Humans possess a remarkable ability to draw inferences from limited data. While some inferences are made intuitively from past experience, others require online reasoning based on novel information given in context. Here, we formalize these strategies, which we term geometric and algebraic inference, and study their computational tradeoffs in humans and neural networks. The geometric strategy gradually shapes representational geometry to reflect underlying relational structure, making subsequent inference cheap. The algebraic strategy flexibly attends to and reasons over novel relations given in context, but costs additional computation at inference time. We identify opposing signatures of these two strategies in transitive inference, a domain widely studied across species, and characterize the conditions under which each is observed. Our framework predicts that the geometric strategy should be dominant when relations are familiar, whereas the algebraic should emerge to support de novo inferences drawn in context. Multiple experiments demonstrate these opposing signatures in humans, neural networks trained from scratch, and large language models (LLMs). Causal interventions in the neural networks reveal that the geometric and algebraic strategies depend on representational geometry and selective attention mechanisms, respectively, and reproduce the effects of analogous neurobiological disruptions in humans. Our framework unifies these effects with disparate phenomena, including content effects observed in human and LLM reasoning and data-distribution effects observed in machine learning. Our findings reconcile competing views of cognitive flexibility that emphasize either representational geometry or algebraic manipulation by showing that each can have unique strengths, and can characterize different aspects of both human and neural network behavior.

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