{"slug": "open-no-more", "title": "Open No More", "summary": "OpenAI released 722 manuscripts solving 372 major open problems in mathematics, including results from theoretical computer science, according to a GitHub overview posted by the company. The release includes a paper by Adam Bouland, Andrew Huang, Anand Natarajan, Itay Shalit and Avishay Tal giving an oracle where BQP is not in IP, and a paper by Eshan Chattopadhyay, Pooya Hatami, Chin Ho Lee, Shachar Lovett, Avishay Tal and Emanuele Viola giving new exponential correlation bounds for polynomials used to build a pseudorandom generator against AC^0[⊕] circuits. Blogger Lance Fortnow said he used Claude to work out and write up details of an alternative proof of Toda's theorem, and advised readers to \"feel free to use AI but verify the proof.", "body_md": "I wrote the post below last week. That was a quaint and quiet time. Last night OpenAI released a [treasure trove](https://github.com/openai/math/blob/main/overview.pdf) of 722 manuscripts solving 372 major open problems in mathematics including from theoretical computer science:\n\nAdam Bouland, Andrew Huang, Anand Natarajan, Itay Shalit and Avishay Tal [posted a paper](https://arxiv.org/abs/2609.25680) giving an oracle where $\\mathrm{BQP}$ is not in $\\mathrm{IP}$ (interactive proofs). Now $\\mathrm{BQP}$ is in $\\mathrm{IP}$ since $\\mathrm{BQP}\\subseteq\\mathrm{PSPACE}=\\mathrm{IP}$, but the $\\mathrm{IP}=\\mathrm{PSPACE}$ proof doesn't relativize and Bouland et al. show you can even get an oracle that puts $\\mathrm{BQP}$ out of $\\mathrm{IP}$.\n\nThe paper also states \"Together with [recent work](https://arxiv.org/abs/2604.11952) due to Scott Aaronson, Anand Natarajan, Avishay Tal, and Ági Villányi, our work also gives the first oracle separation between IP and MIP, answering a question dating back to [Fortnow's thesis](https://bibbase.org/network/publication/fortnow-complexitytheoreticaspectsofinteractiveproofsystems-1989).\" $\\mathrm{MIP}$ is the set of languages with multi-prover interactive proofs.\n\nWhen I saw this paper, I pulled my PhD thesis off the shelf and indeed on page 40 I wrote \"What is the relation between MIP and IP? Is there, for instance, an oracle separating the two classes\".\n\nWhen I wrote the thesis in 1989 we didn't know yet that $\\mathrm{IP}=\\mathrm{PSPACE}$ and $\\mathrm{MIP}=\\mathrm{NEXP}$ so we really didn't have any idea whether multiple provers actually gave you more power than one prover. When László Babai, Carsten Lund and I [proved $\\mathrm{MIP}=\\mathrm{NEXP}$](https://link.springer.com/article/10.1007/BF01200056) a year later, we had strong evidence that $\\mathrm{IP}\\neq\\mathrm{MIP}$ since we believe that $\\mathrm{PSPACE}\\neq\\mathrm{NEXP}$. However since the proof that $\\mathrm{MIP}=\\mathrm{NEXP}$ doesn't relativize either, the question of the oracle separation between $\\mathrm{IP}$ and $\\mathrm{MIP}$ remained open until the Bouland et al. paper.\n\nFinally, Eshan Chattopadhyay, Pooya Hatami, Chin Ho Lee, Shachar Lovett, Avishay Tal and Emanuele Viola gave [new exponential correlation bounds for polynomials](https://arxiv.org/abs/2609.28839). The authors use that bound to give a new pseudorandom generator against $\\mathrm{AC}^0[\\oplus]$ circuits.\n\nWhen I saw the paper I realized one could use this generator to show that $\\text{Almost-}\\oplus\\mathrm{P}=\\mathrm{BPP}^{\\oplus\\mathrm{P}}$, answering a question I had wondered about in the 90s. Here $\\text{Almost-}\\oplus\\mathrm{P}$ is the class of languages $L$ such that $L\\in\\oplus\\mathrm{P}^R$ with probability one for a random oracle $R$. This in turn could be used to give an alternative proof of Toda's theorem. [Ken Regan and Jim Royer](https://doi.org/10.1007/BF01303057) showed that relative to a random oracle the polynomial-time hierarchy is contained in $\\oplus\\mathrm{P}$, so $\\mathrm{PH}\\subseteq\\text{Almost-}\\oplus\\mathrm{P}=\\mathrm{BPP}^{\\oplus\\mathrm{P}}$. It would take me a long time to work out and write up the details so I had [Claude do it for me](https://arxiv.org/abs/2609.35326).\n\nI still have many more open problems, see for example [my survey of open oracle questions](https://dl.acm.org/doi/10.1145/3494656.3494663). I'd be happy to see them solved. Feel free to use AI but verify the proof. You too could get mentioned on this blog.", "url": "https://wpnews.pro/news/open-no-more", "canonical_source": "https://blog.computationalcomplexity.org/2026/10/open-no-more.html", "published_at": "2026-10-07 15:42:22+00:00", "updated_at": "2026-10-07 15:49:18.680320+00:00", "lang": "en", "topics": ["artificial-intelligence", "ai-research", "large-language-models", "ai-tools"], "entities": ["OpenAI", "Adam Bouland", "Andrew Huang", "Anand Natarajan", "Itay Shalit", "Avishay Tal", "Eshan Chattopadhyay", "Claude"], "also_reported_by": [], "alternates": {"html": "https://wpnews.pro/news/open-no-more", "markdown": "https://wpnews.pro/news/open-no-more.md", "text": "https://wpnews.pro/news/open-no-more.txt", "jsonld": "https://wpnews.pro/news/open-no-more.jsonld"}}