The Kolmogorov complexity of random reals

Annals of Pure and Applied Logic 129 (1-3):163-180 (2004)
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Abstract

We investigate the initial segment complexity of random reals. Let K denote prefix-free Kolmogorov complexity. A natural measure of the relative randomness of two reals α and β is to compare complexity K and K. It is well-known that a real α is 1-random iff there is a constant c such that for all n, Kn−c. We ask the question, what else can be said about the initial segment complexity of random reals. Thus, we study the fine behaviour of K for random α. Following work of Downey, Hirschfeldt and LaForte, we say that αK β iff there is a constant such that for all n, . We call the equivalence classes under this measure of relative randomness K-degrees. We give proofs that there is a random real α so that lim supn K−K=∞ where Ω is Chaitin's random real. One is based upon work of Solovay, and the other exploits a new idea. Further, based on this new idea, we prove there are uncountably many K-degrees of random reals by proving that μ=0. As a corollary to the proof we can prove there is no largest K-degree. Finally we prove that if n ≠ m then the initial segment complexities of the natural n- and m-random sets and Ω︀) are different. The techniques introduced in this paper have already found a number of other applications

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Citations of this work

Calibrating randomness.Rod Downey, Denis R. Hirschfeldt, André Nies & Sebastiaan A. Terwijn - 2006 - Bulletin of Symbolic Logic 12 (3):411-491.
The K -Degrees, Low for K Degrees,and Weakly Low for K Sets.Joseph S. Miller - 2009 - Notre Dame Journal of Formal Logic 50 (4):381-391.
The Equivalence of Definitions of Algorithmic Randomness.Christopher Porter - 2021 - Philosophia Mathematica 29 (2):153–194.
Algorithmic Randomness and Measures of Complexity.George Barmpalias - 2013 - Bulletin of Symbolic Logic 19 (3):318-350.
Algorithmic randomness and measures of complexity.George Barmpalias - 2013 - Bulletin of Symbolic Logic 19 (3):318-350.

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References found in this work

[Omnibus Review].Rod Downey - 1997 - Journal of Symbolic Logic 62 (3):1048-1055.
Von Mises' definition of random sequences reconsidered.Michiel van Lambalgen - 1987 - Journal of Symbolic Logic 52 (3):725-755.

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