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You'd think that, but Timothy Gowers (the famous mathematician they worked with) wrote (https://x.com/wtgowers/status/1816509817382735986 https://x.com/wtgowers

You'd think that, but Timothy Gowers (the famous mathematician they worked with) wrote (https://x.com/wtgowers/status/1816509817382735986 https://x.com/wtgowers/status/1816509817382735986) > However, LLMs are not able to autoformalize reliably, so they got them to autoformalize each problem many times. Some of the formalizations were correct, but even the incorrect ones were useful as training data, as often they were easier problems. So didn't actually solve autoformalization, which is why they still needed humans to translate the input IMO 2024 problems. The reason why formalization is harder than you think is that there is no way to know if you got it right. You can use Reinforcement Learning with proofs and have a clear signal from the proof checker. We don't have a way to verify formalizations the same way.

llwu 2y ago
> We don't have a way to verify formalizations the same way. While there is no perfect method, it is possible to use the agent to determine if the statement is false, has contradictory hypotheses, or a suspiciously short proof.
thrdbndndn 2y ago
> However, LLMs are not able to autoformalize reliably, so they got them to autoformalize each problem many times. Some of the formalizations were correct, but even the incorrect ones were useful as training data, as often they were easier problems. A small detail wasn't clear to me: for these incorrectly formalized problems, how do they get the correct answer as ground truth for training? Have a human to manually solve them? (In contrast to problems actually from "a huge database of IMO-type problems", they do have answers for these already).
- adrianN 2y ago
  You write proofs in a formal language that can be machine checked. If the checker is happy, the proof is correct (unless there is a bug in the checker, but that is unlikely).
- raincole 2y ago
  They said the incorrectly formalized ones are usually easier, so I assume they just hire humans to solve them in the old way until the AI is smart enough to solve these easier problems.
  - czl 2y ago
    > I assume the just hire humans to solve… An incorrectly formalized problem is a different problem and a solution to any formalized problem still useful for AI training because such solutions can be mechanically checked for correctness and this does not require the hire of humans. What requires humans is the initial formalization process since that is more a language translation task which requires nuance and judgment and is difficult to mechanically verify.
- summerlight 2y ago
  > A small detail wasn't clear to me: for these incorrectly formalized problems, how do they get the correct answer as ground truth for training? Have a human to manually solve them? Formal proofs can be mechanically checked if it's correct or not. We just don't know what's the answer. Think it as an extremely rigorous type system that typically requires really long type annotations, like annotation itself is a complex program. So if AlphaProof happens to generate a proof that passes this checker, we know that it's correct.
  - thrdbndndn 2y ago
    Ah, thanks. That makes a lot of sense now.
    - thomasahle 2y ago
      One more trick: They look for both proofs and disproofs. So even if they failed the formalization and created a "wrong" theorem, it's just another task.

llwu 2y ago

> We don't have a way to verify formalizations the same way. While there is no perfect method, it is possible to use the agent to determine if the statement is false, has contradictory hypotheses, or a suspiciously short proof.

thrdbndndn 2y ago

> However, LLMs are not able to autoformalize reliably, so they got them to autoformalize each problem many times. Some of the formalizations were correct, but even the incorrect ones were useful as training data, as often they were easier problems. A small detail wasn't clear to me: for these incorrectly formalized problems, how do they get the correct answer as ground truth for training? Have a human to manually solve them? (In contrast to problems actually from "a huge database of IMO-type problems", they do have answers for these already).

adrianN 2y ago
You write proofs in a formal language that can be machine checked. If the checker is happy, the proof is correct (unless there is a bug in the checker, but that is unlikely).
raincole 2y ago
They said the incorrectly formalized ones are usually easier, so I assume they just hire humans to solve them in the old way until the AI is smart enough to solve these easier problems.
- czl 2y ago
  > I assume the just hire humans to solve… An incorrectly formalized problem is a different problem and a solution to any formalized problem still useful for AI training because such solutions can be mechanically checked for correctness and this does not require the hire of humans. What requires humans is the initial formalization process since that is more a language translation task which requires nuance and judgment and is difficult to mechanically verify.
summerlight 2y ago
> A small detail wasn't clear to me: for these incorrectly formalized problems, how do they get the correct answer as ground truth for training? Have a human to manually solve them? Formal proofs can be mechanically checked if it's correct or not. We just don't know what's the answer. Think it as an extremely rigorous type system that typically requires really long type annotations, like annotation itself is a complex program. So if AlphaProof happens to generate a proof that passes this checker, we know that it's correct.
- thrdbndndn 2y ago
  Ah, thanks. That makes a lot of sense now.
  - thomasahle 2y ago
    One more trick: They look for both proofs and disproofs. So even if they failed the formalization and created a "wrong" theorem, it's just another task.