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.

Planning in large state spaces inevitably needs to balance the depth and breadth of the search. It has a crucial impact on the performance of a planner and most manage this interplay implicitly. We present a novel method Shoot Tree Search (STS), which makes it possible to control this trade-off more explicitly. Our algorithm can be understood as an interpolation between two celebrated search mechanisms: MCTS and random shooting. It also lets the user control the bias-variance trade-off, akin to TD(n), but in the tree search context. In experiments on challenging domains, we show that STS can get the best of both worlds consistently achieving higher scores.

Humans excel in solving complex reasoning tasks through a mental process of moving from one idea to a related one. Inspired by this, we propose Subgoal Search (kSubS) method. Its key component is a learned subgoal generator that produces a diversity of subgoals that are both achievable and closer to the solution. Using subgoals reduces the search space and induces a high-level search graph suitable for efficient planning. In this paper, we implement kSubS using a transformer-based subgoal module coupled with the classical best-first search framework. We show that a simple approach of generating $k$-th step ahead subgoals is surprisingly efficient on three challenging domains: two popular puzzle games, Sokoban and the Rubik's Cube, and an inequality proving benchmark INT. kSubS achieves strong results including state-of-the-art on INT within a modest computational budget.