On the Complexity of Computing G\"odel Numbers

TL;DR

This paper investigates the computational complexity of finding G"odel numbers for sequences, classifying it within the Weihrauch hierarchy and revealing differences between topological and computability-theoretic perspectives.

Contribution

It provides a novel classification of the G"odel number computation problem within the Weihrauch framework, connecting it to principles in reverse mathematics.

Findings

The problem is neither continuous nor computable.

It is classified within the Weihrauch hierarchy using closed and compact choice problems.

Topological and computability-theoretic classifications differ significantly.

Abstract

Given a computable sequence of natural numbers, it is a natural task to find a G\"odel number of a program that generates this sequence. It is easy to see that this problem is neither continuous nor computable. In algorithmic learning theory this problem is well studied from several perspectives and one question studied there is for which sequences this problem is at least learnable in the limit. Here we study the problem on all computable sequences and we classify the Weihrauch complexity of it. For this purpose we can, among other methods, utilize the amalgamation technique known from learning theory. As a benchmark for the classification we use closed and compact choice problems and their jumps on natural numbers, and we argue that these problems correspond to induction and boundedness principles, as they are known from the Kirby-Paris hierarchy in reverse mathematics. We provide a topological as well as a computability-theoretic classification, which reveal some significant differences.