K-trivial, K-low and MLR-low Sequences: A Tutorial

A remarkable achievement in algorithmic randomness and algorithmic information theory was the discovery of the notions of K-trivial, K-low and MartinLöf-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 (see, e.g., [3] for introduction and [4] for more detailed exposition). More information about the subject and its history can be found in [2, 1]. 1 K-trivial sets: definition and existence Consider an infinite bit sequence and complexities of its prefixes. If they are small, the sequence is computable or almost computable; if they are big, the sequence looks random. This idea goes back to 1960s and appears in the algorithmic information theory in different forms (Schnorr–Levin criterion of randomness in terms of complexities of prefixes, the notion of algorithmic Hausdorff dimension). The notion of K-triviality is on the low end of this spectrum: we consider sequences that have prefixes of minimal possible prefix complexity: Definition. A bit sequence a0a1a2 . . ., is called K-trivial if it has minimal possible prefix complexity of its prefixes, i.e., if K(a0a1 . . .an−1) = K(n)+O(1). • Note that n can be reconstructed from a0 . . .an−1, so K(a0 . . .an−1) cannot be smaller than K(n)−O(1). • Every computable sequence is K-trivial, since a0 . . .an−1 can be computed given n. • Similar definition for plain complexity has no sense, since this would imply that sequence A is computable (it is enough to have C(a0a1 . . .an−1) ≤ logn+O(1) for computability, see, e.g.,[4, problems 48 and 49]). ∗Poncelet laboratory, CNRS, Moscow, [email protected] †LIRMM, Montpellier, CNRS, UM2, on leave from IITP RAS, Moscow, [email protected]