Incompleteness of AI Safety Verification via Kolmogorov Complexity
Abstract
Ensuring that artificial intelligence (AI) systems satisfy formal safety and policy constraints is a central challenge in safety-critical domains. While limitations of verification are often attributed to combinatorial complexity and model expressiveness, we show that they arise from intrinsic information-theoretic limits. We formalize policy compliance as a verification problem over encoded system behaviors and analyze it using Kolmogorov complexity. We prove an incompleteness result: for any fixed sound computably enumerable verifier, there exists a threshold beyond which true policy-compliant instances cannot be certified once their complexity exceeds that threshold. Consequently, no finite formal verifier can certify all policy-compliant instances of arbitrarily high complexity. This reveals a fundamental limitation of AI safety verification independent of computational resources, and motivates proof-carrying approaches that provide instance-level correctness guarantees.
Keywords
Cite
@article{arxiv.2604.04876,
title = {Incompleteness of AI Safety Verification via Kolmogorov Complexity},
author = {Munawar Hasan},
journal= {arXiv preprint arXiv:2604.04876},
year = {2026}
}