Recursive Self-Improvement

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Algorithmic Unverifiability of Safety for Fixed and Recursively Self-Improving Systems
Jun 26, 2026Jose Pascual Gumbau MezquitaFormal Verification Self-Improving Agents
University Jaume I de Castell´o, Spain
The Red Queen Gödel Machine: Co-Evolving Agents and Their Evaluators
Jun 24, 2026Alex Iacob, Andrej Jovanović, William F. Shen +12Evolutionary Optimization AI Agent Evaluation
University of Cambridge · Flower Labs · NVIDIA +2
Recursive Self-Evolving Agents via Held-Out Selection
Jun 17, 2026Michael Nguyen, Quoc Nguyen, Paul VuongLLM Agent Evaluation LLM Agent Workflow Optimization
School of Information Technology Monash University Malaysia
From AGI to ASI
Jun 10, 2026Tim Genewein, Matija Franklin, Alexander Lerchner +11Self-Improving Agents Recursive Self-Improvement
Google DeepMind · University of Waterloo (work conducted while at Google DeepMind) · Australian National University +1
From 0-to-1 to 1-to-N: Reproducible Engineering Evidence for MetaAI Recursive Self-Design
Jun 8, 2026Dun Li, Jiatao Li, Hongzhi LiLLM Agent Self-Improvement Self-Improving Agents
Department of Computing, The Hong Kong Polytechnic University, Hong Kong SAR, China · Merchant Marine College, Shanghai Maritime University, Shanghai 201306, China · College of Big Data and Artificial Intelligence, Chizhou University, Chizhou 247100, China
A Relative-Computability Theory of Self-Improving Agents
Apr 9, 2026Chien-Ping LuSelf-Improving Agents Recursive Self-Improvement
Polaris: A Gödel Agent Framework for Small Language Models through Experience-Abstracted Policy Repair
Mar 24, 2026Aditya Kakade, Vivek Srivastava, Shirish KarandeSmall Language Models Self-Improving Agents
TCS Research, India
SGM: A Statistical Godel Machine for Risk-Controlled Recursive Self-Modification
Oct 11, 2025Xuening Wu, Shenqin Yin, Yanlan Kang +4Automated Machine Learning Self-Improving Agents
Fudan University · University of California, San Diego · Tongji University

Jun 26, 2026, cs.LOJ/K move · Enter open · S save

Algorithmic Unverifiability of Safety for Fixed and Recursively Self-Improving Systems

Jose Pascual Gumbau Mezquita

University Jaume I de Castell´o, Spain

Formal Verification Self-Improving Agents AI Safety Recursive Self-Improvement

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We establish mathematical limits of algorithmic safety verification for Turing-complete self-modifying systems, the class in which recursive self-improvement takes place, both for a fixed system and across its own modification. Statically, no verifier is sound, complete and tractable: over unbounded domains by Rice's and Gödel's theorems, over all finite configurations by Trakhtenbrot's theorem, and over succinctly described finite environments because verifying a policy against an adversary is coNP-complete and synthesising one is PSPACE-complete. Dynamically, we model one step of self-modification as a computable transformation of code and ask whether a safety property survives it. If the transformation depends only on behaviour, this is Rice's theorem one level up; if it reads the code, as self-modification does, the question is no longer semantic, yet the same s-m-n reduction works inside a class of behaviourally identical programs and inherits the halting degree. One step is never harder than the property; persistence along the whole trajectory can be

Π^0_2

-complete. Certification by a total algorithm is possible only for transformations of restricted expressivity, not merely for systems that stop changing. No tower of supervisors helps, and every total supervisor errs on an undecidable set of systems. For effectively pointwise properties, every faithful bounded scheme that certifies on finite behavioural evidence admits evolution traces certified at every stage while the property is violated. What survives is exact: a monitor that raises an alarm on violation semidecides it, and comparison against a frozen reference keeps the full theory.

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