Computability of continuous solutions of higher-type equations

Abstract

Given a continuous functional f: X → 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 X: 2X → 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. © 2009 Springer Berlin Heidelberg.