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

Karthik Viswanathan, Plamen Dimitrov, Stefano Rocca, Sarayu Sirikonda

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…