Nies and Scholz formalized the notion of an infinite qubitstring and referred to it as a ‘state’. They defined ‘quantum Martin-Löf randomness’ for states. We give a notion of measurement of a state in a computable basis and introduce ‘quantum measurement randomness’, a randomness notion for states. A state is quantum measurement random if measuring it in any computable basis yields a Martin-Löf random bitstring with probability one. Our main result is that quantum Martin-Löf randomness strictly implies quantum measurement randomness. This uses the construction of a quantum measurement random state which is not quantum Martin-Löf random. We prove two general results on which this construction relies: The first concerns Martin-Löf randomness relative to computable measures and extends a result of V. Vovk. The second is a combinatorial result about Kronecker products.

By a celebrated result of Kučera and Slaman [5], the Martin-Löf random left-c.e. reals form the highest left-c.e. Solovay degree. Barmpalias and Lewis-Pye [1] strengthened this result by showing that, for all left-c.e. reals $\alpha $ and $\beta $ such that $\beta $ is Martin-Löf random and all left-c.e. approximations $a_0,a_1,\dots $ and $b_0,b_1,\dots $ of $\alpha $ and $\beta $, respectively, the limit

$$ \begin{align*} \lim\limits_{n\to\infty}\frac{\alpha - a_n}{\beta - b_n} \end{align*} $$

$\alpha $

$\beta $

Here we give an equivalent formulation of the result of Barmpalias and Lewis-Pye in terms of nondecreasing translation functions and generalize their result to the set of all (i.e., not necessarily left-c.e.) reals.

We show that for each computable ordinal $\alpha> 0$ it is possible to find in each Martin-Löf random ${\rm{\Delta }}_2^0 $ degree a sequence R of Cantor-Bendixson rank α, while ensuring that the sequences that inductively witness R’s rank are all Martin-Löf random with respect to a single countably supported and computable measure. This is a strengthening for random degrees of a recent result of Downey, Wu, and Yang, and can be understood as a randomized version of it.

By the complexity of a finite sequence of 0’s and 1’s wemean the Kolmogorov complexity, that is the length of the shortest input to auniversal recursive function which returns the given sequence as output. Byinitial segment complexity of an infinite sequence of 0’s and1’s we mean the asymptotic behavior of the complexity of its finiteinitial segments. In this paper, we construct infinite sequences of0’s and 1’s with given recursive lower bounds on initialsegment complexity which do not compute any infinite sequences of 0’sand 1’s with a significantly larger recursive lower bound on initialsegment complexity. This improves several known results about randomnessextraction and separates many natural degrees in the lattice of Muchnikdegrees.

A Martin-Löf random sequence is an infinite binary sequence with the property that every initial segment σ has prefix-free Kolmogorov complexity K(σ) at least ∣σ∣ − c, for some constant c ϵ ω. Informally, initial segments of Martin-Löf randoms are highly complex in the sense that they are not compressible by more than a constant number of bits. However, all Martin-Löf randoms necessarily have contiguous substrings of arbitrarily low complexity. If we demand that all substrings of a sequence be uniformly complex, then we arrive at the notion of shift-complex sequences. In this paper, we collect some of the existing results on these sequences and contribute two new ones. Rumyantsev showed that the measure of oracles that compute shift-complex sequences is zero. We strengthen this result by proving that the Martin-Löf random sequences that do not compute shift-complex sequences are exactly the incomplete ones, in other words, the ones that do not compute the halting problem. In order to do so, we make use of the characterization by Franklin and Ng of the class of incomplete Martin-Löf randoms via a notion of randomness called difference randomness. Turning to the power of shift-complex sequences as oracles, we show that there are shift-complex sequences that do not compute Martin-Löf random (or even Kurtz random) sequences.

We prove a number of results in effective randomness, using methods in which Π1 0 classes play an essential role. The results proved include the fact that every PA Turing degree is the join of two random Turing degrees, and the existence of a minimal pair of LR degrees below the LR degree of the halting problem.