Distilling Tool Knowledge into Language Models via Back-Translated Traces

Large language models (LLMs) often struggle with mathematical problems that require exact computation or multi-step algebraic reasoning. Tool-integrated reasoning (TIR) offers a promising solution by leveraging external tools such as code interpreters to ensure correctness, but it introduces inference-time dependencies that hinder scalability and deployment. In this work, we propose a new paradigm for distilling tool knowledge into LLMs purely through natural language. We first construct a Solver Agent that solves math problems by interleaving planning, symbolic tool calls, and reflective reasoning. Then, using a back-translation pipeline powered by multiple LLM-based agents, we convert interleaved TIR traces into natural language reasoning traces. A Translator Agent generates explanations for individual tool calls, while a Rephrase Agent merges them into a fluent and globally coherent narrative. Empirically, we show that fine-tuning a small open-source model on these synthesized traces enables it to internalize both tool knowledge and structured reasoning patterns, yielding gains on competition-level math benchmarks without requiring tool access at inference.

SubgoalXL: Subgoal-based Expert Learning for Theorem Proving

Formal theorem proving, a field at the intersection of mathematics and computer science, has seen renewed interest with advancements in large language models (LLMs). This paper introduces SubgoalXL, a novel approach that synergizes subgoal-based proofs with expert learning to enhance LLMs' capabilities in formal theorem proving within the Isabelle environment. SubgoalXL addresses two critical challenges: the scarcity of specialized mathematics and theorem-proving data, and the need for improved multi-step reasoning abilities in LLMs. By optimizing data efficiency and employing subgoal-level supervision, SubgoalXL extracts richer information from limited human-generated proofs. The framework integrates subgoal-oriented proof strategies with an expert learning system, iteratively refining formal statement, proof, and subgoal generators. Leveraging the Isabelle environment's advantages in subgoal-based proofs, SubgoalXL achieves a new state-of-the-art performance of 56.1\% in Isabelle on the standard miniF2F dataset, marking an absolute improvement of 4.9\%. Notably, SubgoalXL successfully solves 41 AMC12, 9 AIME, and 3 IMO problems from miniF2F. These results underscore the effectiveness of maximizing limited data utility and employing targeted guidance for complex reasoning in formal theorem proving, contributing to the ongoing advancement of AI reasoning capabilities. The implementation is available at \url{https://github.com/zhaoxlpku/SubgoalXL}.