Xiao Liang, Zhong-Zhi Li, Zhenghao Lin Microsoft, Eric Hanchen Jiang, Hengyuan Zhang, Yelong Shen, Kai-Wei Chang University of California, Los Angeles and Amazon, Ying Nian Wu 等 10 位作者
Large language models (LLMs) have demonstrated strong reasoning capabilities through step-by-step chain-of-thought (CoT) reasoning. Nevertheless, at the limits of model capability, CoT often proves insufficient, and its strictly sequential nature constrains test-time scalability. A potential alternative is divide-and-conquer (DAC) reasoning, which decomposes a complex problem into subproblems to facilitate more effective exploration of the solution space. Although promising, our analysis reveals a fundamental misalignment between general-purpose post-training and DAC-style inference, which limits the model’s capacity to fully leverage this potential. To bridge this gap and fully unlock LLMs’ reasoning capabilities on the most challenging tasks, we propose an end-to-end reinforcement learning (RL) framework to enhance their DAC-style reasoning capacity. At each step, the policy decomposes a problem into a group of subproblems, solves them sequentially, and addresses the original problem conditioned on the subproblem solutions, with both decomposition and solution integrated into RL training. Under comparable training settings, our DAC-style framework endows the model with a higher performance ceiling and stronger test-time scalability, surpassing CoT by 8.6% in Pass@1 and 6.3% in Pass@32 on competition-level benchmarks. The code is available at the provided link.
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cs.CL
Proceedings of the 64th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers) · 2026 · 正式发表
Boyan Duan, Xiao Liang, Shuai Lu JD.com, Yaoxiang Wang, Yelong Shen, Kai-Wei Chang University of California, Los Angeles and Amazon, Ying Nian Wu, Mao Yang 等 10 位作者
Automated theorem proving in Euclidean geometry, particularly for International Mathematical Olympiad (IMO) level problems, remains a major challenge and an important research focus in Artificial Intelligence. In this paper, we present a highly efficient method for geometry theorem proving that runs entirely on CPUs without relying on neural network–based inference. Our initial study shows that a simple random strategy for adding auxiliary points can achieve ”silver-medal” level human performance on IMO. Building on this, we propose HAGeo, a Heuristic-based method for adding Auxiliary points in Geometric deduction that solves 28 of 30 problems on the IMO-30 benchmark, achieving “gold-medal” level performance and surpassing AlphaGeometry, a competitive neural network–based approach, by a notable margin. To evaluate our method and existing approaches more comprehensively, we further construct HAGeo, a benchmark consisting of 409 geometry problems with human-assessed difficulty levels. Compared with the widely used IMO-30, our benchmark poses greater challenges and provides a more precise evaluation, setting a higher bar for geometry theorem proving.
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cs.CL · cs.AI
Proceedings of the 64th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers) · 2026 · 正式发表