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Even-if Explanations: Formal Foundations, Priorities and Complexity

Gianvincenzo Alfano, Sergio Greco, Domenico Mandaglio, Francesco Parisi, Reza Shahbazian, Irina Trubitsyna

TL;DR

This work tackles the interpretability of local semifactual explanations under the even-if thinking paradigm and contrasts them with counterfactual explanations. It analyzes the computational complexity of semifactual queries across three boolean classifier families—perceptrons, free binary decision diagrams (FBDDs), and multilayer perceptrons (MLPs)—revealing that PTIME solvability holds for perceptrons and FBDDs while MLPs incur higher complexity. A preference-based BCMp framework is introduced to tailor semifactuals and counterfactuals to user priorities, with PTIME algorithms for linear preferences in restricted classes and coNP-level verification results. The results advance user-centric interpretability and provide a foundation for regulation-aware explanations in AI systems.

Abstract

EXplainable AI has received significant attention in recent years. Machine learning models often operate as black boxes, lacking explainability and transparency while supporting decision-making processes. Local post-hoc explainability queries attempt to answer why individual inputs are classified in a certain way by a given model. While there has been important work on counterfactual explanations, less attention has been devoted to semifactual ones. In this paper, we focus on local post-hoc explainability queries within the semifactual `even-if' thinking and their computational complexity among different classes of models, and show that both linear and tree-based models are strictly more interpretable than neural networks. After this, we introduce a preference-based framework that enables users to personalize explanations based on their preferences, both in the case of semifactuals and counterfactuals, enhancing interpretability and user-centricity. Finally, we explore the complexity of several interpretability problems in the proposed preference-based framework and provide algorithms for polynomial cases.

Even-if Explanations: Formal Foundations, Priorities and Complexity

TL;DR

This work tackles the interpretability of local semifactual explanations under the even-if thinking paradigm and contrasts them with counterfactual explanations. It analyzes the computational complexity of semifactual queries across three boolean classifier families—perceptrons, free binary decision diagrams (FBDDs), and multilayer perceptrons (MLPs)—revealing that PTIME solvability holds for perceptrons and FBDDs while MLPs incur higher complexity. A preference-based BCMp framework is introduced to tailor semifactuals and counterfactuals to user priorities, with PTIME algorithms for linear preferences in restricted classes and coNP-level verification results. The results advance user-centric interpretability and provide a foundation for regulation-aware explanations in AI systems.

Abstract

EXplainable AI has received significant attention in recent years. Machine learning models often operate as black boxes, lacking explainability and transparency while supporting decision-making processes. Local post-hoc explainability queries attempt to answer why individual inputs are classified in a certain way by a given model. While there has been important work on counterfactual explanations, less attention has been devoted to semifactual ones. In this paper, we focus on local post-hoc explainability queries within the semifactual `even-if' thinking and their computational complexity among different classes of models, and show that both linear and tree-based models are strictly more interpretable than neural networks. After this, we introduce a preference-based framework that enables users to personalize explanations based on their preferences, both in the case of semifactuals and counterfactuals, enhancing interpretability and user-centricity. Finally, we explore the complexity of several interpretability problems in the proposed preference-based framework and provide algorithms for polynomial cases.
Paper Structure (8 sections, 4 theorems, 1 figure, 1 table, 2 algorithms)

This paper contains 8 sections, 4 theorems, 1 figure, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Contributions
  3. Preliminaries
  4. Classification Models
  5. Complexity Classes
  6. Even-if Explanations
  7. Preferences over Explanations
  8. Final Discussion

Key Result

Theorem 1

MCR is $i)$ in PTIME for FBDDs and perceptrons, and $ii)$ NP-complete for MLPs.

Figures (1)

  • Figure 1: (a) Binary classification model ${\cal M}: step({\bf x}\cdot [-2,2,0]+1)$ of Example \ref{['ex:intro1']} representing the hiring scenario. The binary feature $f_1$ (resp., $f_2$ and $f_3$) represents part-time employment contract (resp., salary lower than 5K$, and on site-working). Crosses (resp., circles) on the corners of the green (resp., red) area correspond to instances where the model outputs $1$ (resp., $0$). (b) FBDD model ${\cal M}=(V,E,\lambda_V,\lambda_E)$ of Example \ref{['ex:FBDD']} with root $\textsf{t}$ and $\lambda_V(\textsf{t})=3$. (c) FBDD model ${\cal M}'=(V',E',\lambda_{V'},\lambda_{E'})$ computed at Line 1 of Algorithm \ref{['alg:FBDD']}. (d) Graph $\cal N$ obtained at Line 2 of Algorithm \ref{['alg:FBDD']}. Squared nodes represent leaf nodes ($\top$ for ${\cal M}(\cdot)=1$, and $\bot$ for ${\cal M}(\cdot)=0$).

Theorems & Definitions (18)

  • Example 1
  • Example 2
  • Definition 1: Counterfactual
  • Example 3
  • Theorem 1: NIPS20
  • Definition 2: Semifactual
  • Theorem 2
  • Definition 3: Preference rule
  • Definition 4: BCMp framework
  • Example 4
  • ...and 8 more