Year: 2026
Type: preprint
Abstract: A longstanding open question in Boolean function complexity asks whether approximate degree composes multiplicatively under block composition. Although a general multiplicative upper bound is known, matching lower bounds have previously been established only for restricted classes of functions. We resolve this question for all total Boolean functions by proving the matching lower bound. Together with Sherstov's upper bound, our result shows that, for every pair of total Boolean functions $f:\{0,1\}^n\to\{0,1\}$ and $g:\{0,1\}^m\to\{0,1\}$, \[ \widetilde{deg}(f\circ g) = Θ\!\left( \widetilde{deg}(f)\,\widetilde{deg}(g) \right), \] where $\widetilde{deg}$ denotes constant-error approximate degree. We also consider the analogous question for exact quantum query complexity and give a counterexample showing that multiplicative composition fails in this setting.
Source: arXiv (Cornell University)
Authors: Pednekar, Samruddhi, Podder, Supartha
Institution: Stony Brook University
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214353741