All Kolmogorov complexity functions are optimal, but are some more optimal?

TL;DR

The paper examines the concept of Kolmogorov complexity, focusing on the existence of optimal programming languages and whether some are more optimal than others, analyzing various proposals for stronger notions of optimality.

Contribution

It analyzes different notions of optimality for Kolmogorov complexity functions and shows that stronger requirements often do not alter the set of complexity functions.

Findings

Most stronger optimality conditions do not change the set of Kolmogorov complexity functions.

Optimal programming languages are essentially equivalent under many proposed stronger criteria.

The paper clarifies the limitations of achieving a universally agreed-upon optimal complexity measure.

Abstract

Kolmogorov (1965) defined the complexity of a string $x$ as the minimal length of a program generating $x$. Obviously this definition depends on the choice of the programming language. Kolmogorov noted that there exist \emph{optimal} programming languages that make the complexity function minimal up to $O(1)$ additive terms, and we should take one of them -- but which one? Is there a chance to agree on some specific programming language in this definition? Or at least should we add some other requirements to optimality? What can we achieve in this way? In this paper we discuss different suggestions of this type that appeared since 1965, specifically a stronger requirement of universality (and show that in many cases this does not change the set of complexity functions).