Automatically designing fast and space-efficient digital circuits is challenging because circuits are discrete, must exactly implement the desired logic, and are costly to simulate. We address these challenges with CircuitVAE, a search algorithm that embeds computation graphs in a continuous space and optimizes a learned surrogate of physical simulation by gradient descent. By carefully controlling overfitting of the simulation surrogate and ensuring diverse exploration, our algorithm is highly sample-efficient, yet gracefully scales to large problem instances and high sample budgets. We test CircuitVAE by designing binary adders across a large range of sizes, IO timing constraints, and sample budgets. Our method excels at designing large circuits, where other algorithms struggle: compared to reinforcement learning and genetic algorithms, CircuitVAE typically finds 64-bit adders which are smaller and faster using less than half the sample budget. We also find CircuitVAE can design state-of-the-art adders in a real-world chip, demonstrating that our method can outperform commercial tools in a realistic setting.

Large language models (LLMs) have shown promise in proving formal theorems using proof assistants such as Lean. However, existing methods are difficult to reproduce or build on, due to private code, data, and large compute requirements. This has created substantial barriers to research on machine learning methods for theorem proving. This paper removes these barriers by introducing LeanDojo: an open-source Lean playground consisting of toolkits, data, models, and benchmarks. LeanDojo extracts data from Lean and enables interaction with the proof environment programmatically. It contains fine-grained annotations of premises in proofs, providing valuable data for premise selection—a key bottleneck in theorem proving. Using this data, we develop ReProver (Retrieval-Augmented Prover): an LLM-based prover augmented with retrieval for selecting premises from a vast math library. It is inexpensive and needs only one GPU week of training. Our retriever leverages LeanDojo's program analysis capability to identify accessible premises and hard negative examples, which makes retrieval much more effective. Furthermore, we construct a new benchmark consisting of 98,734 theorems and proofs extracted from Lean's math library. It features challenging data split requiring the prover to generalize to theorems relying on novel premises that are never used in training. We use this benchmark for training and evaluation, and experimental results demonstrate the effectiveness of ReProver over non-retrieval baselines and GPT-4. We thus provide the first set of open-source LLM-based theorem provers without any proprietary datasets and release it under a permissive MIT license to facilitate further research.