- The "direct connection language" of a circuit family is the set of tuples $\langle t, a, b, y \rangle$, where $a$ and $b$ are node/gate numbers in the $n$th circuit in the family, $t$ is the type of gate $a$, gate $b$ is a child of gate $a$, and $y$ is a padding string of length $n$.
- A circuit family is $DLOGTIME$-uniform if its direct connection language can be recognized in $O(\log(n))$ time by a deterministic multi-tape Turing machine (with an index tape for random access to the input).
$DLOGTIME$-uniformity was introduced by Barrington et al. here, and seems to be the standard lowest uniformity measure used for e.g. constant-depth circuit classes ($AC^0$, $ACC^0$, etc.). Are there circuit uniformities more strict than $DLOGTIME$-uniformity, especially with pre-existing basis in literature? The only one I could find was Rational uniformity from this paper, but it doesn't seem to have caught on. If there are no other measures used in the literature now, I would still accept "theoretical" uniformity measures more strict/sharp/restrictive than $DLOGTIME$-uniformity.