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Computability and Continuity on the Real Arithmetic Hierarchy and the Power of Type-2 Nondeterminism

  • Martin Ziegler
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3526)

Abstract

The sometimes so-called Main Theorem of Recursive Analysis implies that any computable real function is necessarily continuous. We consider three relaxations of this common notion of real computability for the purpose of treating also discontinuous functions f: ℝ→ℝ:

  • non-deterministic computation;

  • relativized computation, specifically given access to oracles like ∅′ or ∅″;

  • encoding input x εℝ and/or output y = f(x) in weaker ways according to the Real Arithmetic Hierarchy.

It turns out that, among these approaches, only the first one provides the required power.

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References

  1. 1.
    Barmpalias, G.: A Transfinite Hierarchy of Reals. Mathematical Logic Quarterly 49(2), 163–172 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Brattka, V., Hertling, P.: Topological Properties of Real Number Representations. Theoretical Computer Science 284, 241–257 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Grzegorczyk, A.: On the Definitions of Computable Real Continuous Functions. Fundamenta Mathematicae 44, 61–77 (1957)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Ho, C.-K.: Relatively Recursive Real Numbers and Real Functions. Theoretical Computer Science 210, 99–120 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Ko, K.-I.: Complexity Theory of Real Functions. Birkhäuser, Basel (1991)zbMATHGoogle Scholar
  6. 6.
    Pour-El, M.B., Richards, J.I.: Computability in Analysis and Physics. Springer, Heidelberg (1989)zbMATHGoogle Scholar
  7. 7.
    Randolph, J.F.: Basic Real and Abstract Analysis. Academic Press, London (1968)zbMATHGoogle Scholar
  8. 8.
    Soare, R.I.: Recursively Enumerable Sets and Degrees. Springer, Heidelberg (1987)Google Scholar
  9. 9.
    Spaan, E., Torenvliet, L., van Emde Boas, P.: Nondeterminism, Fairness and a Fundamental Analogy. In: The Bulletin of the European Association for Theoretical Computer Science (EATCS Bulletin), vol. 37, pp. 186–193 (1989)Google Scholar
  10. 10.
    Thomas, W.: Automata on Infinite Objects. In: Handbook of Theoretical Computer Science (Formal Models and Semantics), vol. B, pp. 133–191. Elsevier, Amsterdam (1990)Google Scholar
  11. 11.
    Turing, A.M.: On Computable Numbers, with an Application to the Entscheidungsproblem. Proc. London Math. Soc. 42(2), 230–265 (1936)zbMATHGoogle Scholar
  12. 12.
    Turing, A.M.: On Computable Numbers, with an Application to the Entscheidungsproblem. A correction. Proc. London Math. Soc. 43(2), 544–546 (1937)zbMATHGoogle Scholar
  13. 13.
    Weihrauch, K.: Computable Analysis. Springer, Heidelberg (2001)Google Scholar
  14. 14.
    Zheng, X., Weihrauch, K.: The Arithmetical Hierarchy of Real Numbers. Mathematical Logic Quarterly 47, 51–65 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Zheng, X.: Recursive Approximability of Real Numbers. Mathematical Logic Quarterly 48(suppl. 1), 131–156 (2002)Google Scholar
  16. 16.
    Zhong, N., Weihrauch, K.: Computability Theory of Generalized Functions. J. ACM 50, 469–505 (2003)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Martin Ziegler
    • 1
  1. 1.Institut for Matematik og DatalogiSyddansk UniversitetOdense MDenmark

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