<ul data-eligibleForWebStory="true">Equality reasoning is a common and abstract concept that can be evaluated regardless of the objects involved.Same-different (SD) tasks are extensively studied to understand abstract reasoning in humans and animals.Neural networks have shown proficiency in abstract reasoning, sparking interest in studying equality reasoning in these models.However, there is little consensus on conclusions regarding equality reasoning in neural networks.A theory of equality reasoning in multi-layer perceptrons (MLP) is developed to clarify the principles in learning SD tasks.Two types of behaviors, conceptual and perceptual, are identified in relation to equality reasoning.Conceptual behavior is task-specific, efficient in learning, and less affected by irrelevant details.Perceptual behavior is highly sensitive to irrelevant details and requires exhaustive training to learn the task.The behavior of an MLP in equality reasoning tasks is driven by learning richness, categorized as rich or lazy regimes.Rich-regime MLPs exhibit conceptual behavior, while lazy-regime MLPs exhibit perceptual behavior.Experimental validation in vision SD tasks shows that rich feature learning enhances success by promoting conceptual behavior.Learning richness in feature learning is identified as a critical factor influencing equality reasoning.The study suggests that equality reasoning in humans and animals could also depend on learning richness in neural circuits.