Uniformity within Parameterized Circuit Classes

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.