OpenAI and Anthropic's internal models almost simultaneously disproved the Exponential Time Hypothesis (ETH). Since the late 1990s, it has been believed that testing whether a logical formula with n variables can be true in the worst case cannot be done faster than exponential time to n. OpenAI released an early print with a faster algorithm that its model found. A day later, Anthropite published a similar result of the Claude model, and the formal proof in Lean 4 was also written by the model, and the program checks it. It turns out that the models are already finding new algorithms where the hypothesis has been held for more than 25 years, and bring the proof to machine verification, so there is no need to take their word for it. As Anthropit writes, the result says nothing about actually solving formulas. OpenAI wrote in early October that it was preparing to release the model behind this repository, so I expect such features in the public API as well.

OpenAI and Anthropic's internal models almost simultaneously disproved the Exponential Time Hypothesis (ETH). Since the late 1990s, it has been believed that testing whether a logical formula with n variables can be true in the worst case cannot be done faster than exponential time to n. OpenAI released an early print with a faster algorithm that its model found. A day later, Anthropite published a similar result of the Claude model, and the formal proof in Lean 4 was also written by the model, and the program checks it. It turns out that the models are already finding new algorithms where the hypothesis has been held for more than 25 years, and bring the proof to machine verification, so there is no need to take their word for it. As Anthropit writes, the result says nothing about actually solving formulas. OpenAI wrote in early October that it was preparing to release the model behind this repository, so I expect such features in the public API as well.