Relation between $AC^0$ and regular languages Let $\mathsf{REG}$ be the class of all regular languages. 

It is known $\mathsf{AC}^0 \not\subset \mathsf{REG}$ and $\mathsf{REG} \not\subset \mathsf{AC}^0$. But is there any characterization for languages in $\mathsf{AC}^0 \cap \mathsf{REG}$?
The following paper seems to contain an answer:

Mix Barrington, D. A., Compton, K., Straubing, H., Therien, D.: Regular languages in $\mathsf{NC}^1$. Journal of Computer and System Sciences 44(3), 478-499 (1992) (link)

One of the characterizations obtained there is as follows. The class $\mathsf{REG} \cap \mathsf{AC}^0 \subset \{0, 1\}^*$ contains exactly those languages that can be obtained from $\{0\}$, $\{1\}$ and $\mathsf{LENGTH}(q)$ for $q &gt; 1$ with a finite number of Boolean operations and concatenations. Here every language $\mathsf{LENGTH}(q)$ contains all strings whose length is divisible by $q$. (There is also a logical characterization and two algebraic ones.)