OpenAI and Anthropic say their AI models disproved the Exponential Time Hypothesis

OpenAI said on Oct 26 that an unreleased internal model had disproved the Exponential Time Hypothesis, a conjecture proposed in 1999 that a classic logic problem cannot be solved much faster than by brute-force search. Hours later on the same day, Anthropic published its own paper claiming a refutation of the hypothesis.

Neither claim has yet passed independent review. A group that includes two of the researchers who proposed the hypothesis is now examining OpenAI's result.

What OpenAI claims

OpenAI's paper, titled "The Exponential Time Hypothesis is false," describes a deterministic algorithm that exactly counts the solutions of fixed-width logic formulas in subexponential time while using only polynomial memory. Applied to 3-SAT, the standard form of the Boolean satisfiability problem, that algorithm would refute the hypothesis.

OpenAI says the model produced the disproof on Oct 16 and that it was checked on Oct 17 in Lean, software that verifies proofs by machine. The 67-page paper and the Lean proof are public in OpenAI's GitHub repository.

OpenAI says the hypothesis has been used in hundreds of papers to establish that problems in graph theory, computational biology, databases and computational geometry resist fast solutions.

Anthropic's rival paper

Anthropic's paper, titled "A 2^o(n) Time Algorithm for 3-SAT" and credited to Claude, presents a deterministic subexponential-time algorithm for 3-SAT and its counting version. Tech Times reported that both labs trace their core discovery to Oct 16.

Before Anthropic's release, Anthropic mathematician Levent Alpöge suggested OpenAI's solutions drew on pre-existing work, TechCrunch reported.

The review

OpenAI asked Russell Impagliazzo and Ramamohan Paturi of UC San Diego, who proposed the hypothesis, and Ryan Williams of MIT to organize an open examination of the result. In an open letter, the three wrote that their involvement "should not be taken as certification of the claim." They signed no nondisclosure agreements and are not paid.

At the group's request, OpenAI agreed to release all raw data behind the result and to fund a conference for the research community. OpenAI also gave $5 million to the Simons Foundation for workshops and conferences on AI-produced results.

A machine check confirms that a formal proof is valid, but experts still have to confirm that the formal statement matches what the paper claims. Xenospectrum reported that OpenAI's own scope document shows the Lean check covers only standardly encoded inputs and proves a weaker running-time bound than the paper states.

What it means for P versus NP

Leslie Goldberg of the University of Oxford told New Scientist that disproving the hypothesis has no direct impact on P versus NP, the open question of whether every problem that is easy to check is also easy to solve. Williams told TechCrunch that every computer scientist should put a little more weight on the possibility that P equals NP.

Williams also said a faster algorithm could imply that cryptographic systems relying on certain key sizes are broken. He said OpenAI science lead Dan Roberts told him the company had checked and was reasonably confident the result did not imply any breaks.

The other results

The same OpenAI announcement claimed a proof of Christopher Sogge's 1991 local smoothing conjecture in every dimension, which would also settle the Kakeya and Stein restriction conjectures. It also claimed existence proofs for four-dimensional quantum field theories, including quantum chromodynamics and pure Yang-Mills theory, which forms part of a Clay Mathematics Institute Millennium Prize problem.

A physics group led by Nobel laureate David Gross is withholding the AI-produced physics material and plans to write up the work itself.