Counting problems over the reals

  • Klaus Meer
Contributed Papers
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1295)


In this paper we introduce a complexity theoretic notion of counting problems over the real numbers. We follow the approaches of Blum, Shub, and Smale


Polynomial Time Function Symbol Counting Function Real Zero Satisfying Assignment 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Allgower, E.L., Georg, K.: Continuation and Path Following. Acta Numerica. (1992) 1–64Google Scholar
  2. 2.
    Benedetti, R., Risler, J.J.: Real algebraic and semi-algebraic sets. Hermann (1990)Google Scholar
  3. 3.
    Blum, L., Cucker, F., Shub, M., Smale, S.: Complexity and Real Computation. Springer-Verlag (to appear)Google Scholar
  4. 4.
    Blum, L., Shub, M., Smale, S.: On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines. Bulletin American Mathematical Society 21 (1989) 1–46Google Scholar
  5. 5.
    Compton, K.J., Grädel, E.: Logical Definability of Counting Functions Proceedings of IEEE Conference on Structure in Complexity Theory (1994) 255–266Google Scholar
  6. 6.
    Cucker, F.:, Meer, K.: Logics which capture complexity classes over the reals. Extended abstract to appear in: Proc. 11th International Symposium on Fundamentals of Computation Theory Krakow, Lecture Notes in Computer Science, Springer 1997Google Scholar
  7. 7.
    Ebbinghaus, H.D., Flum, J.: Finite Model Theory. Springer-Verlag (1995)Google Scholar
  8. 8.
    Fagin, R.: Generalized first-order spectra and polynomial-time recognizable sets. SIAM-AMS Proc. 7 (1974) 43–73Google Scholar
  9. 9.
    Grädel, E., Meer, K.: Descriptive complexity theory over the real numbers. In J. Renegar, M. Shub, and S. Smale (editors): The Mathematics of Numerical Analysis, Lectures in Applied Mathematics 32, American Mathematical Society (1996) 381–404Google Scholar
  10. 10.
    Heintz, J., Roy, M.F., Solerno, P.: On the complexity of semialgebraic sets. Proceedings IFIP 1989, San Francisco, North-Holland (1989) 293–298Google Scholar
  11. 11.
    Heintz, J., Krick, T., Roy, M.F., Solerno, P.: Geometric Problems solvable in single exponential time. In: Proc. 8th Conference AAECC, LNCS 508 (1990) 11–23Google Scholar
  12. 12.
    Meer, K., Michaux, C.: A survey on real structural complexity theory. Bulletin of the Belgian Math. Soc. Simon Stevin 4 (1997) 113–148Google Scholar
  13. 13.
    Pedersen, P.: Multivariate Sturm theory. In: Proc. 9th Conference AAECC, LNCS 539 (1991) 318–331Google Scholar
  14. 14.
    Renegar, J.: On the computational Complexity and Geometry of the first-order Theory of the Reals, I-III. Journal of Symbolic Computation 13 (1992) 255–352Google Scholar
  15. 15.
    Saluja, S., Subrahmanyam, K.V., Thakur, M.N.: Descriptive Complexity of #P Functions. In Proc. 7th IEEE Symposium on Structure in Complexity Theory (1992) 169–184Google Scholar
  16. 16.
    Shub, M., Smale, S.: Complexity of Bezout's Theorem I: Geometric aspects. Journal of the AMS 6 (1993) 459–501Google Scholar
  17. 17.
    Valiant, L.: The complexity of computing the permanent. Theoretical Computer Science 8 (1979) 189–201CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Klaus Meer
    • 1
  1. 1.RWTH Aachen Lehrstuhl C für MathematikAachenGermany

Personalised recommendations