Interpretable DNFs

A classifier is considered interpretable if each of its decisions has an explanation which is small enough to be easily understood by a human user. A DNF formula can be seen as a binary classifier $\kappa$ over boolean domains. The size of an explanation of a positive decision taken by a DNF $\kappa$ is bounded by the size of the terms in $\kappa$, since we can explain a positive decision by giving a term of $\kappa$ that evaluates to true. Since both positive and negative decisions must be explained, we consider that interpretable DNFs are those $\kappa$ for which both $\kappa$ and $\overline{\kappa}$ can be expressed as DNFs composed of terms of bounded size. In this paper, we study the family of $k$-DNFs whose complements can also be expressed as $k$-DNFs. We compare two such families, namely depth-$k$ decision trees and nested $k$-DNFs, a novel family of models. Experiments indicate that nested $k$-DNFs are an interesting alternative to decision trees in terms of interpretability and accuracy.