A decomposability index in logical analysis of data

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[S₀], h(x[S₁])) for some S₀,S₁⊆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.