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A Model-Theoretic Characterization of Constant-Depth Arithmetic Circuits

  • Anselm Haak
  • Heribert Vollmer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9803)

Abstract

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.

Notes

Acknowledgements

We are grateful to Lauri Hella (Tampere) and Juha Kontinen (Helsinki) for helpful discussion, leading in particular to Definition 20. We also thank the anonymous referees for helpful comments.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Institut für Theoretische InformatikLeibniz Universität HannoverHannoverGermany

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