# Busy beavers and Kolmogorov complexity

## TL;DR

This paper explores variations of busy beaver numbers defined via Kolmogorov complexity, analyzing their relationships and bounds depending on the complexity type used, thus connecting computational and descriptive complexity.

## Contribution

It introduces and compares different Kolmogorov complexity-based busy beaver notions, establishing bounds and relationships among them.

## Key findings

- Different versions of Kolmogorov complexity lead to distinct busy beaver-like notions.
- Matching lower and upper bounds are established for these notions.
- The relationships among plain, prefix, and a priori complexities are clarified.

## Abstract

The idea to find the "maximal number that can be named" can be traced back to Archimedes (see his Psammit). From the viewpoint of computation theory the natural question is "which number can be described by at most n bits"? This question led to the definition of the so-called "busy beaver" numbers (introduced by T. Rado). In this note we consider different versions of the busy beaver-like notions defined in terms of Kolmogorov complexity. We show that these versions differ depending on the version of complexity used (plain, prefix, or a priori complexities) and find out how these notions are related, providing matching lower and upper bounds.