  In this paper we give a characterization of both Boolean and arithmetic
circuit classes of logarithmic depth in the vein of descriptive complexity
theory, i.e., the Boolean classes $\textrm{NC}^1$, $\textrm{SAC}^1$ and
$\textrm{AC}^1$ as well as their arithmetic counterparts $\#\textrm{NC}^1$,
$\#\textrm{SAC}^1$ and $\#\textrm{AC}^1$. We build on Immerman's
characterization of constant-depth polynomial-size circuits by formulas of
first-order logic, i.e., $\textrm{AC}^0 = \textrm{FO}$, and augment the logical
language with an operator for defining relations in an inductive way.
Considering slight variations of the new operator, we obtain uniform
characterizations of the three just mentioned Boolean classes. The arithmetic
classes can then be characterized by functions counting winning strategies in
semantic games for formulas characterizing languages in the corresponding
Boolean class.
