Variable wire weights in DLOGTIME-uniform circuits

Question

The definition of a $DLOGTIME$-uniform circuit family is based on a Turing machine that accepts the language $\langle t, a, b \rangle$, where gate $a$ is of type $t$ and has gate $b$ as a child, according to Barrington et. al's paper "On Uniformity Within $NC^1$" (1). However, this definition assumes that whether there is a wire or not between two gates is a binary choice. While this is true for $AND/OR$-based circuit classes like $AC^0$, other circuit classes like $CC^0$ and especially $TC^0$ assume that circuits can have an arbitrary constant or even $O(n)$ number of wires between two gates.

My questions are:

• Is there a standard way in the literature to extend the definition of dlogtime-uniformity with wire weights greater than 1?
• Is there a standard way to add this feature to logical characterizations of these families (e.g. $FO[bit]$)?

Answer 1

After re-reading Barrington et al., it seems that the case of $CC^0$ can be handled by using "group quantifiers", which allow for quantifiers to act over finite groups (i.e. $\mathbb{Z}/m\mathbb{Z}$) directly, while still using a binary representation under the hood so that the original definition of $DLOGTIME$-uniformity can still be used. With group quantifiers, wire weights (or any arbitrary constant-size function) can be simulated with a formula over the bits of the representation. I am still unsure how this might extend to a class like $TC^0$ that operates over an infinite monoid/group (or at least one that grows with $n$), but this answers the core of the question I had.