The computer did find the answers itself. I.e., it found "even integers" for P1, "{1,1}" for P2, and "2" for P6. It then also provided provided a Lean proof in each case.
It would make a lot of sense for the lean-code-formalisation of the problems done by the researchers fed to the AI to be provided. Not assuming bad intent in not providing them, but it would help understand better the results.
formal definition of first theorem already contain answer of the problem
"{α : ℝ | ∃ k : ℤ, Even k ∧ α = k}" (which mean set of even real numbers).if they say that they have translated first problem into formal definition then it is very interesting how they initially formalized problem without including answer in it
I think maybe parent comment is referring to it essentially just employing a zerg rush but with the speed and reaction time of an AI?
Not 100% sure...
Unrelated, iirc the starcraft functionality was an early example of generalizing a pretrained NN, alphaGO, and showing that it could adapt to learn and defeat games across strategic domains, especially after it learned so much strategy from the most difficult, widely played, and most strategically-varied physical game available.
Exactly, a problem and its answer are just different ways of describing the same object. Every step of a proof is a transformation/translation of the same object. It would be disingenuous to say that some heavy lifting isn't done in formalizing a problem but it seems that step is also performed by a machine:
"We established a bridge between these two complementary spheres by fine-tuning a Gemini model to automatically translate natural language problem statements into formal statements, creating a large library of formal problems of varying difficulty."
I'm confused, is the formalization by Gemini or "manually"? Which is it?
"AlphaProof is a system that trains itself to prove mathematical statements in the formal language Lean. It couples a pre-trained language model with the AlphaZero reinforcement learning algorithm, which previously taught itself how to master the games of chess, shogi and Go."
Huh, so MCTS to find the ‘best’ token using a (relatively) small, quick language model? Sounds like an interesting approach to small model text generation too…
Yeah I am not clear the degree to which this system and LLMs are related. Are they related? Or is AlphaProof a complete tangent to CHatGPT and its ilk?
It's a math Language Model. Not even sure it's a Large Language Model. (Maybe shares a foundational model with an English LLM; I don't know)
It learns mathematical statements, and generates new mathematical statements, then uses search techniques to continue. Similar to Alpha Go's neural network, what makes it new and interesting is how the NN/LLM part makes smart guesses that drastically prune the search tree, before the brute-force search part.
(This is also what humans do to solve math probrems. But humans are really, really slow at brute-force search, so we really almost entirely on the NN pattern-matching analogy-making part.)
These kind of LLMs are also very interesting for software engineering. It's just a matter of replacing Lean with something that is more oriented towards proving software properties.
For example, write a formal specification of a function in Dafny on Liquid Haskell and get the LLM to produce code that is formally guaranteed to be correct. Logic-based + probability-based ML.
My reading of it is that it uses the same architecture as one of the Gemini models but does not share any weights with it. (i.e it's not just a finetune)
This is really interesting. I would have expected the understanding to be that humans make a guess, test it, and learn from what did or did not work. The lessons learned from the prior tests would impact future guesses.
Do you know if a system like the OP is learning from failed tests to guide future tests, or is it a truly a brute force search as if it were trying to mine bitcoin?
This quote from the article sounds like it learns from failed tests:
>We trained AlphaProof for the IMO by proving or disproving millions of problems, covering a wide range of difficulties and mathematical topic areas over a period of weeks leading up to the competition. The training loop was also applied during the contest, reinforcing proofs of self-generated variations of the contest problems until a full solution could be found.
yeah but it doesn't understand the exact syntax on an absolute level, does it...? I understood this to be the same as any language model applied to programming languages (aka Formal Languages). Is that mistaken?
As far as I understand, and I may be wrong here, the system is composed of two networks: Gemini and AlphaZero. Gemini, being an ordinary LLM with some fine-tunes, only does translation from natural to formal language. Then, AlphaZero solves the problem. AlphaZero, unburdened with natural language and only dealing with "playing a game in the proof space" (where the "moves" are commands to the Lean theorem prover), does not hallucinate in the same way an LLM does because it is nothing like an LLM.
Yes, but the problem space means that invalid outputs can be quickly identified - whereas general programming isn’t necessarily amenable to rapid checks.