A Model-Theoretic Characterization of Constant-Depth Arithmetic Circuits

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9803)


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

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Institut für Theoretische InformatikLeibniz Universität HannoverHannoverGermany

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