OpenAI has built a neural theorem prover for Lean that was able to solve a range of problems from AMC12 and AIME competitions as well as problems from the IMO.
The prover uses a language model to find proofs of formal statements. Whenever a new proof is found, the prover uses this as training data which improves the neural network and enables it to constantly find solutions to more complex problems.
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Formal mathematics involves two major challenges that make a simple application of reinforcement learning unlikely to succeed: Infinite action space and Lack of self-play.
To overcome the infinite action space challenge, the natural theorem prover uses sampling actions from a language model as it searches for proof. Language models have the capacity to generate tactic calls and mathematical terms often required as arguments. To address the lack of self-play, the creators replaced unsupervised curriculum with an auxiliary set of problem statements of varying difficulty. The training procedure was able to solve a curriculum of increasingly difficult problems when provided with a varied set of difficulties in auxiliary problems.
Although, the neural theorem prover was able to generate exciting results proving the potential deep learning models capability of non-trivial mathematical reasoning when interacting with a formal system. But, the system is still far away from providing consistent results when compared with the performance of the best students who participate in these challenging olympiad problems.
