Kolmogorov complexity as a language

TL;DR

This survey explores how Kolmogorov complexity can serve as a language to reformulate and gain new insights into various notions across different fields, by comparing complexity-based and traditional approaches.

Contribution

It provides numerous examples demonstrating the utility of Kolmogorov complexity as a unifying language for reformulating concepts and statements.

Findings

Kolmogorov complexity offers a powerful framework for reformulating notions.

Many classical concepts can be reinterpreted through complexity measures.

The survey highlights the advantages of using complexity-based approaches.

Abstract

The notion of Kolmogorov complexity (=the minimal length of a program that generates some object) is often useful as a kind of language that allows us to reformulate some notions and therefore provide new intuition. In this survey we provide (with minimal comments) many different examples where notions and statements that involve Kolmogorov complexity are compared with their counterparts not involving complexity.