A Model-Theoretic Characterization of Constant-Depth Arithmetic Circuits

Anselm Haak; Heribert Vollmer

doi:10.1016/j.apal.2019.04.006

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arXiv · 1603.09531

A Model-Theoretic Characterization of Constant-Depth Arithmetic Circuits

Anselm Haak·Heribert Vollmer

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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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BibTeX RIS

Anselm Haak, Heribert Vollmer. 2016-03-31. A Model-Theoretic Characterization of Constant-Depth Arithmetic Circuits. https://doi.org/10.1016/j.apal.2019.04.006

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