Decomposing Finite Closure Operators by Attribute Exploration

There are examples of algorithms which allow the e cient computation of the concepts of a given formal context. However, when one wants to apply those algorithms to an arbitrary closure operator, which might not be given by a formal context, those algorithms cannot be applied directly. Therefore we want to discuss the problem of nding an appropriate formal context for a given closure operator, and present an application of attribute exploration to achieve this goal. The occurrence of high performance algorithms for the fast computation of concepts of a formal context (as [1,8,9]) makes it possible to deal with large amounts of data (if one is interested in its formal concepts). These algorithms are mostly variants of Kuznetsov's well-known Close-by-One algorithm (as for example described in [6]). In contrast to Ganter's NextClosure [3], which computes the xpoints of a given closure operator, Close-by-One computes the concepts of a given formal context. But still both algorithms get a closure operator as input and compute it's xpoints. The only di erence is in the representation of the closure operator c: Close-by-One requires it to be given as a formal context, whereas NextClosure does not place any restrictions on the representation of c. We want to take this asymmetry of representing a closure operator as motivation to consider the following question: Can we e ectively (and also maybe e ciently) compute for a given closure operator c on a setM a formal context whose intents are precisely the closed sets of c? We shall formalise this idea as decomposing closure operators and shall show a neat application of attribute exploration in solving this problem. We also shall discuss some complexity aspects and show some experimental evaluation. 1 Formal Concept Analysis and Closure Operators To better understand the following discussion, we brie y introduce the necessary de nitions from Formal Concept Analysis and the theory of closure operators. The basic structure used by Formal Concept Analysis is the one of a formal context. Let G and M be two sets and let I ⊆ G ×M . We will call the triple (G,M, I) a formal context. Intuitively, the set G is the set of objects, the set M