Algorithmic solution of higher type equations

作者：Escardo Martin^*

来源：Journal of Logic and Computation, 2013, 23(4): 839-854.

DOI：10.1093/logcom/exr048

• Higher type computability
• Kleene-Kreisel spaces of continuous functionals
• exhaustible set
• searchable set
• computationally compact set
• QCB space
• admissible representation
• topology in the theory of computation

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 article, we give applications to the computation of solutions of higher type equations. Given a continuous functional f : X -%26gt; Y and y is an element of Y, we wish to compute x is an element of 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.

• 出版日期2013-8