Computability of Continuous Solutions of Higher-Type Equations
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.
KeywordsHigher-type computability Kleene–Kreisel spaces of continuous functionals exhaustible set
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- 2.Escardó, M.: Exhaustible sets in higher-type computation. Log. Methods Comput. Sci. 4(3), 3:3, 37 (2008)Google Scholar