Computability of Continuous Solutions of Higher-Type Equations

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


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.


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


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© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

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

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