Uniformity within Parameterized Circuit Classes
Year: 2025
Type: preprint
Abstract: We study uniformity conditions for parameterized Boolean circuit families. Uniformity conditions require that the infinitely many circuits in a circuit family are in some sense easy to construct from one shared description. For shallow circuit families, logtime-uniformity is often desired but quite technical to prove. Despite that, proving it is often left as an exercise for the reader -- even for recently introduced classes in parameterized circuit complexity, where uniformity conditions have not yet been explicitly studied. We formally define parameterized versions of linear-uniformity, logtime-uniformity, and FO-uniformity, and prove that these result in equivalent complexity classes when imposed on $\text{para-}\textsf{AC}^0$ and $\text{para-}\textsf{AC}^{0\uparrow}$. Overall, we provide a convenient way to verify uniformity for shallow parameterized circuit classes, and thereby substantiate claims of uniformity in the literature.
Source: arXiv (Cornell University)
Authors: Hegeman, Steef, Martens, Jan, Laarman, Alfons
Institution: Leiden University
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7081569447
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