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The Limits of AI Explainability: An Algorithmic Information Theory Approach

Shrisha Rao

TL;DR

This work establishes a rigorous, algorithmic-information-theory framework for the fundamental limits of AI explainability by framing explanations as simpler representations bounded by Kolmogorov complexity. It introduces key constructs such as the Explanation Error Function $\\varepsilon_f(k)$, the Explanation Complexity $\\kappa_f(\\delta)$, and the Complexity Gap Theorem, deriving Lipschitz-function bounds $\\kappa_f(\\delta)=O((L/\\delta)^d\\log(L/\\delta))$ and highlighting the exponential dependence on input dimension. The paper also unpacks local vs global explainability, connects the theory to practical model classes (linear models, decision trees, neural networks), and integrates an information-theoretic perspective via rate-distortion bounds. Finally, it analyzes regulatory implications, revealing a trilemma that makes simultaneous unrestricted capability, human-interpretable explanations, and negligible error infeasible, and proposes principled governance strategies grounded in feasibility regions and purpose-specific requirements.

Abstract

This paper establishes a theoretical foundation for understanding the fundamental limits of AI explainability through algorithmic information theory. We formalize explainability as the approximation of complex models by simpler ones, quantifying both approximation error and explanation complexity using Kolmogorov complexity. Our key theoretical contributions include: (1) a complexity gap theorem proving that any explanation significantly simpler than the original model must differ from it on some inputs; (2) precise bounds showing that explanation complexity grows exponentially with input dimension but polynomially with error tolerance for Lipschitz functions; and (3) a characterization of the gap between local and global explainability, demonstrating that local explanations can be significantly simpler while maintaining accuracy in relevant regions. We further establish a regulatory impossibility theorem proving that no governance framework can simultaneously pursue unrestricted AI capabilities, human-interpretable explanations, and negligible error. These results highlight considerations likely to be relevant to the design, evaluation, and oversight of explainable AI systems.

The Limits of AI Explainability: An Algorithmic Information Theory Approach

TL;DR

This work establishes a rigorous, algorithmic-information-theory framework for the fundamental limits of AI explainability by framing explanations as simpler representations bounded by Kolmogorov complexity. It introduces key constructs such as the Explanation Error Function , the Explanation Complexity , and the Complexity Gap Theorem, deriving Lipschitz-function bounds and highlighting the exponential dependence on input dimension. The paper also unpacks local vs global explainability, connects the theory to practical model classes (linear models, decision trees, neural networks), and integrates an information-theoretic perspective via rate-distortion bounds. Finally, it analyzes regulatory implications, revealing a trilemma that makes simultaneous unrestricted capability, human-interpretable explanations, and negligible error infeasible, and proposes principled governance strategies grounded in feasibility regions and purpose-specific requirements.

Abstract

This paper establishes a theoretical foundation for understanding the fundamental limits of AI explainability through algorithmic information theory. We formalize explainability as the approximation of complex models by simpler ones, quantifying both approximation error and explanation complexity using Kolmogorov complexity. Our key theoretical contributions include: (1) a complexity gap theorem proving that any explanation significantly simpler than the original model must differ from it on some inputs; (2) precise bounds showing that explanation complexity grows exponentially with input dimension but polynomially with error tolerance for Lipschitz functions; and (3) a characterization of the gap between local and global explainability, demonstrating that local explanations can be significantly simpler while maintaining accuracy in relevant regions. We further establish a regulatory impossibility theorem proving that no governance framework can simultaneously pursue unrestricted AI capabilities, human-interpretable explanations, and negligible error. These results highlight considerations likely to be relevant to the design, evaluation, and oversight of explainable AI systems.
Paper Structure (21 sections, 33 theorems, 135 equations)

This paper contains 21 sections, 33 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Theoretical Model and Fundamental Limits of Explainability
  3. Foundational Definitions
  4. Core Explainability Measures
  5. Fundamental Limits
  6. Function Classes and Practical Implications
  7. Smooth Function Explainability
  8. Model-Specific Explainability
  9. Relating Theoretical and Practical Complexity
  10. Linear Models
  11. Decision Trees
  12. Neural Networks
  13. Information-Theoretic Perspective
  14. Local vs. Global Explainability
  15. Practical Guidelines
  16. ...and 6 more sections

Key Result

Theorem 2.6

If $U_1$ and $U_2$ are two universal Turing machines, then there exists a constant $c_{U_1,U_2}$ such that for any function $g$: where $K_{U_i}(g)$ denotes the Kolmogorov complexity of $g$ with respect to machine $U_i$.

Theorems & Definitions (100)

  • Definition 2.1: AI System
  • Definition 2.2: Explanation
  • Definition 2.3: Kolmogorov Complexity
  • Definition 2.4: Computable Real Functions
  • Definition 2.5
  • Theorem 2.6: Invariance Theorem
  • proof
  • Remark 2.7
  • Definition 2.8: Interpretability Class
  • Lemma 2.9: Size of Interpretability Class
  • ...and 90 more