OpenAI’s AI And A Complexity Theory Claim: What’s Proven So Far?

🔍 Read the full analysis: OpenAI’s AI And A Complexity Theory Claim: What’s Proven So Far? on ThorstenMeyerAI.com

Before you orderOffer from Amazon

Get the latest gadgets delivered free with Prime

• Fast, free delivery on millions of items
• Prime Video, Amazon Music and more included
• Member-only deals all year

Start your free Prime trial →Free trial for eligible customers · Cancel anytime

As an affiliate, we earn on qualifying purchases.Hide

OpenAI said October 26 that an unreleased internal AI model produced a proof refuting the Exponential Time Hypothesis (ETH), a widely used conjecture about the difficulty of solving certain logic problems. The claim and a machine-checked Lean formalization are public, but independent expert review has not yet concluded, and the formalization checks a slightly weaker runtime bound than the preprint’s sharpest claim.

The preprint’s central result is an algorithm that exactly counts satisfying assignments for any fixed-width k-SAT formula in subexponential time and polynomial memory. For 3-SAT, that would contradict ETH, which holds that no algorithm can solve the problem in subexponential time. The paper also says the result refutes the stronger Strong Exponential Time Hypothesis and three related hypotheses. These are claims in the preprint, not conclusions independently endorsed by the field.

OpenAI says its system found the disproof on October 16, and that a Lean formalization followed the next day. The preprint is dated October 17 and was revised October 25. The source article reports that the public Lean repository contains 3,977 files and about 710,000 lines, based on its own count; the formal statement itself is much shorter. Lean can check that a formal proof follows from its stated definitions and rules, but that check does not establish that the paper’s strongest bound has been formalized or that experts agree with the interpretation.

According to OpenAI’s announcement, an outside group will examine the computer-science result, with Russell Impagliazzo, Ramamohan Paturi and Ryan Williams among those leading the review. Impagliazzo and Paturi were two of ETH’s original authors. OpenAI says it has agreed to fund a conference on the result. The source article’s review of the formalization says Lean covers a slightly weaker runtime bound than the preprint’s sharpest one; the gap matters when judging exactly what has been mechanically verified.

What an ETH Refutation Would Change

If the preprint’s result survives scrutiny, it would overturn a major assumption used by theoretical computer scientists to show that many problems cannot be solved quickly. Researchers use ETH and related hypotheses to establish conditional lower bounds: statements of the form that an algorithm cannot run faster unless a widely held complexity assumption fails. A valid refutation would require revisiting results that rely on those assumptions, although it would not automatically invalidate every such result.

The proposed algorithm is still subexponential, not polynomial-time. The source describes its runtime as roughly 2 to the power of n divided by the square root of log n. That grows more slowly than a conventional exponential bound, but it remains far slower than polynomial time. OpenAI itself says practical improvements would require significant further research. The claim is therefore a possible change to theoretical understanding, not evidence that ordinary software or commercial optimization will immediately become faster.

The result also draws attention to a developing proof workflow: an AI system proposes a mathematical argument, a proof assistant checks a formal version, and domain specialists assess the claims and their implications. Formal checking can catch errors within the formalized proof, but it does not replace expert evaluation of how the formal result corresponds to the paper, which bound it establishes, or whether the result is correctly presented.

Products Worth Considering

Samsung 27-Inch Odyssey QD-OLED G8 4K 240Hz Gaming Monitor

AmazonView Latest Price

DJI Osmo Mobile 8 Standard Combo Phone Gimbal Stabilizer

AmazonView Latest Price

Jackery HomePower 3600 Plus Portable Power Station, 3584Wh, 3600W

AmazonView Latest Price

As an affiliate, we earn on qualifying purchases.

The Hypothesis Behind the Claim

k-SAT asks whether variables can be assigned true or false values so that every rule in a formula is satisfied; each rule involves at most k variables. A direct search over n variables has 2^n possible assignments. Researchers have developed algorithms that reduce the work, but ETH proposes that for 3-SAT the exponential character cannot be eliminated: solving it requires time of the form 2^(c·n) for some fixed positive c.

Impagliazzo, Paturi and Francis Zane introduced ETH in 2001. It is stronger than the unresolved P vs. NP question, which asks whether every problem whose proposed solutions can be checked efficiently can also be solved efficiently. Refuting ETH would not prove P = NP or settle P vs. NP; a subexponential algorithm can still take superpolynomial time. The source article also notes that views about ETH are not unanimous: OpenAI described it as widely believed, while computer scientist Scott Aaronson wrote in a 2017 survey that there was no consensus in the field that it is true.

The announcement is part of a wider OpenAI release of mathematical and scientific work. The company said it had three results, in mathematics, computer science and physics, and described the manuscripts as raw research data. The source article points to an earlier release on October 6 of more than 700 AI-generated manuscripts, followed by corrections to some submissions. That history gives independent checking added relevance, but it does not by itself confirm or disprove the present ETH claim.

What Review Must Still Establish

Independent review is pending. The public preprint and Lean code make the claims open to examination, but the source material does not report a completed assessment or endorsement by the outside reviewers. It is not yet clear whether reviewers will accept the paper’s argument, its interpretation of the result, or its claimed implications for the stronger and related hypotheses.

A second unresolved point is the difference between the paper’s strongest runtime bound and the one checked in Lean. The source says the formalization establishes a slightly weaker bound, but does not provide enough detail here to quantify that difference or explain whether it affects the paper’s main conclusion. Until reviewers compare the manuscript with the formal statement, readers should distinguish the paper’s full claim from the result checked by the proof assistant.

It also remains uncertain whether any version of the proposed algorithm could lead to practical gains. The source provides no implementation or benchmark showing improved performance on real-world SAT instances. A theoretical asymptotic result alone does not establish such an effect, and OpenAI says further research would be needed.

The Independent Review Process

The outside group led by Impagliazzo, Paturi and Williams is expected to examine the computer-science result. OpenAI has said it will support a conference on the work, which could provide a venue for specialists to question the proof, the formalization and the consequences claimed in the preprint. No review deadline or conference date is specified in the supplied material.

The next meaningful milestones are a public account of the reviewers’ findings and clarification of how the Lean-checked statement relates to the paper’s sharpest bound. Until those steps are reported, the accurate description is that OpenAI has announced a potentially major result, while independent verification and the practical significance remain unresolved.

Key Questions

Has ETH been disproved?

Not independently confirmed. OpenAI says its internal model produced a proof, and the preprint claims to refute ETH. Outside experts have not yet completed the review described in the source material.

What has Lean checked?

OpenAI published a Lean formalization, and the source article says it checks a slightly weaker runtime bound than the preprint’s sharpest claim. Lean checks the formal proof as encoded; reviewers still need to assess how that statement relates to the full paper.

Does this solve P vs. NP?

No. The proposed result concerns ETH and a subexponential algorithm for k-SAT. It does not establish a polynomial-time algorithm and does not settle the P vs. NP question.

Will this make computers solve SAT problems faster now?

There is no evidence in the supplied material of an immediate practical speed-up. The claimed algorithm remains superpolynomial, and OpenAI says practical improvements would require significant further research.

Who is reviewing the result?

OpenAI says an outside group will examine the computer-science work, with Russell Impagliazzo, Ramamohan Paturi and Ryan Williams leading the review. The supplied material does not give a timetable for its findings.

Source: ThorstenMeyerAI.com