AI Claims to Break the Exponential Time Hypothesis, Shaking Theoretical Computer Science

A basic assumption that has guided theoretical computer science for nearly three decades may have just fallen, and an AI model is said to have done it. OpenAI says its models have disproved the exponential time hypothesis (ETH), and rival Anthropic says one of its models reached an almost identical result on the same day. Researchers are calling it a potential breakthrough, but they also stress that the proofs have not yet been fully confirmed by humans.

How does AI impact theoretical computer science?

According to Science, OpenAI released solutions to three problems spanning math, computer science and physics on Monday, Oct. 26. The one widely seen as most consequential was an apparent refutation of the ETH, proposed in 1999 by Russell Impagliazzo and Ramamohan Paturi of the University of California San Diego.

Hours after OpenAI posted its results, Anthropic said one of its models had found an almost identical ETH proof on Oct. 16, the same day as OpenAI, though by a different route, Science reported.

Reactions from researchers were striking:

• “Clearly this is a major breakthrough,” Holger Dell of the IT University of Copenhagen told Science.
• “This came as a complete surprise, a shock,” Fedor Fomin of the University of Bergen told New Scientist. “Nobody I know expected it.”
• “Assuming that it is confirmed, this is an enormous breakthrough,” said Leslie Goldberg of the University of Oxford, per New Scientist.

What is the exponential time hypothesis?

The ETH concerns a classic puzzle called satisfiability. Science describes it as a set of on/off switches and a list of rules: is there a way to set every switch so all rules are satisfied? Each added switch doubles the number of possible combinations, so the work grows exponentially. Clever shortcuts exist, but none had escaped that exponential climb, and the ETH held that none ever would.

The puzzle matters beyond theory because many practical problems, from chip design to scheduling to cryptography, can be translated into this format, Science noted.

OpenAI says its models devised an algorithm that beats that assumed speed limit, New Scientist reported.

Is the result confirmed?

Not fully. Both companies say Lean, a proof-checking programming language, formally verified their results, but no human has yet fully confirmed the proofs, according to Science. “As far as I’m concerned, this is not 100% certain,” Impagliazzo, who is part of a team OpenAI asked to review the work, told the magazine.

Stefan Szeider of TU Wien told New Scientist he expects the result to withstand scrutiny, but said researchers still need to check that the formally proven statement means what its authors say it means.

The review team, which also includes Paturi and MIT’s Ryan Williams, plans to have a working group produce a fully human-readable proof, followed by a virtual review conference before the end of the year and possibly an in-person one in 2027, Science reported.

What does it mean for P versus NP?

The ETH is linked to the famous P versus NP problem, which asks whether every problem whose answer is easy to check is also easy to solve. Experts quoted by both outlets cautioned against overreading the news. “This doesn’t have a direct impact on P versus NP, so we should not view it as progress towards P versus NP,” Goldberg told New Scientist. Science reported that the new results only just break the ETH and leave P versus NP far out of reach.

Practical impact is also limited for now. Anthropic’s paper says its approach is unlikely to beat brute force on any puzzle with fewer than about 10 million switches, according to Science.

Why are researchers worried about how it was released?

The bigger shake-up may be to the research literature. Anthropic counted more than 800 papers that prove results only by assuming the ETH or a stronger version, Science reported. Fomin told New Scientist a textbook he wrote would become obsolete if the finding holds.

How the result reached the public has also drawn debate. OpenAI asked a hand-picked group of academics to shepherd the work, which some found uneasy. An independent advisory group of mathematicians argued that companies should not handpick reviewers for their own results, Science reported. Magnus Wahlström of Royal Holloway described the trend to New Scientist as “mathematics done by press.”

Others welcomed the disruption. “This result destroys a lot, but it also opens many new opportunities and new questions that nobody expected,” Fomin said.

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