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)


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.



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.


  1. 1.
    Agrawal, M., Allender, E., Datta, S.: On TC\(^0\), AC\(^0\), and arithmetic circuits. J. Comput. Syst. Sci. 60(2), 395–421 (2000)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Barrington, D.A.M., Immerman, N., Straubing, H.: On uniformity within NC\(^1\). J. Comput. Syst. Sci. 41, 274–306 (1990)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Caussinus, H., McKenzie, P., Thérien, D., Vollmer, H.: Nondeterministic NC\(^1\) computation. J. Comput. Syst. Sci. 57, 200–212 (1998)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Dawar, A.: The nature and power of fixed-point logic with counting. ACM SIGLOG News 2(1), 8–21 (2015)Google Scholar
  5. 5.
    Ebbinghaus, H., Flum, J., Thomas, W.: Mathematical Logic. Undergraduate Texts in Mathematics. Springer, New York (1994)CrossRefMATHGoogle Scholar
  6. 6.
    Fagin, R.: Generalized first-order spectra and polynomial-time recognizable sets. In: Karp, R.M. (ed.) Complexity of Computation, vol. 7, pp. 43–73. SIAM-AMS Proceedings (1974)Google Scholar
  7. 7.
    Grädel, E.: Model-checking games for logics of imperfect information. Theoret. Comput. Sci. 493, 2–14 (2013)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Gurevich, Y., Lewis, H.: A logic for constant-depth circuits. Inf. Control 61, 65–74 (1984)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Hesse, W.: Division is in uniform TC\({}^0\). In: Orejas, F., Spirakis, P.G., Leeuwen, J. (eds.) ICALP 2001. LNCS, vol. 2076, pp. 104–114. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  10. 10.
    Immerman, N.: Languages that capture complexity classes. SIAM J. Comput. 16, 760–778 (1987)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Immerman, N.: Descriptive Complexity. Graduate Texts in Computer Science. Springer, New York (1999)CrossRefMATHGoogle Scholar
  12. 12.
    Libkin, L.: Elements of Finite Model Theory. Springer, New York (2012)MATHGoogle Scholar
  13. 13.
    Saluja, S., Subrahmanyam, K.V., Thakur, M.N.: Descriptive complexity of #P functions. J. Comput. Syst. Sci. 50(3), 493–505 (1995)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Vollmer, H.: Introduction to Circuit Complexity - A Uniform Approach. Texts in Theoretical Computer Science. An EATCS Series. Springer, New York (1999)CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Institut für Theoretische InformatikLeibniz Universität HannoverHannoverGermany

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