Theory of Computing Systems

, Volume 43, Issue 3–4, pp 603–624

# 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.

### References

1. 1.
Aberth, O.: Computable Calculus. Academic, San Diego (2001)
2. 2.
Ambos-Spies, K., Weihrauch, K., Zheng, X.: Weakly computable real numbers. J. Complex. 16(4), 676–690 (2000)
3. 3.
Bridges, D.S.: Constructive mathematics: a foundation for computable analysis. Theor. Comput. Sci. 219(1–2), 95–109 (1999)
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)
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)
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)
8. 8.
Grzegorczyk, A.: Some classes of recursive functions. Rozprawy Mat. 4, 46 (1953)
9. 9.
Ko, K.-I.: Complexity Theory of Real Functions. Progress in Theoretical Computer Science. Birkhäuser, Boston (1991)
10. 10.
Lempp, S., Lerman, M.: A general framework for priority arguments. Bull. Symb. Log. 1(2), 189–201 (1995)
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)
12. 12.
Myhill, J.: Criteria of constructibility for real numbers. J. Symb. Log. 18(1), 7–10 (1953)
13. 13.
Pour-El, M.B., Richards, J.I.: Computability in Analysis and Physics. Perspectives in Mathematical Logic. Springer, Berlin (1989)
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)
16. 16.
Robinson, R.M.: Review of “Peter, R., Rekursive Funktionen”. J. Symb. Log. 16, 280–282 (1951)
17. 17.
Soare, R.I.: Cohesive sets and recursively enumerable Dedekind cuts. Pac. J. Math. 31, 215–231 (1969)
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)
20. 20.
Turing, A.M.: On computable numbers, with an application to the “Entscheidungsproblem”. Proc. Lond. Math. Soc. 42(2), 230–265 (1936)
21. 21.
Weihrauch, K.: Computable Analysis, An Introduction. Springer, Berlin (2000)
22. 22.
Zheng, X., Rettinger, R.: Weak computability and representation of reals. Math. Log. Q. 50(4/5), 431–442 (2004)
23. 23.
Zheng, X., Rettinger, R., Gengler, R.: Closure properties of real number classes under CBV functions. Theory Comput. Syst. 38, 701–729 (2005)
24. 24.
Zheng, X., Lu, D., Bao, K.: Divergence bounded computable real numbers. Theory Comput. Sci. 351, 27–38 (2006)