[
    {
        "id": "osp-21216",
        "type": "article-journal",
        "title": "Function-Structured Reinforcement Learning with Executable Verifiers for Mathematical Reasoning",
        "author": [
            {
                "family": "Liu",
                "given": "Zihan"
            },
            {
                "family": "Xie",
                "given": "Xurong"
            }
        ],
        "URL": "https://omanscience.com/en/articles/function-structured-reinforcement-learning-with-executable-verifiers-for-mathematical-reasoning",
        "language": "en",
        "issued": {
            "date-parts": [
                [
                    2026
                ]
            ]
        },
        "abstract": "Algorithmic mathematical reasoning requires reliable decomposition, computation, and aggregation. Final-answer rewards provide limited guidance on intermediate errors, while successful execution does not guarantee mathematical correctness. This work proposes Function-Structured Graph Reinforcement Learning (FSG-RL), connecting subproblem graphs and Python implementations with multi-verifier feedback. The policy first learns to generate code from function graphs through supervised fine-tuning (SFT). Group Relative Policy Optimization (GRPO) then optimizes the policy using answer-gated rewards and span-level credit assignment. The framework also supports teacher supervision and structured memory. A benchmark curated from Grade School Math 8K (GSM8K), MathQA, MATH, and Omni-MATH pairs public function graphs with private verification specifications. Under a unified evaluation protocol, GRPO improves final-answer accuracy from 43.25% to 67.50% and full solution success from 32.25% to 52.25% over SFT. Continued reinforcement learning (RL) with teacher supervision yields additional gains. The gains extend beyond producing correctly formatted code, supporting verifier-guided reinforcement learning for mathematical reasoning. Code is available at https://github.com/ZihanLiummyycc/FSG-RL."
    }
]