Uncomputability Below the Real Halting Problem

  • Klaus Meer
  • Martin Ziegler
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3988)


Most of the existing work in real number computation theory concentrates on complexity issues rather than computability aspects. Though some natural problems like deciding membership in the Mandelbrot set or in the set of rational numbers are known to be undecidable in the Blum-Shub-Smale (BSS) model of computation over the reals, there has not been much work on different degrees of undecidability. A typical question into this direction is the real version of Post’s classical problem: Are there some explicit undecidable problems below the real Halting Problem?

In this paper we study three different topics related to such questions: First an extension of a positive answer to Post’s problem to the linear setting. We then analyze how additional real constants increase the power of a BSS machine. And finally a real variant of the classical word problem for groups is presented which we establish reducible to and from (that is, complete for) the BSS Halting problem.


Word Problem Turing Machine Real Constant Real Group Cylindrical Algebraic Decomposition 
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. [Bak75]
    Baker, A.: Transcendental Number Theory. Camb. Univ. Press, Cambridge (1975)CrossRefMATHGoogle Scholar
  2. [BCSS98]
    Blum, L., Cucker, F., Shub, M., Smale, S.: Complexity and Real Computation. Springer, Heidelberg (1998)CrossRefMATHGoogle Scholar
  3. [BMM00]
    Ben-David, S., Meer, K., Michaux, C.: A note on non-complete problems in NP . Journal of Complexity 16(1), 324–332 (2000)MathSciNetCrossRefMATHGoogle Scholar
  4. [Boo58]
    Boone, W.W.: The word problem. Proc. Nat. Acad. Sci. U.S.A 44, 265–269 (1958)MathSciNetCrossRefMATHGoogle Scholar
  5. [BSS89]
    Blum, L., Shub, M., Smale, S.: On a Theory of Computation and Complexity over the Real Numbers: \(\mathcal{NP}\)-Completeness, Recursive Functions, and Universal Machines. Bulletin of the American Mathematical Society (AMS Bulletin) 21, 1–46 (1989)MathSciNetCrossRefMATHGoogle Scholar
  6. [Cha99]
    Chapuis, O., Koiran, P.: Saturation and stability in the theory of computation over the reals. Annals of Pure and Applied Logic 99, 1–49 (1999)MathSciNetCrossRefMATHGoogle Scholar
  7. [CG97]
    Cucker, F., Grigoriev, D.Y.: On the power of real turing machines over binary inputs. SIAM Journal on Computing 26(1), 243–254 (1997)MathSciNetCrossRefMATHGoogle Scholar
  8. [CK95]
    Cucker, F., Koiran, P.: Computing over the Real with Addition and Order: Higher Complexity Classes. Journal of Complexity 11, 358–376 (1995)MathSciNetCrossRefMATHGoogle Scholar
  9. [Cuc92]
    Cucker, F.: The arithmetical hierarchy over the reals. Journal of Logic and Computation 2(3), 375–395 (1992)MathSciNetCrossRefMATHGoogle Scholar
  10. [Dei85]
    Deimling, K.: Nonlinear Functional Analysis. Springer, Heidelberg (1985)CrossRefMATHGoogle Scholar
  11. [DJK05]
    Derksen, H., Jeandel, E., Koiran, P.: Quantum automata and algebraic groups. J. Symbolic Computation 39, 357–371 (2005)MathSciNetCrossRefMATHGoogle Scholar
  12. [FK00]
    Fournier, H., Koiran, P.: Lower Bounds Are Not Easier over the Reals. In: Welzl, E., Montanari, U., Rolim, J.D.P. (eds.) ICALP 2000. LNCS, vol. 1853, pp. 832–843. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  13. [Fri57]
    Friedberg, R.M.: Two recursively enumerable sets of incomparable degrees of unsolvability. Proc. Natl. Acad. Sci. 43, 236–238 (1957)CrossRefMATHGoogle Scholar
  14. [Koi93]
    Koiran, P.: A weak version of the Blum-Shub-Smale model. In: Proceedings FOCS 1993, pp. 486–495 (1993)Google Scholar
  15. [Koi94]
    Koiran, P.: Computing over the Reals with Addition and Order. Theoretical Computer Science 133, 35–48 (1994)MathSciNetCrossRefMATHGoogle Scholar
  16. [Koi96]
    Koiran, P.: Elimination of Constants from Machines over Algebraically Closed Fields. Journal of Complexity 13, 65–82 (1997)MathSciNetCrossRefMATHGoogle Scholar
  17. [LS77]
    Lyndon, R.C., Schupp, P.E.: Combinatorial Group Theory. Springer, Heidelberg (1977)MATHGoogle Scholar
  18. [Mic90]
    Michaux, C.: Machines sur les réels et problèmes \(\mathcal{NP}\)-complets. Séminaire de Structures Algébriques Ordonnées, Prépublications de l’equipe de logique mathématique de Paris 7 (1990)Google Scholar
  19. [Mic91]
    Michaux, C.: Ordered rings over which output sets are recursively enumerable. In: Proceedings of the AMS, vol. 111, pp. 569–575 (1991)Google Scholar
  20. [Muc58]
    Muchnik, A.A.: Solution of Post’s reduction problem and of certain other problems in the theory of algorithms. Trudy Moskov Mat. Obsc. 7, 391–405 (1958)MathSciNetGoogle Scholar
  21. [MZ05]
    Meer, K., Ziegler, M.: An explicit solution to Post’s problem over the reals. In: Liśkiewicz, M., Reischuk, R. (eds.) FCT 2005. LNCS, vol. 3623, pp. 467–478. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  22. [MZ06]
    Meer, K., Ziegler, M.: On the word problem for a class of groups with infinite presentations (preprint 2006)Google Scholar
  23. [Nov59]
    Novikov, P.S.: On the algorithmic unsolvability of the word problem in group theory. Trudy Mat. Inst. Steklov 44, 1–143 (1959)Google Scholar
  24. [Pos44]
    Post, E.L.: Recursively enumerable sets of positive integers and their decision problems. Bull. Amer. Math. Soc. 50, 284–316 (1944)MathSciNetCrossRefMATHGoogle Scholar
  25. [Tuc80]
    Tucker, J.V.: Computability and the algebra of fields. J. Symbolic Logic 45, 103–120 (1980)MathSciNetCrossRefMATHGoogle Scholar
  26. [TZ00]
    Tucker, J.V., Zucker, J.I.: Computable functions and semicomputable sets on many sorted algebras. In: Abramskz, S., Gabbay, D., Maibaum, T. (eds.) Handbook of Logic for Computer Science. Logic and Algebraic Methods, vol. V, pp. 317–523. Oxford University Press, Oxford (2000)Google Scholar
  27. [Wei01]
    Weihrauch, K.: Computable Analysis. Springer, Heidelberg (2001)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Klaus Meer
    • 1
  • Martin Ziegler
    • 2
  1. 1.IMADASyddansk UniversitetOdense MDenmark
  2. 2.University of PaderbornGermany

Personalised recommendations