Order-Invariant Answers, Order-Sensitive Representations in Mathematical Reasoning

2026-09-23

AI Summary

研究探讨了数学推理中规则顺序变化对模型内部表示的影响，通过合成函数组合问题和测量置换信噪比发现，高准确率模型能更清晰地区分不同规则顺序。

Abstract

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.

Problem

• mathematical reasoning
• internal representations
• rule orderings
• answer invariance
• representation invariance

Innovation

• Order-Invariant Answers
• Order-Sensitive Representations
• Permutation Signal-to-Noise Ratio (SNR)
• Mathematical Reasoning