Theory of Computing Systems

, Volume 43, Issue 3–4, pp 603–624 | Cite as

Classification of Computably Approximable Real Numbers

Article

Abstract

A real number is called computably approximable if it is the limit of a computable sequence of rational numbers. Therefore the complexity of these real numbers can be classified by considering the convergence speed of computable sequences. In this paper we introduce a natural way to measure the convergence speed by counting the number of jumps of given sizes that appear after certain stages. Bounding the number of such kind of big jumps by different bounding functions, we introduce various classes of real numbers with different levels of approximation quality. We discuss further their mathematical properties as well as computability theoretical properties.

Keywords

Computable real numbers Computably approximable real numbers Hierarchy 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aberth, O.: Computable Calculus. Academic, San Diego (2001) MATHGoogle Scholar
  2. 2.
    Ambos-Spies, K., Weihrauch, K., Zheng, X.: Weakly computable real numbers. J. Complex. 16(4), 676–690 (2000) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bridges, D.S.: Constructive mathematics: a foundation for computable analysis. Theor. Comput. Sci. 219(1–2), 95–109 (1999) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Calude, C.S., Hertling, P.H., Khoussainov, B., Wang, Y.: Recursively enumerable reals and Chaitin Ω numbers. Theor. Comput. Sci. 255, 125–149 (2001) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Downey, R.G.: Some computability-theoretic aspects of reals and randomness. In: The Notre Dame Lectures. Lect. Notes Log., vol. 18, pp. 97–147. Assoc. Symb. Logic, Urbana (2005) Google Scholar
  6. 6.
    Ershov, Y.L.: A certain hierarchy of sets. i, ii, iii. Algebra i Logika 7(1), 47–73 (1968); 7(4), 15–47 (1968); 9, 34–51 (1970) (Russian) MATHMathSciNetGoogle Scholar
  7. 7.
    Friedberg, R.M.: Two recursively enumerable sets of incomparable degrees of unsolvability (solution of Post’s problem, 1944). Proc. Nat. Acad. Sci. U.S.A. 43, 236–238 (1957) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Grzegorczyk, A.: Some classes of recursive functions. Rozprawy Mat. 4, 46 (1953) MathSciNetGoogle Scholar
  9. 9.
    Ko, K.-I.: Complexity Theory of Real Functions. Progress in Theoretical Computer Science. Birkhäuser, Boston (1991) MATHGoogle Scholar
  10. 10.
    Lempp, S., Lerman, M.: A general framework for priority arguments. Bull. Symb. Log. 1(2), 189–201 (1995) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Muchnik, A.A.: On the unsolvability of the problem of reducibility in the theory of algorithms. Dokl. Akad. Nauk SSSR (N.S.) 108, 194–197 (1956) MATHMathSciNetGoogle Scholar
  12. 12.
    Myhill, J.: Criteria of constructibility for real numbers. J. Symb. Log. 18(1), 7–10 (1953) MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Pour-El, M.B., Richards, J.I.: Computability in Analysis and Physics. Perspectives in Mathematical Logic. Springer, Berlin (1989) MATHGoogle Scholar
  14. 14.
    Rettinger, R., Zheng, X., Gengler, R., von Braunmühl, B.: Weakly computable real numbers and total computable real functions. In: Proceedings of COCOON 2001, Guilin, China, August 20–23, 2001. Lecture Notes in Comput. Sci., vol. 2108, pp. 586–595. Springer, Berlin (2001) Google Scholar
  15. 15.
    Rice, H.G.: Recursive real numbers. Proc. Am. Math. Soc. 5, 784–791 (1954) MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Robinson, R.M.: Review of “Peter, R., Rekursive Funktionen”. J. Symb. Log. 16, 280–282 (1951) CrossRefGoogle Scholar
  17. 17.
    Soare, R.I.: Cohesive sets and recursively enumerable Dedekind cuts. Pac. J. Math. 31, 215–231 (1969) MATHMathSciNetGoogle Scholar
  18. 18.
    Soare, R.I.: Recursively Enumerable Sets and Degrees. A Study of Computable Functions and Computably Generated Sets. Perspectives in Mathematical Logic. Springer, Berlin (1987) Google Scholar
  19. 19.
    Specker, E.: Nicht konstruktiv beweisbare Sätze der Analysis. J. Symb. Log. 14(3), 145–158 (1949) MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Turing, A.M.: On computable numbers, with an application to the “Entscheidungsproblem”. Proc. Lond. Math. Soc. 42(2), 230–265 (1936) MATHGoogle Scholar
  21. 21.
    Weihrauch, K.: Computable Analysis, An Introduction. Springer, Berlin (2000) MATHGoogle Scholar
  22. 22.
    Zheng, X., Rettinger, R.: Weak computability and representation of reals. Math. Log. Q. 50(4/5), 431–442 (2004) MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Zheng, X., Rettinger, R., Gengler, R.: Closure properties of real number classes under CBV functions. Theory Comput. Syst. 38, 701–729 (2005) MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Zheng, X., Lu, D., Bao, K.: Divergence bounded computable real numbers. Theory Comput. Sci. 351, 27–38 (2006) MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Department of Computer ScienceJiangsu UniversityZhenjiangChina
  2. 2.Theoretische InformatikBTU CottbusCottbusGermany

Personalised recommendations