A Model-Theoretic Characterization of Constant-Depth Arithmetic Circuits

Conference paper

DOI: 10.1007/978-3-662-52921-8_15

Volume 9803 of the book series Lecture Notes in Computer Science (LNCS)
Cite this paper as:
Haak A., Vollmer H. (2016) A Model-Theoretic Characterization of Constant-Depth Arithmetic Circuits. In: Väänänen J., Hirvonen Å., de Queiroz R. (eds) Logic, Language, Information, and Computation. WoLLIC 2016. Lecture Notes in Computer Science, vol 9803. Springer, Berlin, Heidelberg

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.

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Institut für Theoretische InformatikLeibniz Universität HannoverHannoverGermany