On Weak-Space Complexity over Complex Numbers

  • Pushkar S. Joglekar
  • B. V. Raghavendra Rao
  • Siddhartha Sivakumar
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10472)


Defining a feasible notion of space over the Blum-Shub-Smale (BSS) model of algebraic computation is a long standing open problem. In an attempt to define a right notion of space complexity for the BSS model, Naurois [CiE 2007] introduced the notion of weak-space. We investigate the weak-space bounded computations and their plausible relationship with the classical space bounded computations. For weak-space bounded, division-free computations over BSS machines over complex numbers with \(\mathop {=}\limits ^{?}0\) tests, we show the following:

  1. 1.

    The Boolean part of the weak log-space class is contained in deterministic log-space, i.e., \(\mathsf{BP}(\mathsf{LOGSPACE_W}) \subseteq \mathsf{DLOG}\);

  2. 2.

    There is a set \(L\in \) \(\mathsf {NC}^{1}_{\mathbb {C}}\) that cannot be decided by any deterministic BSS machine whose weak-space is bounded above by a polynomial in the input length, i.e., \({\mathsf {NC}}^1_{\mathbb {C}} \nsubseteq \mathsf{PSPACE_W}\).


The second result above resolves the first part of Conjecture 1 stated in [6] over complex numbers and exhibits a limitation of weak-space. The proof is based on the structural properties of the semi-algebraic sets contained in \(\mathsf{PSPACE_W}\) and the result that any polynomial divisible by a degree-\(\omega (1)\) elementary symmetric polynomial cannot be sparse. The lower bound on the sparsity is proved via an argument involving Newton polytopes of polynomials and bounds on number of vertices of these polytopes, which might be of an independent interest.



We thank the anonymous reviewers for this and an earlier version of the paper for suggestions that helped to improve the presentation of proofs.


  1. 1.
    Allender, E., Bürgisser, P., Kjeldgaard-Pedersen, J., Miltersen, P.B.: On the complexity of numerical analysis. SIAM J. Comput. 38(5), 1987–2006 (2009)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Blum, L., Cucker, F., Shub, M., Smale, S.: Complexity and Real Computation. Springer, New York (1997). doi: 10.1007/978-1-4612-0701-6 MATHGoogle Scholar
  3. 3.
    Blum, L., Shub, M., Smale, S.: On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines. Bull. (New Ser.) Am. Math. Soc. 21(1), 1–46 (1989)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Cucker, F.: P\({}_{ \text{ R }}\) != NC\({}_{ \text{ R }}\). J. Complex. 8(3), 230–238 (1992)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Cucker, F., Grigoriev, D.: On the power of real turing machines over binary inputs. SIAM J. Comput. 26(1), 243–254 (1997)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    de Naurois, P.J.: A measure of space for computing over the reals. In: Beckmann, A., Berger, U., Löwe, B., Tucker, J.V. (eds.) CiE 2006. LNCS, vol. 3988, pp. 231–240. Springer, Heidelberg (2006). doi: 10.1007/11780342_25 CrossRefGoogle Scholar
  7. 7.
    Forbes, M.A.: Personal communicationGoogle Scholar
  8. 8.
    Forbes, M.A., Shpilka, A., Tzameret, I., Wigderson, A.: Proof complexity lower bounds from algebraic circuit complexity. CoRR, abs/1606.05050 (2016)Google Scholar
  9. 9.
    Fournier, H., Koiran, P.: Are lower bounds easier over the reals? In: Proceedings of 30th Annual ACM Symposium on Theory of Computing, STOC 1998, New York, NY, USA, pp. 507–513. ACM (1998)Google Scholar
  10. 10.
    Gao, S.: Absolute irreducibility of polynomials via Newton polytopes. J. Algebra 237(1), 501–520 (1997)MathSciNetMATHGoogle Scholar
  11. 11.
    Gruenbaum, B.: Convex Polytopes. Interscience Publisher, New York (1967)Google Scholar
  12. 12.
    Joglekar, P., Raghavendra Rao, B.V., Sivakumar, S.: On weak-space complexity over complex numbers. In: Electronic Colloquium on Computational Complexity (ECCC), vol. 24, p. 87 (2017)Google Scholar
  13. 13.
    Koiran, P.: Computing over the reals with addition and order. Theoret. Comput. Sci. 133(1), 35–47 (1994)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Koiran, P.: Elimination of constants from machines over algebraically closed fields. J. Complex. 13(1), 65–82 (1997)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Koiran, P.: A weak version of the Blum, Shub, and Smale model. J. Comput. Syst. Sci. 54(1), 177–189 (1997)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Koiran, P., Perifel, S.: VPSPACE and a transfer theorem over the complex field. Theor. Comput. Sci. 410(50), 5244–5251 (2009)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Koiran, P., Perifel, S.: VPSPACE and a transfer theorem over the reals. Comput. Complex. 18(4), 551–575 (2009)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Koiran, P., Portier, N., Tavenas, S., Thomassé, S.: A tau-conjecture for Newton polygons. Found. Comput. Math. 15(1), 185–197 (2015)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Mahajan, M., Raghavendra Rao, B.V.: Small space analogues of valiant’s classes and the limitations of skew formulas. Comput. Complex. 22(1), 1–38 (2013)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Meer, K., Michaux, C.: A survey on real structural complexity theory. Bull. Belg. Math. Soc. Simon Stevin 4(1), 113–148 (1997)MathSciNetMATHGoogle Scholar
  21. 21.
    Michaux, C.: Une remarque à propos des machines sur \(\mathbb{R}\) introduites par Blum, Shub et Smale. Comptes Rendus de l’Académie des Sciences de Paris 309(7), 435–437 (1989)MathSciNetMATHGoogle Scholar
  22. 22.
    Morandi, P.: Field and Galois Theory. Graduate Texts in Mathematics. Springer, Cham (1996). doi: 10.1007/978-1-4612-4040-2 CrossRefMATHGoogle Scholar
  23. 23.
    Ostrowski, A.M.: On multiplication and factorization of polynomials, i. lexicographic ordering and extreme aggregates of terms. Aequationes Mathematicae 13, 201–228 (1975)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Schneider, R.: Convex Bodies: The Brunn-Minkowski Theory. Cambridge University Press, Cambridge (1993)CrossRefMATHGoogle Scholar
  25. 25.
    Shafarevich, I.R.: Basic Algebraic Geometry, 3rd edn. Springer, Berlin (2013). doi: 10.1007/978-3-642-96200-4 CrossRefMATHGoogle Scholar
  26. 26.
    Shpilka, A., Yehudayoff, A.: Arithmetic circuits: a survey of recent results and open questions. Found. Trends\(\textregistered \) Theoret. Comput. Sci. 5(3–4), 207–388 (2010)Google Scholar
  27. 27.
    Tzamaret, I.: Studies in algebraic and propositional proof complexity. Ph.D. thesis, Tel Aviv University (2008)Google Scholar
  28. 28.
    Valiant, L.G.: The complexity of computing the permanent. Theor. Comput. Sci. 8, 189–201 (1979)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Ziegler, G.M.: Lectures on Polytopes. Springer, New York (1995). doi: 10.1007/978-1-4613-8431-1 CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  • Pushkar S. Joglekar
    • 1
  • B. V. Raghavendra Rao
    • 2
  • Siddhartha Sivakumar
    • 2
  1. 1.Vishwakarma Institute of TechnologyPuneIndia
  2. 2.Indian Institute of Technology MadrasChennaiIndia

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