Arithmetic Branching Programs with Memory

  • Stefan Mengel
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8087)


We extend the well known characterization of the arithmetic circuit class VP ws as the class of polynomials computed by polynomial size arithmetic branching programs to other complexity classes. In order to do so we add additional memory to the computation of branching programs to make them more expressive. We show that allowing different types of memory in branching programs increases the computational power even for constant width programs. In particular, this leads to very natural and robust characterizations of VP and VNP by branching programs with memory.


Directed Acyclic Graph Arithmetic Circuit Polynomial Size Input Gate Output Gate 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Barrington, D.A.: Bounded-width polynomial-size branching programs recognize exactly those languages in NC1. Journal of Computer and System Sciences 38(1), 150–164 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ben-Or, M., Cleve, R.: Computing algebraic formulas using a constant number of registers. SIAM J. Comput. 21(1), 54–58 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Brent, R.P.: The complexity of multiple-precision arithmetic. In: Brent, R.P., Andersson, R.S. (eds.) The Complexity of Computational Problem Solving, pp. 126–165. Univ. of Queensland Press (1976)Google Scholar
  4. 4.
    Briquel, I., Koiran, P.: A dichotomy theorem for polynomial evaluation. In: Královič, R., Niwiński, D. (eds.) MFCS 2009. LNCS, vol. 5734, pp. 187–198. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  5. 5.
    Bürgisser, P.: Completeness and reduction in algebraic complexity theory. Springer (2000)Google Scholar
  6. 6.
    Kintali, S.: Realizable paths and the NL vs L problem. Electronic Colloquium on Computational Complexity (ECCC) 17, 158 (2010)Google Scholar
  7. 7.
    Koiran, P.: Arithmetic circuits: The chasm at depth four gets wider. Theor. Comput. Sci. 448, 56–65 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Malod, G., Portier, N.: Characterizing Valiant’s algebraic complexity classes. J. Complexity 24(1), 16–38 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Mengel, S.: Arithmetic Branching Programs with Memory, arXiv:1303.1969 (2013)Google Scholar
  10. 10.
    Nisan, N.: Lower bounds for non-commutative computation. In: Proceedings of the Twenty-Third Annual ACM Symposium on Theory of Computing, p. 418. ACM (1991)Google Scholar
  11. 11.
    Skyum, S., Valiant, L.G.: A complexity theory based on boolean algebra. J. ACM 32(2), 484–502 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Toda, S.: Classes of arithmetic circuits capturing the complexity of computing the determinant. IEICE Transactions on Information and Systems 75(1), 116–124 (1992)Google Scholar
  13. 13.
    Valiant, L.G., Skyum, S., Berkowitz, S., Rackoff, C.: Fast parallel computation of polynomials using few processors. SIAM J. Comput. 12(4), 641–644 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Valiant, L.G.: Completeness classes in algebra. In: Proceedings of the Eleventh Annual ACM Symposium on Theory of Computing, pp. 249–261. ACM (1979)Google Scholar
  15. 15.
    Weber, V., Schwentick, T.: Dynamic complexity theory revisited. Theory Comput. Syst. 40(4), 355–377 (2007)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Stefan Mengel
    • 1
  1. 1.Institute of MathematicsUniversity of PaderbornPaderbornGermany

Personalised recommendations