Formally Verified Autoresearch for Analyzing the Expressivity of Multi-Head Attention

Abstract:Multi-head attention is a central component of the transformer architecture, yet how its expressivity depends on the number of heads remains poorly understood. We study a one-layer, attention-only transformer with additive positional embeddings and define the head complexity $H^\ast(f)$ of a Boolean function $f$ as the minimum number of attention heads required to represent it. We derive an exact linear-fractional normal form for head complexity, prove a general threshold-degree lower bound, and give an explicit ten-bit function for which this bound is strict. For symmetric Boolean functions, we independently show that head complexity equals the number of times the function changes value between consecutive Hamming-weight levels, even when positional embeddings are allowed. This agrees with the concurrent characterization of Hsu and Xu (arXiv:2608.04243) in a model without positional embeddings. The symmetric lower bound combines our general threshold-degree bound with the classical result of Aspnes et al. (1994). These results were discovered and formalized by a frontier model through conjecture generation, theorem proving, and machine verification, while human researchers asked the high-level questions and reviewed the generated theorems. We make the Lean formalization and research traces publicly available to support agent-assisted theorem proving and formal verification in mechanistic interpretability. This work provides a proof of concept for using the capabilities of frontier models, grounded by formal verification, to develop an understanding of transformer architectures.

25 pages, 1 figure. Oral presentation at the AutoMLR Workshop at NeurIPS 2026

Computational Complexity (cs.CC); Artificial Intelligence (cs.AI); Machine Learning (cs.LG)

arXiv:2610.25341 [cs.CC]

(or arXiv:2610.25341v1 [cs.CC] for this version)

https://doi.org/10.48550/arXiv.2610.25341 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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