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On the Complexity of Global Necessary Reasons to Explain Classification

Marco Calautti, Enrico Malizia, Cristian Molinaro

TL;DR

The paper addresses explaining classifier decisions via global necessary reasons expressed in a logic-based language, focusing on minimality under two preorders. It conducts a thorough complexity analysis for three classifier families—$\mathsf{BDD}$, $\mathsf{PRC}$, and $\mathsf{MLP}$—across two minimality notions. Key findings show that $\mathsf{PRC}$ and $\mathsf{BDD}$ yield low-complexity results (logspace or NL-complete), while $\mathsf{MLP}$ incurs higher complexity (co-$\mathsf{NP}$-complete for existence and $\mathsf{DP}$-complete for minimality). The work also shows that minimal global necessary reasons can be computed efficiently with appropriate oracles, motivating SAT-based strategies for harder cases. Together, these results establish a foundational understanding of global explanations and point to directions for future expressive languages and practical algorithms.

Abstract

Explainable AI has garnered considerable attention in recent years, as understanding the reasons behind decisions or predictions made by AI systems is crucial for their successful adoption. Explaining classifiers' behavior is one prominent problem. Work in this area has proposed notions of both local and global explanations, where the former are concerned with explaining a classifier's behavior for a specific instance, while the latter are concerned with explaining the overall classifier's behavior regardless of any specific instance. In this paper, we focus on global explanations, and explain classification in terms of ``minimal'' necessary conditions for the classifier to assign a specific class to a generic instance. We carry out a thorough complexity analysis of the problem for natural minimality criteria and important families of classifiers considered in the literature.

On the Complexity of Global Necessary Reasons to Explain Classification

TL;DR

The paper addresses explaining classifier decisions via global necessary reasons expressed in a logic-based language, focusing on minimality under two preorders. It conducts a thorough complexity analysis for three classifier families—, , and —across two minimality notions. Key findings show that and yield low-complexity results (logspace or NL-complete), while incurs higher complexity (co--complete for existence and -complete for minimality). The work also shows that minimal global necessary reasons can be computed efficiently with appropriate oracles, motivating SAT-based strategies for harder cases. Together, these results establish a foundational understanding of global explanations and point to directions for future expressive languages and practical algorithms.

Abstract

Explainable AI has garnered considerable attention in recent years, as understanding the reasons behind decisions or predictions made by AI systems is crucial for their successful adoption. Explaining classifiers' behavior is one prominent problem. Work in this area has proposed notions of both local and global explanations, where the former are concerned with explaining a classifier's behavior for a specific instance, while the latter are concerned with explaining the overall classifier's behavior regardless of any specific instance. In this paper, we focus on global explanations, and explain classification in terms of ``minimal'' necessary conditions for the classifier to assign a specific class to a generic instance. We carry out a thorough complexity analysis of the problem for natural minimality criteria and important families of classifiers considered in the literature.
Paper Structure (13 sections, 17 theorems, 12 equations, 1 table, 2 algorithms)

This paper contains 13 sections, 17 theorems, 12 equations, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Classification.
  4. Computational Complexity.
  5. Global Necessary Reasons
  6. Complexity Analysis
  7. The Case of Perceptrons
  8. The case of BDDs
  9. The case of MLPs
  10. Computing Minimal Global Necessary Reasons
  11. Related Work
  12. Summary and Outlook
  13. Detailed Proofs

Key Result

Lemma 4.0

Let $\mathcal{M}$ be a classifier, let $c$ be a class, and let $\phi$ be a global necessary reason. Then, for each preorder ${\preccurlyeq} {} \in \{ \leq,\subseteq \}$, $\phi$ is not$\preccurlyeq$- minimal iff there exists a literal $\ell$ such that $\phi \not \models \ell$ and $\ell$ is a global n

Theorems & Definitions (28)

  • Definition 3.1: Global necessary reasons
  • Definition 3.2: Minimal global necessary reason
  • Lemma 4.0
  • Theorem 4.1
  • Theorem 4.2
  • Theorem 4.3
  • Lemma 4.3
  • Theorem 4.4
  • Theorem 4.5
  • Theorem 4.6
  • ...and 18 more