OpenAI, Anthropic models disprove longstanding Exponential Time Hypothesis

OpenAI and Anthropic have both released AI-driven results claiming to refute the Exponential Time Hypothesis, a long-standing conjecture in computational complexity theory.

On Oct. 26, OpenAI announced that its models produced a proof that the Exponential Time Hypothesis is false. The hypothesis, proposed in the early 2000s, asserts that 3-SAT cannot be solved in subexponential time - meaning that some common logic puzzles cannot be solved much faster than brute-forcing every possible answer.

As the hypothesis has been widely accepted and used in hundreds of papers to derive conditional computational hardness results across a wide variety of fields, disproving it would invalidate a significant portion of those results and force researchers to reassess the computational difficulty of problems in areas including algorithms, graph theory, and computational biology.

The AI firm said its models generated the natural-language disproof on Oct. 16 and formalized it in the Lean mathematical proof assistant the following day, both of which are available in a public GitHub repository.

The work is being examined by an external group led by Russell Impagliazzo and Ramamohan Paturi of UC San Diego - two of the original proposers of the hypothesis - and Ryan Williams of MIT, OpenAI said.

On X, Williams stated that researchers are still figuring out the practical implications of the discovery.

"At the current rate of progress, it's plausible we could see a serious update on practical SAT solving within the next 12 months. Or we might not see anything like that. I really don't know," Williams said.

Separately, Anthropic released a paper and Lean 4 formalization stating that a deterministic algorithm solves 3-SAT in a way that would refute the Exponential Time Hypothesis. The company said an internal Claude research model discovered the core algorithm on Oct. 16 - the same day OpenAI models made the discovery - using 14 million output tokens with no human input.

"The result is purely asymptotic and says nothing about solving formulas in practice," Anthropic noted.

Meanwhile, OpenAI also unveiled two other AI-generated results: a proof of the local smoothing conjecture in all dimensions in harmonic analysis, and a mathematically rigorous existence proof for certain quantum field theories, including Quantum Chromodynamics and pure Yang-Mills theory.