Formula-One Prompting：一种可组合的方程优先前缀用于应用数学
Formula-One Prompting: A Composable Equation-First Prefix for Applied Mathematics
AI总结：
提出公式提示（FP）和Formula-One提示（F-1），通过先形式化问题中的控制方程再求解，在多个应用数学基准上优于思维链和程序思维提示，平均提升5.76和8.42个百分点。
摘要：
本文介绍了公式提示（FP）和Formula-One提示（F-1），两种单次调用方法，在解决应用数学问题之前先引出控制方程。思维链（CoT）和程序思维（PoT）提示通过引出预训练期间学到的推理轨迹或类似代码的结构来改进数学推理。这提出了一个诊断性问题：哪些有用的预训练模式仍然未被充分引出？使用infini-gram-mini，我们扫描了81.7万亿预训练令牌，发现在精心策划的语料库（如DataComp-LM）中，以方程为中心的语言出现频率比代码高121倍，比逐步叙述高3.79倍，但标准提示方法并未明确引出方程形式化。FP要求模型在求解前先形式化问题的控制方程；F-1扩展了FP，增加了一个可组合的第二阶段，在同一调用中选择直接、CoT或PoT风格的求解。在五个推理模型和四个应用数学基准（金融、物理、密码学、竞赛数学）上，F-1平均优于CoT 5.76个百分点，优于PoT 8.42个百分点，在FinanceMath上取得最大提升13.30个百分点，同时以仅68个提示令牌的开销占据准确率-令牌效率前沿。变体消融实验表明，方程形式化前缀（而非策略菜单）是主要驱动因素：在前缀之上添加CoT或PoT不会带来进一步收益，且73.3%的剩余失败发生在第一阶段方程正确之后。
英文摘要：
This paper introduces Formula Prompting (FP) and Formula-One Prompting (F-1), two single-call methods that elicit governing equations before solving applied-math problems. Chain-of-Thought (CoT) and Program-of-Thought (PoT) prompting improve mathematical reasoning by eliciting reasoning traces or code-like structures learned during pretraining. This suggests a diagnostic question: which useful pretraining patterns remain under-elicited? Using infini-gram-mini, we scan 81.7 trillion pretraining tokens and find that, in curated corpora such as DataComp-LM, equation-centered language appears 121x more often than code and 3.79x more often than step-by-step narration, yet standard prompting methods do not explicitly elicit equation formulation. FP asks the model to formalize a problem's governing equations before solving; F-1 extends FP with a composable Phase 2 that selects Direct, CoT, or PoT-style solving in the same call. Across five reasoning models and four applied-math benchmarks (finance, physics, cryptography, competition math), F-1 outperforms CoT by 5.76 pp and PoT by 8.42 pp on average, with the largest gain of 13.30 pp on FinanceMath, while topping the accuracy-token efficiency frontier at only 68 prompt tokens of overhead. Variant ablations identify the equation-formalization prefix, not the strategy menu, as the primary driver: adding CoT or PoT on top of the prefix yields no further gain, and 73.3% of remaining failures occur downstream of a correct Phase-1 equation.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
