Arithmetic Branching Programs with Memory

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

Abstract

We extend the well known characterization of the arithmetic circuit class VPws 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.

References

  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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Malod, G., Portier, N.: Characterizing Valiant’s algebraic complexity classes. J. Complexity 24(1), 16–38 (2008)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

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

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