An Algebraic View of the Expressivity of Recurrent Language Models

Abstract:What formal languages can a recurrent neural language model recognize? Formal results in the literature conflict: some authors report Turing-completeness, while others show equivalence to regular languages. The reason for this discrepancy is that the underlying arithmetic model differs. The paper develops a unified algebraic account of the expressivity of recurrent neural networks, starting with a formal account of various arithmetic models. This account reduces expressivity to an algebraic question, e.g., whether a network's syntactic monoid divides a certain wreath product. As a case study, the paper revisits diagonal state-space models: the same architecture cannot implement an even-modulus counter once floating-point recurrences are enforced, yet realizes every even-modulus counter under unsigned-integer quantization.

Accepted at the 43rd International Conference on Machine Learning (ICML 2026). 28 pages, 2 figures

Formal Languages and Automata Theory (cs.FL); Computation and Language (cs.CL); Machine Learning (cs.LG)

arXiv:2606.01765 [cs.FL]

(or arXiv:2606.01765v3 [cs.FL] for this version)

https://doi.org/10.48550/arXiv.2606.01765 Focus to learn more arXiv-issued DOI via DataCite

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