Monotone extensions of boolean data sets

Abstract

As a form of knowledge acquisition from data, we consider the problem of computing “monotone” extensions of a pair of data sets (T, F), where T (resp., F) ⊑ {0, 1}ⁿ is a set of positive (resp., negative) examples, and an extension is a Boolean function that is consistent with (T, F). A motivation of this study comes from an observation that real world data are often monotone (or at least approximately monotone), and in such cases it is natural to build monotone extensions. We define five types of monotone extensions called error-free, best-fit, consistent, robust and most-robust extensions to deal with various cases, in which (T, F) may contain errors and/or incomplete data. We provide polynomial time algorithms for constructing error-free, best-fit, consistent and robust extensions. For most-robust extensions, we show that the problem is solvable in polynomial time if AS(a) ≤ 1 holds for all a ∈ T∪F, where AS(a) denotes the set of missing bits in a vector a, but is NP-hard even if AS(a) < 2 holds for all a ∈ T∪F. We also give an approximability result for computing a most-robust extension.