Formal Concept Analysis of Higher Order

The second order formal context is a formal context such that its object and attribute sets are disjoint unions of object and attribute sets of external formal contexts. Every subset of object or attribute set will be evaluated from concept lattice of corresponding external formal context. The paper provides a method how to compute such second order formal concepts by using of bonds between external formal contexts or by using of heterogeneous formal contexts methods. Last part of the paper shows how this structure generalizes homogenic fuzzy formal context and its derivation operators. 1 Motivation example Imagine the following situation as a motivation. Lets have a group of people that everybody knows each other (schoolmates, co-workers, etc.). All of them are going to travel somewhere together and you (as an organizer of the trip) would like to know their requirements for accommodation. Consider the following formal context. Set of objects represents a group of co-workers (Anna, Bob, Cyril, David, Erik). Set of attributes expresses their requirements (TV, Wellness, Closeness to a city center, Restaurant in a hotel). An example of such context is in the following table. Lets denote the following table as P as preferences. P TV W Ce R Anna • • • Bob ◦ ◦ • • Cyril ◦ • • ◦ David • • ◦ Erik • • ◦ • A particular formal concept of the given context describes a set of co-workers such that these people together have a common requirements for accommodation. In addition, there are another two formal contexts. The first one describes a friendship relation inside the group of such people (denoted as F). The second one describes a situation about hotels and services they offer (denoted as H). ? Partially supported by grant VEGA 1/0832/12 and APVV-0035-10. c © paper author(s), 2013. Published in Manuel Ojeda-Aciego, Jan Outrata (Eds.): CLA 2013, pp. 117–128, ISBN 978–2–7466–6566–8, Laboratory L3i, University of La Rochelle, 2013. Copying permitted only for private and academic purposes. F Anna Bob Cyril David Erik Anna • ◦ ◦ Bob ◦ • • ◦ ◦ Cyril ◦ • • • • David ◦ • • • Erik ◦ • • • H TV W C R H1 • • • • H2 • • • H3 • H4 • • • All contexts are filled by truth degrees from the following set {• = true, ◦ = middle, ” ” = false}. Computing of L-concepts is based on Lukasiewicz logic. Now, we aim at connecting such table data with known intercontextual mechanisms in order to obtain closed sets of friends from F that are able to stay in any of a closed set of hotels from H. The hotels of this closed set offer as much requirements from P as it gets.