OpenAI's GPT-5.6 Pro Solves 30-Year-Old Complexity Theory Problem
Key Takeaways
- ▸OpenAI's GPT-5.6 Pro generated a novel proof proving Completeness of Canonical Closure Representations is coNP-complete, resolving an open problem unsolved for 30 years across multiple mathematical domains
- ▸The theorem establishes computational barriers for Horn logic, formal concept analysis, and database theory, proving polynomial-time algorithms cannot exist for several important problems unless P = NP
- ▸This achievement demonstrates large language models' emerging capability to solve fundamental theoretical problems in mathematics, opening new possibilities for AI-augmented research in academia
Summary
A three-decade-old open problem in complexity theory has been resolved with the assistance of OpenAI's GPT-5.6 Pro. The problem, which concerns the completeness of canonical closure representations across Horn logic, formal concept analysis, convex geometries, and database theory, was finally proven to be coNP-complete. Mathematician Mikhail Babin used GPT-5.6 Pro through ChatGPT to generate the main proof and independently verified the mathematical claims, publishing the findings in a peer-reviewed format.
The proof establishes that unless P = NP, complete canonical lists cannot be generated in polynomial time even for acyclic convex geometries—a finding with significant algorithmic implications. The theorem makes several related problems coNP-complete, including Characteristic Models Identification in Horn formulas, the Duquenne-Guigues basis in formal concept analysis, and functional dependency equivalence in relational databases. These results demonstrate fundamental computational barriers that extend across multiple mathematical and computational domains.
This milestone represents a watershed moment in AI-assisted mathematical research. The successful resolution of a longstanding open problem signals that large language models have reached a level of sophistication sufficient to tackle complex theoretical mathematics, potentially reshaping how cutting-edge research is conducted across pure mathematics and computer science.
Editorial Opinion
The resolution of this long-standing theoretical problem marks a significant inflection point in mathematics and AI. While human verification remains essential, the fact that GPT-5.6 Pro could generate a rigorous proof of this magnitude suggests we're entering an era where AI systems function as genuine research collaborators on fundamental problems. This has profound implications not only for the practical speed of mathematical discovery but also for how we conceive of artificial intelligence's role in expanding human knowledge at the frontier of theoretical computer science.
