A Galois connection between Turing jumps and limits

TL;DR

This paper establishes a formal Galois connection between Turing jumps and limits in computability, characterizing classes of limit computable functions and their relation to halting problem computability across various spaces.

Contribution

It introduces a Galois connection framework linking Turing jumps and limits, extending to represented and metric spaces, and characterizes functions computable relative to the halting problem.

Findings

Limit computable functions can be characterized by Turing jumps or limits.

Functions computable relative to the halting problem have a computable modulus of continuity.

Lipschitz continuous limit computable functions are also halting problem computable.

Abstract

Limit computable functions can be characterized by Turing jumps on the input side or limits on the output side. As a monad of this pair of adjoint operations we obtain a problem that characterizes the low functions and dually to this another problem that characterizes the functions that are computable relative to the halting problem. Correspondingly, these two classes are the largest classes of functions that can be pre or post composed to limit computable functions without leaving the class of limit computable functions. We transfer these observations to the lattice of represented spaces where it leads to a formal Galois connection. We also formulate a version of this result for computable metric spaces. Limit computability and computability relative to the halting problem are notions that coincide for points and sequences, but even restricted to continuous functions the former class is strictly larger than the latter. On computable metric spaces we can characterize the functions that are computable relative to the halting problem as those functions that are limit computable with a modulus of continuity that is computable relative to the halting problem. As a consequence of this result we obtain, for instance, that Lipschitz continuous functions that are limit computable are automatically computable relative to the halting problem. We also discuss 1-generic points as the canonical points of continuity of limit computable functions, and we prove that restricted to these points limit computable functions are computable relative to the halting problem. Finally, we demonstrate how these results can be applied in computable analysis.