Solving Olympiad-level mathematical problems represents a significant advancement in machine intelligence and automated reasoning.

Asuccessful autoformalization system could advance the fields of formal verification, program synthesis, and artificial intelligence. While the long-term goal of autoformalization seemed elusive for a long time, we show large language models provide new prospects towards this goal.

In theorem proving, the task of selecting useful premises from alarge library to unlock the proof of a given conjecture is crucially important. This presents a challenge foralltheorem provers,especially theonesbasedonlanguage models, due to their relative inability to reason over huge volumes of premises in text form.

Prior work has combined chain-of-thought prompting in large language models (LLMs) with programmatic representations to perform effective and transparent reasoning. While such an approach works well for tasks that only require forward reasoning (e.g., straightforward arithmetic), it is less effective for constraint solving problems that require more sophisticated planning and search. In this paper, we propose a new satisfiability-aided language modeling (SatLM) approach for improving the reasoning capabilities of LLMs. We use an LLM to generate a declarative task specification rather than an imperative program and leverage an off-the-shelf automated theorem prover to derive the final answer. This approach has two key advantages. The declarative specification is closer to the problem description than the reasoning steps are, so the LLM can parse it out of the description more accurately. Furthermore, by offloading the actual reasoning task to an automated theorem prover, our approach can guarantee the correctness of the answer with respect to the parsed specification and avoid planning errors in the solving process. We evaluate SATLM on 8 different datasets and show that it consistently outperforms program-aided LMs in the imperative paradigm. In particular, SATLM outperforms program-aided LMs by 23% on a challenging subset of the GSM arithmetic reasoning dataset; SATLM also achieves a new SoTA on LSAT and BoardgameQA, surpassing previous models that are trained on the respective training sets.

We consider the task of automated theorem proving, a key AI task. Deep learning has shown promise for training theorem provers, but there are limited human-written theorems and proofs available for supervised learning. To address this limitation, we propose to learn a neural generator that automatically synthesizes theorems and proofs for the purpose of training a theorem prover. Experiments on real-world tasks demonstrate that synthetic data from our approach improves the theorem prover and advances the state of the art of automated theorem proving in Metamath.

We propose a novel approach to interactive theorem-proving (ITP) using deep reinforcement learning. The proposed framework is able to learn proof search strategies as well as tactic and arguments prediction in an end-to-end manner. We formulate the process of ITP as a Markov decision process (MDP) in which each state represents a set of potential derivation paths. This structure allows us to introduce a novel backtracking mechanism which enables the agent to efficiently discard (predicted) dead-end derivations and restart the derivation from promising alternatives. We implement the framework in the HOL theorem prover. Experimental results show that the framework using learned search strategies outperforms existing automated theorem provers (i.e., hammers) available in HOL when evaluated on unseen problems. We further elaborate the role of key components of the framework using ablation studies.

In theorem proving, the task of selecting useful premises from a large library to unlock the proof of a given conjecture is crucially important. This presents a challenge for all theorem provers, especially the ones based on language models, due to their relative inability to reason over huge volumes of premises in text form. This paper introduces Thor, a framework integrating language models and automated theorem provers to overcome this difficulty. In Thor, a class of methods called hammers that leverage the power of automated theorem provers are used for premise selection, while all other tasks are designated to language models. Thor increases a language model's success rate on the PISA dataset from $39\%$ to $57\%$, while solving $8.2\%$ of problems neither language models nor automated theorem provers are able to solve on their own. Furthermore, with a significantly smaller computational budget, Thor can achieve a success rate on the MiniF2F dataset that is on par with the best existing methods. Thor can be instantiated for the majority of popular interactive theorem provers via a straightforward protocol we provide.