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A Model-Theoretic Characterization of Constant-Depth Arithmetic Circuits (1603.09531v2)

Published 31 Mar 2016 in cs.CC and cs.LO

Abstract: We study the class $\textrm{AC}0$ 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 class $\textrm{AC}0$, we remedy this situation and develop such a characterization of $\textrm{AC}0$. Our characterization can be interpreted as follows: Functions in $\textrm{AC}0$ 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 $\textrm{TC}0$, the class of languages accepted by constant-depth polynomial-size majority circuits.

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