Quantitative Aspects of Speed-Up and Gap Phenomena

  • Klaus Ambos-Spies
  • Thorsten Kräling
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5532)


We show that, for any abstract complexity measure in the sense of Blum and for any computable function f (or computable operator F), the class of problems which are f-speedable (or F-speedable) does not have effective measure 0. On the other hand, for sufficiently fast growing f (or F), the class of the nonspeedable problems does not have effective measure 0 too. These results answer some questions raised by Calude and Zimand in [CZ96] and [Zim06]. We also give a short quantitative analysis of Borodin and Trakhtenbrot’s Gap Theorem which corrects a claim in [CZ96] and [Zim06].


Quantitative Aspect Computable Function Category Concept Baire Space Abstract Complexity 
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. [AS96]
    Ambos-Spies, K.: Resource-bounded genericity. In: Cooper, S., Slaman, T., Wainer, S. (eds.) Computability, enumerability, unsolvability. Directions in recursion theory, pp. 1–60. Cambridge University Press, Cambridge (1996)Google Scholar
  2. [ASM97]
    Ambos-Spies, K., Mayordomo, E.: Resource-bounded measure and randomness. In: Sorbi, A. (ed.) Complexity, Logic and Recursion Theory, pp. 1–47. Dekker, New York (1997)Google Scholar
  3. [ASR97]
    Ambos-Spies, K., Reimann, J.: Effective Baire category concepts. In: Proc. Sixth Asian Logic Conference 1996, pp. 13–29. Singapore University Press (1997)Google Scholar
  4. [Blu67]
    Blum, M.: A machine-independent theory of the complexity of recursive functions. Journal of the ACM 14(2), 322–336 (1967)zbMATHCrossRefMathSciNetGoogle Scholar
  5. [Bor72]
    Borodin, A.: Computational complexity and the existence of complexity gaps. Journal of the ACM 19(1), 158–174 (1972)zbMATHCrossRefMathSciNetGoogle Scholar
  6. [CZ96]
    Calude, C., Zimand, M.: Effective category and measure in abstract complexity theory. Theoretical Computer Science 154(2), 307–327 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  7. [Lu92]
    Lutz, J.: Almost everywhere high nonuniform complexity. Journal of Computer and System Sciences 44, 220–258 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  8. [May94]
    Mayordomo, E.: Almost every set in exponential time is P-bi-immune. Theoretical Computer Science 136, 487–506 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  9. [Me73]
    Mehlhorn, K.: On the size of sets of computable functions. In: Proceedings of the 14th IEEE Symposium on Switching and Automata Theory, pp. 190–196 (1973)Google Scholar
  10. [MF72]
    Meyer, A.R., Fischer, P.C.: Computational speed-up by effective operators. Journal of Symbolic Logic 37(1), 55–68 (1972)zbMATHCrossRefMathSciNetGoogle Scholar
  11. [Sch73]
    Schnorr, C.: Process complexity and effective random tests. Journal of Computer and System Sciences 7, 376–388 (1973)zbMATHCrossRefMathSciNetGoogle Scholar
  12. [Tra67]
    Trakhtenbrot, B.A.: Complexity of algorithms and computations. Course Notes, Novosibirsk (in Russian) (1967)Google Scholar
  13. [Zim06]
    Zimand, M.: Computational Complexity: A Quantitative Perspective. Elsevier, Amsterdam (2006)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Klaus Ambos-Spies
    • 1
  • Thorsten Kräling
    • 1
  1. 1.Institut für InformatikUniversity of HeidelbergHeidelbergGermany

Personalised recommendations