Computability of Continuous Solutions of Higher-Type Equations

  • Martín Escardó
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5635)

Abstract

Given a continuous functional \(f \colon X \to Y\) and y ∈ Y, we wish to compute x ∈ X such that f(x) = y, if such an x exists. We show that if x is unique and X and Y are subspaces of Kleene–Kreisel spaces of continuous functionals with X exhaustible, then x is computable uniformly in f, y and the exhaustion functional \(\forall_X \colon 2^X \to 2\). We also establish a version of the above for computational metric spaces X and Y, where is X computationally complete and has an exhaustible set of Kleene–Kreisel representatives. Examples of interest include functionals defined on compact spaces X of analytic functions.

Keywords

Higher-type computability Kleene–Kreisel spaces of continuous functionals exhaustible set 

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References

  1. 1.
    Normann, D.: Recursion on the countable functionals. Lec. Not. Math., vol. 811. Springer, Heidelberg (1980)MATHGoogle Scholar
  2. 2.
    Escardó, M.: Exhaustible sets in higher-type computation. Log. Methods Comput. Sci. 4(3), 3:3, 37 (2008)Google Scholar
  3. 3.
    Weihrauch, K.: Computable analysis. Springer, Heidelberg (2000)CrossRefMATHGoogle Scholar
  4. 4.
    Bauer, A.: A relationship between equilogical spaces and type two effectivity. MLQ Math. Log. Q. 48(suppl. 1), 1–15 (2002)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bishop, E., Bridges, D.: Constructive Analysis. Springer, Berlin (1985)CrossRefMATHGoogle Scholar
  6. 6.
    Simpson, A.: Lazy functional algorithms for exact real functionals. In: Brim, L., Gruska, J., Zlatuška, J. (eds.) MFCS 1998. LNCS, vol. 1450, pp. 323–342. Springer, Heidelberg (1998)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Martín Escardó
    • 1
  1. 1.School of Computer ScienceUniversity of BirminghamUK

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