We extend the familiar program of understanding circuit complexity in terms of regular languages to visibly counter languages. Like the regular languages, the visibly counter languages are NC1- complete. We investigate what the visibly counter languages in certain constant depth circuit complexity classes are. We have initiated this in a previous work for AC0. We present characterizations and decidability results for various logics and circuit classes. In particular, our approach yields a way to understand TC0, where the regular approach fails.