Structure within the class of K-trivial sets
University of Auckland
The K-trivial sets are antirandom in the sense that the initial segment complexity in terms of prefix-free Kolmogorov complexity K grows as slowly as possible. Since 2002, many alternative characterisations of this class have been found: properties such as low for K, low for Martin-Löf (ML) randomness, and basis for ML randomness, which state in one way or the other that the set is close to computable.
Initially the class looked amorphous. More recently, internal structure has been found. Bienvenu et al (JEMS, 2016) showed that there is a “smart” K-trivial set, in the sense that any ML-random computing it must compute all K-trivials. Greenberg, Miller and Nies (submitted) showed that there is a dense hierarchy of subclasses. Even more recent results with Turetsky combine the two approaches via cost functions.