Algorithmic Randomness and Measures of Complexity

Abstract

We survey recent advances on the interface between computability theory and algorithmic randomness, with special attention on measures of relative complexity. We focus on (weak) reducibilities that measure (a) the initial segment complexity of reals and (b) the power of reals to compress strings, when they are used as oracles. The results are put into context and several connections are made with various central issues in modern algorithmic randomness and computability.

Universality of quantum Turing machines with deterministic control

“…In consequence of Kleene's s-m-n theorem, this classical notion of Kolmogorov complexity is robust in the sense that it does depend (up to a constant) on the adopted universal Turing machine. For a recent survey see [1]. When the object is a quantum state, it is not yet clear how one should proceed.…”

“…For a comparison between plain and prefix-free classical Kolmogorov complexities see, for example, [1]. In the quantum case, the definition proposed in [19] is prefix free since it is based on a prefix-free variant of the Deutsch machine, while the definition in [3] is plain.…”