Theory of Computing Systems

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

The Arithmetic Complexity of Tensor Contraction



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 VP, 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 


  1. 1.
    Beaudry, M., Holzer, M.: The complexity of tensor circuit evaluation. Comput. Complex. 16(1), 60–111 (2007)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bürgisser, P.: Completeness and Reduction in Algebraic Complexity Theory, vol. 7. Springer, Berlin Heidelberg New York (2000)CrossRefMATHGoogle Scholar
  3. 3.
    Cook, S.A., McKenzie, P.: Problems complete for deterministic logarithmic space. Journal of Algorithms 8(3), 385–394 (1987)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Damm, C., Holzer, M., McKenzie, P.: The complexity of tensor calculus. Comput. Complex. 11(1-2), 54–89 (2002)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Gottlob, G., Leone, N., Scarcello, F.: The complexity of acyclic conjunctive queries. J. ACM 48(3), 431–498 (2001)MathSciNetCrossRefMATHGoogle 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 MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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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