Incompleteness of AI Safety Verification via Kolmogorov Complexity

The verification of artificial intelligence safety is fundamentally limited by intrinsic information-theoretic boundaries, rather than just computational complexity or model expressiveness. Research has demonstrated that these limitations stem from the inherent incompleteness of verifying policy compliance in AI systems, which can be formalized as a problem of verifying encoded system behavior. This incompleteness is rooted in the concept of Kolmogorov complexity, which measures the complexity of an object as the length of its shortest description. As a result, ensuring that AI systems adhere to formal safety and policy constraints is a challenging task that cannot be overcome by simply increasing computational power or model sophistication¹. This has significant implications for the development and deployment of AI systems in safety-critical domains, where regulatory compliance is paramount. So what matters to practitioners is that understanding these fundamental limits can inform the development of more effective verification strategies and provide a competitive advantage in navigating evolving regulatory requirements.