A decomposability index in logical analysis of data

Abstract

Logical analysis of data (LAD) is one of the methodologies for extracting knowledge in the form of a Boolean function f from a given pair of data sets ( T, F) on attributes set S of size n, in which T (resp., F) ⊆{0,1} n denotes a set of positive (resp., negative) examples for the phenomenon under consideration. In this paper, we consider the case in which extracted knowledge f has a decomposable structure; f( x)= g( x[ S 0], h( x[ S 1])) for some S 0, S 1⊆ S and Boolean functions g and h, where x[ I] denotes the projection of vector x on I. In order to detect meaningful decomposable structures, however, it is considered that the sizes | T| and | F| must be sufficiently large. In this paper, based on probabilistic analysis, we provide an index for such indispensable number of examples to detect decomposability; we claim that there exist many deceptive decomposable structures of ( T, F) if |T| |F|⩽2 n−1 . The computational results on synthetically generated data sets and real-world data sets show that the above index gives a good lower bound on the indispensable data size.