[Paper Review] A unified logical framework for explanations in classifier systems

Xinghan Liu, Emiliano Lorini|arXiv (Cornell University)|May 30, 2021

Explainable Artificial Intelligence (XAI)4 citations

TL;DR

This paper introduces a unified modal logic framework for explaining binary classifier decisions, enabling formalization of abductive, contrastive, and counterfactual explanations, as well as biases. It establishes completeness for two proof systems, proves NEXPTIME-completeness for satisfiability in the infinite-variable case and polynomial time in the finite-variable case, and extends the logic with dynamic and epistemic operators for model updates and uncertainty representation.

ABSTRACT

Recent years have witnessed a renewed interest in Boolean function in explaining binary classifiers in the field of explainable AI (XAI). The standard approach of Boolean function is propositional logic. We present a modal language of a ceteris paribus nature which supports reasoning about binary input classifiers and their properties. We study a family of classifier models, axiomatize it as two proof systems regarding the cardinality of the language and show completeness of our axiomatics. Moreover, we prove that satisfiability checking problem for our modal language is NEXPTIME-complete in the infinite-variable case, while it becomes polynomial in the finite-variable case. We furthermore identify an interesting NP fragment of our language in the infinite-variable case. We leverage the language to formalize counterfactual conditional as well as a variety of notions of explanation including abductive, contrastive and counterfactual explanations, and biases. Finally, we present two extensions of our language: a dynamic extension by the notion of assignment enabling classifier change and an epistemic extension in which the classifier's uncertainty about the actual input can be represented.

Motivation & Objective

• To provide a formal, logical foundation for explaining decisions made by binary classifier systems, addressing the lack of a unified framework in explainable AI (XAI).
• To formalize diverse explanation types—abductive, contrastive, counterfactual, and bias-related—within a single modal logic framework.
• To establish proof systems for the logic and prove their completeness with respect to two different language cardinalities (finite and infinite variables).
• To analyze the computational complexity of satisfiability checking in the proposed modal language, distinguishing between finite and infinite variable cases.
• To extend the logic with dynamic and epistemic operators to model classifier updates and uncertainty about inputs.

Proposed method

• Proposes a modal language based on a Kripke-style semantics over classifier models (CMs), where worlds represent input instances and accessibility relations encode perturbations (feature value changes).
• Defines key modal operators: $[ ext{Atm}] heta$ for 'after perturbing all atoms in Atm, θ holds', and $ ext{closest}_C(s, heta, X)$ to capture the nearest worlds under a perturbation set.
• Introduces formal definitions for explanation types: abductive explanation ($\mathsf{AXp}$) as a minimal term implying the output; contrastive explanation ($\mathsf{CXp}$) as a counterfactual condition; and bias detection ($\mathsf{Bias}$) via invariant failure under perturbations.
• Develops two proof systems—one for finite-variable and one for infinite-variable fragments—proving their completeness with respect to the semantics.
• Introduces an epistemic extension $\mathcal{L}^{\mathit{epi}}$ to represent classifier uncertainty, and a dynamic extension with assignment operators to model classifier updates.
• Uses model checking and bounded model construction to show that satisfiability is polynomial in the finite-variable case and NEXPTIME-complete in the infinite-variable case.

Experimental results

Research questions

• RQ1 Can a single modal logic framework unify the formalization of diverse explanation types in binary classifier systems?
• RQ2 What are the proof-theoretic and model-theoretic properties of the proposed logic, and are the proof systems complete?
• RQ3 What is the computational complexity of satisfiability checking in the proposed modal language, and how does it vary with variable finiteness?
• RQ4 Can the logic express counterfactual reasoning and detect biases in classifier decisions?
• RQ5 How can the logic be extended to model classifier updates and uncertainty about inputs?

Key findings

• The proposed modal logic framework successfully unifies the formalization of abductive, contrastive, counterfactual, and bias-related explanations within a single logical system.
• The logic is axiomatized in two proof systems—one for finite and one for infinite variable cases—both proven to be sound and complete.
• Satisfiability checking is NEXPTIME-complete in the infinite-variable case and polynomial-time decidable in the finite-variable case.
• An NP-complete fragment is identified within the infinite-variable case, enabling efficient reasoning for a significant subset of formulas.
• The epistemic extension allows modeling of classifier uncertainty, and the dynamic extension supports reasoning about classifier changes via assignment operators.
• The framework validates key explanation concepts: e.g., $\mathsf{CXp}(\lambda, c)$ is logically equivalent to a counterfactual conditional stating that perturbing $\lambda$'s atoms leads to a different output $c$.