Equality reasoning is ubiquitous and purely abstract: sameness or difference may be evaluated no matter the nature of the underlying objects. As a result, same-different tasks (SD) have been extensively studied as a starting point for understanding abstract reasoning in humans and across animal species. With the rise of neural networks (NN) that exhibit striking apparent proficiency for abstractions, equality reasoning in NNs has also gained interest. Yet despite extensive study, conclusions about equality reasoning vary widely and with little consensus. To clarify the underlying principles in learning SD, we develop a theory of equality reasoning in multi-layer perceptrons (MLP). Following observations in comparative psychology, we propose a spectrum of behavior that ranges from conceptual to perceptual outcomes. Conceptual behavior is characterized by task-specific representations, efficient learning, and insensitivity to spurious perceptual details. Perceptual behavior is characterized by strong sensitivity to spurious perceptual details, accompanied by the need for exhaustive training to learn the task. We develop a mathematical theory to show that an MLP's behavior is driven by learning richness. Rich-regime MLPs exhibit conceptual behavior, whereas lazy-regime MLPs exhibit perceptual behavior. We validate our theoretical findings in vision SD experiments, showing that rich feature learning promotes success by encouraging hallmarks of conceptual behavior. Overall, our work identifies feature learning richness as a key parameter modulating equality reasoning, and suggests that equality reasoning in humans and animals may similarly depend on learning richness in neural circuits.

翻译：等价推理具有普遍性且纯粹抽象：无论底层对象的性质如何，都可以评估其相同性或差异性。因此，相同-差异任务（SD）已被广泛研究，作为理解人类及跨物种抽象推理的起点。随着神经网络（NN）在抽象能力方面展现出惊人的显著熟练度，NN中的等价推理也引起了关注。然而，尽管研究广泛，关于等价推理的结论差异巨大且缺乏共识。为厘清SD学习的基本原理，我们建立了多层感知机（MLP）中等价推理的理论框架。借鉴比较心理学中的观察，我们提出了一个从概念性行为到感知性行为连续变化的行为谱系。概念性行为以任务特异性表征、高效学习以及对虚假感知细节的不敏感性为特征；感知性行为则表现为对虚假感知细节的强烈敏感性，并需要大量训练才能掌握任务。我们建立了数学理论以证明MLP的行为由学习丰富度驱动：丰富机制下的MLP呈现概念性行为，而惰性机制下的MLP呈现感知性行为。我们在视觉SD实验中验证了理论发现，表明丰富的特征学习通过促进概念性行为的典型特征来提升任务成功率。总体而言，我们的研究将特征学习丰富度确定为调控等价推理的关键参数，并提示人类与动物的等价推理可能同样依赖于神经回路中的学习丰富度。