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Interpretable DNFs

Martin C. Cooper, Imane Bousdira, Clément Carbonnel

TL;DR

This work reframes interpretability for binary boolean classifiers as requiring bounded abductive and contrastive explanations, tying interpretability to the ability to express both a function κ and its complement ¬κ as $k$-DNFs. It proves a fundamental bound: any $k$-AXp-interpretable classifier admits a $k$-DNF with at most $k^k$ terms, and introduces a graph-theoretic condition based on induced matchings that guarantees $k$-AXp-interpretability. The authors propose nested $k$-DNFs, a concrete, structured family of $k$-DNFs that are interpretable and expressive (every $k$-var boolean function is nestable), and they provide a practical heuristic to learn them. Empirical results on diverse datasets show that nested $k$-DNFs can match or exceed depth-$k$ decision trees in accuracy while yielding smaller, more interpretable representations, highlighting nested $k$-DNFs as a viable alternative to trees for interpretable learning. Limitations include non-invariance under complementation and bounds on literals, suggesting avenues for richer, future interpretable DNFs and improved learning algorithms.

Abstract

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 $κ$ over boolean domains. The size of an explanation of a positive decision taken by a DNF $κ$ is bounded by the size of the terms in $κ$, since we can explain a positive decision by giving a term of $κ$ that evaluates to true. Since both positive and negative decisions must be explained, we consider that interpretable DNFs are those $κ$ for which both $κ$ and $\overlineκ$ 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.

Interpretable DNFs

TL;DR

This work reframes interpretability for binary boolean classifiers as requiring bounded abductive and contrastive explanations, tying interpretability to the ability to express both a function κ and its complement ¬κ as -DNFs. It proves a fundamental bound: any -AXp-interpretable classifier admits a -DNF with at most terms, and introduces a graph-theoretic condition based on induced matchings that guarantees -AXp-interpretability. The authors propose nested -DNFs, a concrete, structured family of -DNFs that are interpretable and expressive (every -var boolean function is nestable), and they provide a practical heuristic to learn them. Empirical results on diverse datasets show that nested -DNFs can match or exceed depth- decision trees in accuracy while yielding smaller, more interpretable representations, highlighting nested -DNFs as a viable alternative to trees for interpretable learning. Limitations include non-invariance under complementation and bounds on literals, suggesting avenues for richer, future interpretable DNFs and improved learning algorithms.

Abstract

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 over boolean domains. The size of an explanation of a positive decision taken by a DNF is bounded by the size of the terms in , since we can explain a positive decision by giving a term of that evaluates to true. Since both positive and negative decisions must be explained, we consider that interpretable DNFs are those for which both and can be expressed as DNFs composed of terms of bounded size. In this paper, we study the family of -DNFs whose complements can also be expressed as -DNFs. We compare two such families, namely depth- decision trees and nested -DNFs, a novel family of models. Experiments indicate that nested -DNFs are an interesting alternative to decision trees in terms of interpretability and accuracy.

Paper Structure

This paper contains 10 sections, 9 theorems, 7 equations, 1 figure, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Short explanations imply few explanations
  4. Induced matchings and nested $k$-DNF
  5. Expressivity of nested $k$-DNFs
  6. Experiments
  7. Heuristic algorithm
  8. Datasets
  9. Results
  10. Conclusion and future work

Key Result

Lemma 1

A function $\kappa$ that is $k$-AXp-interpretable is also $k$-CXp-interpretable.

Figures (1)

  • Figure 1: The landscape of $k$-AXp-interpretable classifiers $\kappa$

Theorems & Definitions (25)

  • Definition 1
  • Definition 2
  • Lemma 1
  • proof
  • Definition 3
  • Proposition 1
  • proof
  • Example 1
  • Example 2
  • Theorem 1
  • ...and 15 more