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Comment by fmap | Hacker News Reader

Agreed, this is a big step forward. Geometry problems are in a different class, since you can translate them into systems of polynomial equations and use well known computer algebra algorithms to solve them.

By contrast, this kind of open ended formalization is something where progress used to be extremely slow and incremental. I worked in an adjacent field 5 years ago and I cannot stress enough that these results are simply out of reach for traditional automated reasoning techniques.

Real automatic theorem proving is also useful for a lot more than pure mathematics. For example, it's simple to write out an axiomatic semantics for a small programming language in lean and pose a question of the form "show that there exists a program which satisfies this specification".

If this approach scales it'll be far more important than any other ML application that has come out in the last few years.

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> Agreed, this is a big step forward. Geometry problems are in a different class, since you can translate them into systems of polynomial equations and use well known computer algebra algorithms to solve them.

The blog post indicates the opposite. The geometry problem in the IMO problem set was solved by AlphaGeometry 2, which is an LLM based on Google's Gemini. LLMs are considered relatively general systems. But the other three solved problems were proved by AlphaProof, which is a narrow RL system that is literally based on AlphaZero, the Go and Chess AI. Only its initial (bootstrapping) human training data (proofs) were formalized and augmented by an LLM (Gemini).

AlphaGeometry is not an LLM

It is an LLM combined with a symbolic deduction engine.

AlphaZero is more general than a Go and Chess AI, right? Isn't it a general self-play algorithm?

I think that it can only play “games” for which it has perfect information about the state of the “game”.

Only slightly more general. It only works for games that are zero-sum, deterministic, have no hidden information, and discrete game state and moves. Other examples include connect-4.

So finding Lean proofs can be conceptualized as a zero-sum game?

Another basic requirement is that valid moves / inference steps and the winning condition can be efficiently verified using some non-AI algorithm. Otherwise there would not be a reward signal for the reinforcement learning algorithm. This is different from answering most natural language questions, where the answer can't be checked trivially.

I don't think AlphaZero is related to this work, apart from both being NN-based. AlphaZero and its training pipeline fundamentally only works for "chess-like" two-player games, where the agent can play against itself and slowly improve through MCTS.

"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."

Theorem proving can be formulated as a game, see e.g., https://plato.stanford.edu/entries/logic-games/ and interactive theorem provers can verify that a proof is correct (and related sub problems, such as that a lemma application is valid).

Conceptually, if you're trying to show a conjunction, then it's the other player's turn and they ask you for a proof of a particular case. If you're trying to show a disjunction then it's your turn and you're picking a case. "Forall" is a potentially infinite conjunction, "exists" is a potentially infinite disjunction.

In classical logic this collapses somewhat, but the point is that this is still a search problem of the same kind. If you want to feel this for yourself, try out some proofs in lean or coq. :)

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