K-trivial, K-low and MLR-low sequences: a tutorial

TL;DR

This tutorial explains the equivalence of K-trivial, K-low, and MLR-low sequences in algorithmic randomness, providing detailed proofs and related results for readers familiar with algorithmic information theory.

Contribution

It offers a comprehensive exposition of the proof of equivalence among key notions in algorithmic randomness, based on a course lecture.

Findings

Proves the equivalence of K-trivial, K-low, and MLR-low sequences

Provides detailed explanations suitable for learners in the field

Summarizes related results in the area of algorithmic randomness

Abstract

A remarkable achievement in algorithmic randomness and algorithmic information theory was the discovery of the notions of K-trivial, K-low and Martin-Lof-random-low sets: three different definitions turns out to be equivalent for very non-trivial reasons. This paper, based on the course taught by one of the authors (L.B.) in Poncelet laboratory (CNRS, Moscow) in 2014, provides an exposition of the proof of this equivalence and some related results. We assume that the reader is familiar with basic notions of algorithmic information theory.