发表机构
Princeton University（普林斯顿大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究数学推理中答案不变性与表示不变性的区别，发现模型对重排规则顺序的表示区分度与准确性正相关，Spearman相关达0.86。
AI 中文摘要
在不改变含义的情况下重新排列一组数学规则，应当保持正确答案不变，但模型的内部表示是否也必须保持不变？我们通过合成多步函数组合问题来研究这一问题，每个问题在多种规则顺序下呈现，且具有相同的正确答案。我们测量准确性和排列信噪比（SNR），该指标量化了顺序模式相对于问题实例间变化的表示区分度。在从1B到8B参数的16个语言模型中，我们发现一种模式：更准确地解决重排问题的模型，对不同的规则顺序表示得更清晰。在所有评估的合成设置中，层平均排列信噪比与准确性呈正秩相关，Spearman相关系数达到0.86。这些发现凸显了答案不变性与表示不变性之间的区别：成功的数学规则组合可以伴随等价规则顺序之间的不同内部表示。这促使我们区分答案不变性与表示不变性，并为超越答案准确性的数学推理提供了一种表示视角。
英文摘要
Reordering a set of mathematical rules without changing its meaning should preserve the correct answer, but must a model's internal representations stay invariant too? We investigate this question using synthetic multi-step function-composition problems, each presented under multiple rule orderings with the same correct answer. We measure accuracy and permutation signal-to-noise ratio (SNR), which quantifies how distinctly ordering patterns are represented relative to variation across problem instances. Across 16 language models ranging from 1B to 8B parameters, we find a pattern: models that solve reordered problems more accurately represent different rule orderings more distinctly. Layer-averaged permutation SNR is positively rank-correlated with accuracy in every synthetic setting we evaluate, with Spearman correlations reaching 0.86. These findings highlight a distinction between answer invariance and representation invariance: successful mathematical rule composition can accompany distinct internal representations between equivalent rule orderings. This motivates distinguishing answer invariance from representation invariance, and offers a representational perspective on mathematical reasoning beyond answer accuracy alone.
CommentsNeurIPS 2026 Workshop: The 6th Workshop on Mathematical Reasoning and AI
