[cubefox]

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.

[fmap]

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. :)

[sapiogram]

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.

[adroniser]

"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."