Randomness for computable measures and initial segment complexity

Abstract

We study the possible growth rates of the Kolmogorov complexity of initial segments of sequences that are random with respect to some computable measure on 2 ω , the so-called proper sequences. Our main results are as follows: (1) We show that the initial segment complexity of a proper sequence X is bounded from below by a computable function (that is, X is complex) if and only if X is random with respect to some computable, continuous measure. (2) We prove that a uniform version of the previous result fails to hold: there is a family of complex sequences that are random with respect to a single computable measure such that for every computable, continuous measure µ, some sequence in this family fails to be random with respect to µ. (3) We show that there are proper sequences with extremely slow-growing initial segment complexity, that is, there is a proper sequence the initial segment complexity of which is infinitely often below every computable function, and even a proper sequence the initial segment complexity of which is dominated by all computable functions. (4) We prove various facts about the Turing degrees of such sequences and show that they are useful in the study of certain classes of pathological measures on 2 ω , namely diminutive measures and trivial measures.

Rank and Randomness

“…We point out that the measure constructed in the proof of Theorem 1.1 must necessarily have atoms. For, as shown by the authors in previous work [9], if a sequence X is random with respect to a computable, continuous measure, then it must be complex: that is, there must be a computable, unbounded, nondecreasing function f(n) such that K(X n) ≥ f(n) for all n ∈ , where K(•) denotes prefixfree Kolmogorov complexity. As shown by Binns [3], every Π 0 1 class containing a complex element is perfect.…”

“…One particular instance is given by sequences that are random with respect to a computable, countably supported measure µ on 2 ω , that is, a computable measure for which there is some countable collection (X i ) i∈ω such that µ i∈ω {X i } = 1. The behavior of such sequences has been studied by Bienvenu and Porter [2], Porter [12], as well as Hölzl and Porter [7].…”

On the Interplay Between Effective Notions of Randomness and Genericity

“…Recall that a sequence X is complex if there is some computable order g such that K(X↾n) ≥ g(n). As shown explicitly in [HP17], one can equivalently define a sequence to be complex in terms of a priori complexity, i.e., X is complex if and only if there is some computable order h such that KA(X↾n) ≥ h(n). We use this second characterization to derive the following: Proposition 18.…”

Effective Aspects of Bernoulli Randomness

“…Algorithmic randomness with respect to biased probability measures on the Cantor space 2 ω has been studied in two separate strands: with respect to computable measures, such as in [BM09], [BP12], and [HP17], and with respect to non-computable measures, such as in [DM13] and [RS18] (see the recent survey [Por19] for an overview of both approaches). In this article, we study the interaction of these two strands in the context of Bernoulli measures on 2 ω .…”