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

Abstract

A remarkable achievement in algorithmic randomness and algorithmic information theory was the discovery of the notions of Ktrivial, K-low and Martin-Löf-random-low sets: three different definitions turn out to be equivalent for very non-trivial reasons [1,3,5]. This survey, 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 (see, e.g., [7] for introduction and [8] for more detailed exposition). More information about the subject and its history can be found in [2,6].