [email protected] Yiyu Yao [email protected] 1
School of Mathematics, Northwest University, Xi’an 710069, Shaanxi, People’s Republic of China
Department of Computer Science, University of Regina, Regina, SK S4S 0A2, Canada
2
of a concept by a pair of a set of objects as the extension (called the extent) of the concept and a set of attributes as the intension (called the intent) of the concept with respect to a formal context. The extent and intent of a concept uniquely determine each other. Any object in the extent has every at
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