When are two algorithms the same? Towards addressing Hilbert's 24th problem

TL;DR

This paper explores when two algorithms or proofs are considered equivalent by leveraging concepts from Recursion Theory and Kolmogorov Complexity, addressing a long-standing open problem inspired by Hilbert's 24th problem.

Contribution

It introduces a minimalistic framework for assessing algorithmic equivalence using Kolmogorov Complexity within Recursion Theory, inspired by Hilbert's open problem.

Findings

Proposes a formal approach to algorithm equivalence

Utilizes Kolmogorov Complexity to measure similarity

Links proof equivalence to program equivalence

Abstract

The informal question of when two theorem proofs are "essentially the same" goes back to David Hilbert, who considered adding it (or something largely equivalent) to his famous list of open problems, but eventually decided to leave it out. Given that the notion of a formal proof is closely related to that of a (computer) program, i.e. a recursive function, it may be useful to ask the same question with regard to programs instead. Here we propose a minimalistic approach to this question within Recursion Theory, building heavily on the use of Kolmogorov Complexity.