Algorithmic solution of higher type equations

Abstract

In recent work we developed the notion of exhaustible set as a higher-type computational counter-part of the topological notion of compact set. In this paper we give applications to the computation of solutions of higher-type equations. 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 exhaustibility condition. We also establish a version of this 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 evaluation functionals defined on compact spaces X of bounded sequences of Taylor coefficients with values on spaces Y of real analytic functions defined on a compact set. A corollary is that it is semi-decidable whether a function defined on such a compact set fails to be analytic, and that the Taylor coefficients of an analytic function can be computed extensionally from the function.