We present PutnamBench, a new multilingual benchmark for evaluating the ability of neural theorem-provers to solve competition mathematics problems. PutnamBench consists of 1697 hand-constructed formalizations of 640 theorems sourced from the William Lowell Putnam Mathematical Competition, the premier undergraduate-level mathematics competition in North America. All the theorems have formalizations in Lean 4 and Isabelle; a substantial subset also has Coq formalizations. Proving the theorems requires significant problem-solving ability and proficiency in a broad range of topics taught in undergraduate mathematics courses. We use PutnamBench to evaluate several established neural and symbolic theorem-provers. These approaches can only solve a handful of the PutnamBench problems, establishing the benchmark as a difficult open challenge for research on neural theorem-proving. PutnamBench is available at https://github.com/trishullab/PutnamBench.

We present an in-context learning agent for formal theorem-proving in environments like Lean and Coq. Current state-of-the-art models for the problem are finetuned on environment-specific proof data. By contrast, our approach, called COPRA, repeatedly asks a high-capacity, general-purpose large language model (GPT-4) to propose tactic applications from within a stateful backtracking search. Proposed tactics are executed in the underlying proof environment. Feedback from the execution is used to build the prompt for the next model query, along with selected information from the search history and lemmas retrieved from an external database. We evaluate our implementation of COPRA on the miniF2F benchmark for Lean and a set of Coq tasks from the CompCert project. On these benchmarks, COPRA significantly outperforms few-shot invocations of GPT-4. It also compares favorably against finetuning-based approaches, outperforming REPROVER, a state-of-the-art finetuned approach for Lean, in terms of the pass@1 metric. Our code and data are available at https://github.com/trishullab/copra.

Imitation Learning (IL) is a promising paradigm for teaching robots to perform novel tasks using demonstrations. Most existing approaches for IL utilize neural networks (NN), however, these methods suffer from several well-known limitations: they 1) require large amounts of training data, 2) are hard to interpret, and 3) are hard to repair and adapt. There is an emerging interest in programmatic imitation learning (PIL), which offers significant promise in addressing the above limitations. In PIL, the learned policy is represented in a programming language, making it amenable to interpretation and repair. However, state-of-the-art PIL algorithms assume access to action labels and struggle to learn from noisy real-world demonstrations. In this paper, we propose PLUNDER, a novel PIL algorithm that integrates a probabilistic program synthesizer in an iterative Expectation-Maximization (EM) framework to address these shortcomings. Unlike existing PIL approaches, PLUNDER synthesizes probabilistic programmatic policies that are particularly well-suited for modeling the uncertainties inherent in real-world demonstrations. Our approach leverages an EM loop to simultaneously infer the missing action labels and the most likely probabilistic policy. We benchmark PLUNDER against several established IL techniques, and demonstrate its superiority across five challenging imitation learning tasks under noise. PLUNDER policies achieve 95% accuracy in matching the given demonstrations, outperforming the next best baseline by 19%. Additionally, policies generated by PLUNDER successfully complete the tasks 17% more frequently than the nearest baseline.