Theory of Computing Systems

, Volume 58, Issue 4, pp 506–527 | Cite as

The Arithmetic Complexity of Tensor Contraction

  • Florent Capelli
  • Arnaud Durand
  • Stefan Mengel


We investigate the algebraic complexity of tensor calculus. We consider a generalization of iterated matrix product to tensors and show that the resulting formulas exactly capture V P, the class of polynomial families efficiently computable by arithmetic circuits. This gives a natural and robust characterization of this complexity class that despite its naturalness is not very well understood so far.


Algebraic complexity Arithmetic circuits Tensor calculus 



We thank Yann Strozecki for a detailed and helpful feedback on an early version of this paper. We also thank Hervé Fournier, Guillaume Malod and Sylvain Perifel for helpful discussions.


  1. 1.
    Beaudry, M., Holzer, M.: The complexity of tensor circuit evaluation. Comput. Complex. 16(1), 60–111 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bürgisser, P.: Completeness and Reduction in Algebraic Complexity Theory, vol. 7. Springer, Berlin Heidelberg New York (2000)CrossRefzbMATHGoogle Scholar
  3. 3.
    Cook, S.A., McKenzie, P.: Problems complete for deterministic logarithmic space. Journal of Algorithms 8(3), 385–394 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Damm, C., Holzer, M., McKenzie, P.: The complexity of tensor calculus. Comput. Complex. 11(1-2), 54–89 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Durand, A., Mengel, S.: The complexity of weighted counting for acyclic conjunctive queries. J. Comput. Syst. Sci. 80(1), 277–296 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Gottlob, G., Leone, N., Scarcello, F.: The complexity of acyclic conjunctive queries. J. ACM 48(3), 431–498 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Koiran, P.: Arithmetic circuits: the chasm at depth four gets wider. Theor. Comput. Sci. 448(0), 56–65 (2012). doi: 10.1016/j.tcs.2012.03.041 MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Lautemann, C., McKenzie, P., Schwentick, T., Vollmer, H.: The descriptive complexity approach to logcfl. J. Comput. Syst. Sci. 62(4), 629–652 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Malod, G.: Circuits arithmétiques et calculs tensoriels. Journal of the Institute of Mathematics of Jussieu 7, 869–893 (2005)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Malod, G., Portier, N.: Characterizing Valiant’s algebraic complexity classes. J. Complex. 24(1), 16–38 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Marcus, M.: Finite Dimensional Multilinear Algebra, vol. 1. M. Dekker, New York (1973)Google Scholar
  12. 12.
    Mengel, S.: Characterizing arithmetic circuit classes by constraint satisfaction problems - (extended abstract). In: ICALP (1), pp. 700–711 (2011)Google Scholar
  13. 13.
    Mengel, S.: Arithmetic branching programs with memory. In: Chatterjee, K., Sgall, J. (eds.) Mathematical Foundations of Computer Science 2013. no. 8087 in Lecture Notes in Computer Science, pp. 667–678. Springer, Berlin Heidelberg New York (2013)Google Scholar
  14. 14.
    Nisan, N.: Lower bounds for non-commutative computation. In: Proceedings of the 23rd Annual ACM Symposium on Theory of Computing, pp. 410–418. ACM (1991)Google Scholar
  15. 15.
    Valiant, L.: Completeness classes in algebra. In: Proceedings of the 11th Annual ACM Symposium on Theory of Computing, pp. 249–261. ACM (1979)Google Scholar
  16. 16.
    Valiant, L., Skyum, S., Berkowitz, S., Rackoff, C.: Fast parallel computation of polynomials using few processors. SIAM J. Comput. 12, 641 (1983)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.IMJ UMR 7586 - LogiqueUniversité Paris DiderotParisFrance
  2. 2.LIX UMR 7161, Ecole PolytechniquePalaiseauFrance

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