https://arxiv.org/abs/2408.08152 - "DeepSeek-Prover-V1.5: Harnessing Proof Assistant Feedback for Reinforcement Learning and Monte-Carlo Tree Search"

https://github.com/deepseek-ai/DeepSeek-Prover-V1.5

TL;DR

DeepSeek-Prover-V1.5 is an improved open-source language model for theorem proving in Lean 4.

The paper continues pre-training DeepSeekMath-Base, a math foundation model, and then does supervised fine-tuning on a dataset of incomplete proofs in Lean 4, followed by RL with feedback sourced from Lean 4. DeepSeek-Prover-V1.5 finds proofs using truncate-and-resume, which combines existing proof generation techniques, and a novel Monte-Carlo-Tree-Search (MCTS) algorithm for superior performance.

Overview

Definitions:

Tactic - a command used to manipulate the current proof state
Proof State/Tactic State - Current state of a proof at a given time, includes a list of goals and context.
(tactic state and proof state is often used interchangeably, tactic state is more specific as proof state can also be the state of a proof with a missing step)

Language models trained for theorem proving typically do one of two things:

Proof Step Generation, in which successive tactics are predicted by the prover and verified with a proof assistant. The tactic state is updated and search techniques are used to construct proofs. Each proof step is therefore valid. This method is not computationally efficient because each proof step has to be verified by the proof assistant.
Whole Proof Generation, in which the entire proof is generated in a single step based on the theorem statement. If a proof-step is invalid, the model generates a new proof in the next attempt. This is computationally efficient as a proof assistant is not required, but risks more invalid steps.

DeepSeek-Prover-V1.5 combines proof-step and whole-proof generation, naming the combination truncate-and-resume. Truncate-and-resume starts with whole-proof generation. If Lean detects an error in the proof, all code succeeding the error is removed (truncate), and the valid proof steps are used as a prompt for the next proof segment (resume). This truncate and resume mechanism is used in conjunction with MCTS to search over possible proof states. A reward-free exploration algorithm that uses intrinsic motivation to explore the tactic space is introduced to address reward sparsity.

Pre-Training and SFT

Pre-training:

Further pre-t

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