OpenAI has released solutions to three blockbuster math problems that have long vexed researchers. One of them was the surprise rejection of the exponential time hypothesis (ETH), an enduring hypothesis in computer science that plays an important role in how we understand what is difficult to calculate.
As with previous deliveries of proofs from the border laboratory, the scientific communities reacted with a mixture of enthusiasm and dismay. Anthropic mathematician Levent Alpöge suggested that OpenAI was basing its solutions on pre-existing work, and hours later Anthropic then published its own rebuttal of ETH.
However, the OpenAI process is evolving. This time, the laboratory structured its versions differently for each of the three disciplines involved: the mathematicians submitted their work on GitHub; computer scientists formed a working group, demanded that the evidence be shared publicly and that OpenAI fund a conference to discuss it; physicists have formed a working group that will determine how their results will be published.
MIT Professor Ryan Williams is the computer scientist organizing his field’s response to this new discovery. Williams, whose work focuses on these questions of computational complexity, has criticized OpenAI’s approach in the past, calling its early October proof archive a “DDOS attack on mathematicians.”
However, he agreed to collaborate with Frontier Labs to help come up with ideas vital to the field, he told TechCrunch. The lab asked him to help interpret the solution, alongside computer scientists Russell Impagliazzo and Ramamohan Paturi of UC San Diego, who first published ETH in 1999.
Their first step was to publish an open letter calling for feedback from scientists interested in these issues and to discuss standards surrounding AI contributions to their work.
“Some parts (of OpenAI’s proof) are, maybe you could say, routine, but then yes, there are two or three parts that look like aliens, and it was exactly those parts that I saw that made me think, wow, this is something that the whole community needs to work on together,” Williams told TechCrunch.
Do you disagree with P vs. NP?
ETH is part of a broader class of problems known as P versus NP, which have driven computer science and many other fields for decades. Reductively, researchers attempt to understand whether a problem whose solution can be easily verified (NP) can also be easily solved (P).
A simple example is the game Sodoku: once a solution to a puzzle is in place, it can be quickly checked, but finding the solution takes time. This dynamic is true for a range of important problems, and this idea underlies cryptographic systems: when you have the key to a coded message, you can reveal it, but in reality, cracking an effective code takes too long to be useful.
However, scientists have yet to prove whether this is still true. “P=NP means that not only can you check Sudoku solutions when they exist, you can simply generate them,” Williams said. “P=NP claims that this is true for every problem you can quickly check – if you can check it quickly, you can solve it quickly. So it seems too good to be true.”
This is why most scientists assume that P ≠ NP. ETH has provided proof of this. It basically says that even the fastest computer algorithm needs significantly more time to find a solution as the number of variables increases in a specific class of tricky logic puzzle.
OpenAI and Anthropic’s claims are that their models have developed algorithms that solve these puzzles in “subexponential” time, that is, faster than speeds previously thought possible. If true, they might have proven EFT wrong – disproved it, in scientific terms – and might have paved the way for algorithms that typically run P = NP.
“All computer scientists should probably give a little more weight to the plausibility of P = NP, but everyone’s weight may vary,” Williams said. “I think it challenges many of the intuitions we thought we had about… how tractable NP hard problems are. I also think it’s probably not the final answer either. I don’t think OpenAI’s algorithm can be the best algorithm you can get.”
The work that awaits computer scientists
ETH’s durability has led to its inclusion in many scientific papers over the past decade that will need to be revisited if AI labs claim to have proven it wrong, Williams said.
“That’s part of why I asked or pushed OpenAI to support a conference on this topic: it’s going to affect the work of many researchers, and we need to understand, okay, what happens next in addition to understanding the algorithms. We need to understand how the foundations should change from what we had before?” he said.
Before scientific concerns, there are questions of cybersecurity. Williams said an improved algorithm could imply that cryptosystems relying on certain key sizes could be considered flawed. So one of the first questions he asked Dan Roberts, OpenAI’s chief scientist, “was whether they had verified that this affected cryptosystems…was something insecure now? And apparently, they checked, and they were reasonably sure that it didn’t involve any breakage.”
This new AI paradigm of scientific discovery is still a work in progress, but Williams says he is optimistic despite real concerns, particularly about training the next generation of scientists. Being a computer scientist who does high-level mathematics may have made the adaptation easier for him than for pure mathematicians.
Still, he says AI “forces people to be a little more ambitious about the kind of work they do and to think more deeply about the question: Okay, if all the things I was working on are now low-hanging fruit for AI, how can I raise the bar for the kinds of scientific questions I want to ask?”
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