Rectifying Mono-Label Boolean Classifiers

TL;DR

This paper introduces a unique rectification operator for mono-label Boolean classifiers that ensures compliance with background knowledge, with efficient algorithms for circuit and decision tree representations.

Contribution

It proves the uniqueness of the rectification operator and provides algorithms for computing rectified classifiers in linear and polynomial time for circuits and decision trees.

Findings

Unique rectification operator satisfying postulates.

Linear-time computation for Boolean circuit classifiers.

Polynomial-time computation for decision tree classifiers.

Abstract

We elaborate on the notion of rectification of a Boolean classifier $\Sigma$. Given $\Sigma$ and some background knowledge $T$, postulates characterizing the way $\Sigma$ must be changed into a new classifier $\Sigma \star T$ that complies with $T$ have already been presented. We focus here on the specific case of mono-label Boolean classifiers, i.e., there is a single target concept and any instance is classified either as positive (an element of the concept), or as negative (an element of the complementary concept). In this specific case, our main contribution is twofold: (1) we show that there is a unique rectification operator $\star$ satisfying the postulates, and (2) when $\Sigma$ and $T$ are Boolean circuits, we show how a classification circuit equivalent to $\Sigma \star T$ can be computed in time linear in the size of $\Sigma$ and $T$; when $\Sigma$ and $T$ are decision trees, a decision tree equivalent to $\Sigma \star T$ can be computed in time polynomial in the size of $\Sigma$ and $T$.