Our basis for addressing the lack of self-play is the observation that the key role of self-play in 2-player games is to provide an unsupervised curriculum.

Language models have the capability to generate the tactic calls as well as the original mathematical terms often required as arguments. In our work, we address the infinite action space problem by sampling actions from a language model as we search for a proof. This asymmetry prevents naive application of the self-play algorithms that were successful with 2-player games. When faced with a statement that is just too hard, there is no obvious reframing that will let the prover generate intermediary easier statements to tackle first.

$geometry olympiad problems$

(ii) Lack of self-play: conversely to 2-player games, a prover is not playing against an opponent but against a set of statements to prove.…”, or a cut, the introduction and the chaining of a lemma in the middle of a proof). At each step of a proof search, the model must choose not from a well-behaved finite set of actions, but a complex and infinite set of tactics, involving exogenous mathematical terms that have to be generated (e.g., generating a mathematical statement to be used as a witness, an object used in steps such as “there exists an $x$ s.t. (i) Infinite action space: not only does formal mathematics have an extremely large search space (like Go for example), it also has an infinite action space.$$3a^2b^2c^2 + sum_ (a^2b + a^2c – b^2c)^2$$įormal mathematics involves two main challenges that make a naive application of reinforcement learning unlikely to succeed. These tactics take mathematical terms as arguments and each tactic call will transform the current statement to prove, into statements that are easier to prove, until nothing is left to prove.Īfter cancelling terms appearing on both sides, we are left to prove that: As demonstrated in the trivial example below, proving a formal statement requires generating a sequence of proof steps, each proof step consisting in a call to a tactic. We iteratively search for new proofs and re-train our neural network on the newly discovered proofs, and after 8 iterations, our prover ends up being vastly superior when tested on miniF2F.įormal mathematics is an exciting domain to study because of (i) its richness, letting you prove arbitrary theorems which require reasoning, creativity and insight and (ii) its similarity to games-where AI has been spectacularly successful-in that it has an automated way of determining whether a proof is successful (i.e., verified by the formal system). Initially our neural prover is weak and can only prove a few of them. Our approach, which we call statement curriculum learning, consists of manually collecting a set of statements of varying difficulty levels (without proof) where the hardest statements are similar to the benchmark we target. We achieved a new state-of-the-art (41.2% vs 29.3%) on the miniF2F benchmark, a challenging collection of high-school olympiad problems.

Each time we find a new proof, we use it as new training data, which improves the neural network and enables it to iteratively find solutions to harder and harder statements. The prover uses a language model to find proofs of formal statements. We built a neural theorem prover for Lean that learned to solve a variety of challenging high-school olympiad problems, including problems from the AMC12 and AIME competitions, as well as two problems adapted from the IMO.

Geometry olympiad problems