Transformations of Concept Graphs: An Approach to Empirical Induction

Let R be a binary relation, as illustrated in Figure 1(a), with m rows and n columns. Such a relation can represent many phenomena and there exists an extensive literature on relational algebras [1]. In this paper we take a more limited view and simply regard R as an observation of a set of attributes A associated with a set of objects O. In our formulation, objects are denoted by numbered rows and attributes denoted by lettered columns. \Formal Concept Analysis" [5] has been developed by Rudolf Wille [15], Bernard Ganter and their colleagues at Darmstadt. In their approach the construction and visual display of concept lattices, that is partially ordered sets of concepts, is crucial. The nodes of the concept lattice correspond to abstract concepts of the phenomenon being modelled and relationships within the lattice are re ective of relationships in the external world. Their book has numerous examples, and their method has found application in industrial applications (reported by Ganter & Wille) and in code re-engineering [8,14]. What makes it interesting to this workshop is an investigation of how these concept lattices are transformed with the advent of new information. We begin with a very brief overview of concept lattices. Let R be a binary relation between any two sets O and A, as in Figure 1(a). We regard O as a set of objects and A as a set of attributes. But, they can be arbitrary sets. For example, Lindig and Snelting [8] apply concept analysis to legacy code by creating a relation R between P , a set of procedures, and V , a set of global variables. By the closure, ' R ofO with respect toR, we mean a maximal set of objects which share the same attributes as all o 2 O. Similarly, 'R 1 operating on a set A of attributes picks up any other attributes that are common to all objects which satisfy each a 2 A. 1 Ganter and Wille [5] show that 'R and ? Research supported in part by DOE grant DE-FG05-95ER25254. 1 More formally, the Galois closure, 'R, on O with respect to R consists of those closed