A model-theoretic characterization of constant-depth arithmetic circuits

We study the class #AC0 of functions computed by constant-depth polynomial-size arithmetic circuits of unbounded fan-in addition and multiplication gates. No model-theoretic characterization for arithmetic circuit classes is known so far. Inspired by Immerman's characterization of the Boolean circuit class AC0, we remedy this situation and develop such a characterization of #AC0. Our characterization can be interpreted as follows: Functions in #AC0 are exactly those functions counting winning strategies in first-order model checking games. A consequence of our results is a new model-theoretic characterization of TC0, the class of languages accepted by constant-depth polynomial-size majority circuits.