{"slug": "i-guess-we-live-in-the-p-np-world-after-all", "title": "I guess we live in the P = NP world after all", "summary": "A blog post argues that recent AI advances, including OpenAI's claimed solution to the Navier-Stokes Millennium Prize Problem, suggest the P = NP question may no longer matter because AI is automating mathematical creativity. The post cites Terence Tao's recorded remark that P = NP would probably be the last Millennium Prize Problem to fall, and notes the author has personally had AIs answer publishable computer science questions. The author concludes that if AIs automate creativity and quickly prove P = NP via a polynomial-time algorithm for SAT, that would be the funniest possible world.", "body_md": "# I guess we live in the P = NP world after all\n\n## Abstract\n\nOne of the most important questions in computer science is whether it is intrinsically more time-consuming to come up with an answer than to verify one. This is the so-called question: if they’re equal it’s as “easy” to come up with an answer as it is to verify the answer. That would be pretty weird: doesn’t coming up with answers require more creativity? So most computer scientists are pretty sure . But maybe not! It’s a big question. Just recently it seemed to me, maybe it doesn’t matter anymore?\n\n## 1 A Magic Answer Box\n\nImagine we had a machine that could answer any question. Better than that though, it gives you an incontrovertible proof backing its answer up.\n\nA machine like that would be easy to build: It’s just a computer! It doesn’t even need any fancy AI. Here’s how it works: You give it a (mathematical) statement , and some ginormous number, . It tells you whether there is a proof of of length at most . If there is, it can hand you the proof.\n\nAnd it’s easy! The magic answer box just generates all possible strings of length at most , and checks if any of them are valid proofs of . Voilà! Creativity is automated.\n\n## 2 Complexity\n\nThe only issue with the magic answer box is that it’s slow. The only general algorithm we know that is guaranteed to work is very, very slow. Sure, checking one is easier. But we don’t know an efficient algorithm for finding the proofs. It all sounds like the question!\n\nAs one famous meat-mathematician once wrote to another:\n\n“The mental work of a mathematician concerning Yes-or-No questions could be completely replaced by a machine.”\n\nSo, being true would be crazy - we might then have an efficient algorithm for automating mathematics!\n\n## 3 One Million Dollars\n\nWith the recent smashing of Navier-Stokes by [OpenAI](https://openai.com/index/navier-stokes-solution/), I’ve had the\n[Millennium Prize Problems](https://www.claymath.org/millennium-problems/) on my mind.\nThey serve as dramatic lighthouses for mathematical endeavour.\nThese are the amazing, rock hard problems that you get a million dollars for settling (with a proof!).\n\nThe difficulty with having them on my mind though, is that I don’t know what any of them really mean. Except,\nof course for the  question. That one, I know a little about. In recent weeks, I’ve heard people\nsay that it is likely the hardest of them all. Even the magician mathematician Terence\nTao is [on record](https://www.youtube.com/watch?v=PtsrAw1LR3E&t=2727s) saying that it would probably be the last problem to fall.\n\nSo I’ve dared to wonder recently: maybe I’ll know for sure whether soon.\n\n## 4 Brainworm\n\nOf course, you can guess the dreadful imposition that comes blacking in one’s mind: We have automated creativity. An AI has settled Navier-Stokes. I personally have had AIs answer many publishable questions in computer science. I can’t get it out of my head, it turns out, we live in the world!\n\n## 5 Of Course Not Quite\n\nNow, sure, yes, . Or maybe not. And Gödel’s magic-answer-box machine is exhaustive, and AI is not. Maybe that makes all the difference. The AIs might fail on some problems, and the magic-answer-box would be guaranteed not to “miss” any proof. But it does make me wonder. Perhaps every natural mathematical question whose answer is within reach of human understanding is also within reach of AI. Maybe of course, and not perhaps at all.\n\n## 6 I See a Darkness\n\nSo why do I care if  anymore? It feels like the only millennium problem I understood, is maybe just another\ncuriosity now. Perhaps the funniest world would\nbe the one where AIs automate creativity, and quickly prove , via an  time algorithm for [SAT](https://en.wikipedia.org/wiki/Satisfiability).", "url": "https://wpnews.pro/news/i-guess-we-live-in-the-p-np-world-after-all", "canonical_source": "https://nicknash.github.io/posts/pnp/", "published_at": "2026-09-18 18:02:11+00:00", "updated_at": "2026-09-18 18:25:46.867019+00:00", "lang": "en", "topics": ["artificial-intelligence", "ai-research"], "entities": ["OpenAI", "Navier-Stokes", "Millennium Prize Problems", "Terence Tao", "P = NP", "SAT"], "alternates": {"html": "https://wpnews.pro/news/i-guess-we-live-in-the-p-np-world-after-all", "markdown": "https://wpnews.pro/news/i-guess-we-live-in-the-p-np-world-after-all.md", "text": "https://wpnews.pro/news/i-guess-we-live-in-the-p-np-world-after-all.txt", "jsonld": "https://wpnews.pro/news/i-guess-we-live-in-the-p-np-world-after-all.jsonld"}}