Human language and logic abilities are computationally quantified within the well-studied grammar-automata hierarchy. We identify three hierarchical tiers and two corresponding transitions and show their correspondence to specific abilities in transformer-based language models (LMs). These emergent abilities have often been described in terms of scaling; we show that it is the transition between tiers, rather than scaled size itself, that determines a system's capabilities. Specifically, humans effortlessly process language yet require critical training to perform arithmetic or logical reasoning tasks; and LMs possess language abilities absent from predecessor systems, yet still struggle with logical processing. We submit a novel benchmark of computational power, provide empirical evaluations of humans and fifteen LMs, and, most significantly, provide a theoretically grounded framework to promote careful thinking about these crucial topics. The resulting principled analyses provide explanatory accounts of the abilities and shortfalls of LMs, and suggest actionable insights into the expansion of their logic abilities.

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