We prove that there are uncountably many sets that are low for the class of Schnorr random reals. We give a purely recursion theoretic characterization of these sets and show that they all have Turing degree incomparable to 0'. This contrasts with a result of Kučera and Terwijn [5] on sets that are low for the class of Martin-Löf random reals.

We present a faithful axiomatization of von Mises' notion of a random sequence, using an abstract independence relation. A byproduct is a quantifier elimination theorem for Friedman's "almost all" quantifier in terms of this independence relation.

As a natural example of a 1-random real, Chaitin proposed the halting probability Ω of a universal prefix-free machine. We can relativize this example by considering a universal prefix-free oracle machine U. Let [Formula: see text] be the halting probability of UA; this gives a natural uniform way of producing an A-random real for every A ∈ 2ω. It is this operator which is our primary object of study. We can draw an analogy between the jump operator from computability theory (...) and this Omega operator. But unlike the jump, which is invariant under the choice of an effective enumeration of the partial computable functions, [Formula: see text] can be vastly different for different choices of U. Even for a fixed U, there are oracles A =* B such that [Formula: see text] and [Formula: see text] are 1-random relative to each other. We prove this and many other interesting properties of Omega operators. We investigate these operators from the perspective of analysis, computability theory, and of course, algorithmic randomness. (shrink)

We study lowness for genericity. We show that there exists no Turing degree which is low for 1-genericity and all of computably traceable degrees are low for weak 1-genericity.

We study and compare two combinatorial lowness notions: strong jump-traceability and well-approximability of the jump, by strengthening the notion of jump-traceability and super-lowness for sets of natural numbers. A computable non-decreasing unbounded function h is called an order function. Informally, a set A is strongly jump-traceable if for each order function h, for each input e one may effectively enumerate a set Te of possible values for the jump JA, and the number of values enumerated is at most h. A′ (...) is well-approximable if can be effectively approximated with less than h changes at input x, for each order function h. We prove that there is a strongly jump-traceable set which is not computable, and that if A′ is well-approximable then A is strongly jump-traceable. For r.e. sets, the converse holds as well. We characterize jump-traceability and strong jump-traceability in terms of Kolmogorov complexity. We also investigate other properties of these lowness properties. (shrink)

We call A weakly low for K if there is a c such that $K^A(\sigma)\geq K(\sigma)-c$ for infinitely many σ; in other words, there are infinitely many strings that A does not help compress. We prove that A is weakly low for K if and only if Chaitin's Ω is A-random. This has consequences in the K-degrees and the low for K (i.e., low for random) degrees. Furthermore, we prove that the initial segment prefix-free complexity of 2-random reals is infinitely (...) often maximal. This had previously been proved for plain Kolmogorov complexity. (shrink)

Kuera, A., On relative randomness, Annals of Pure and Applied Logic 63 61–67. It is the aim of the paper to answer a question raised by M. van Lambalgen and D. Zambella whether there can be a nonrecursive set A having the property that there is a set B such that B is 1-random relative to A and simultaneously A is recursive in B. We give apositive answer to this question as well as further information about relative randomness.

An infinite binary sequence X is Kolmogorov–Loveland random if there is no computable non-monotonic betting strategy that succeeds on X in the sense of having an unbounded gain in the limit while betting successively on bits of X. A sequence X is KL-stochastic if there is no computable non-monotonic selection rule that selects from X an infinite, biased sequence.One of the major open problems in the field of effective randomness is whether Martin-Löf randomness is the same as KL-randomness. Our first (...) main result states that KL-random sequences are close to Martin-Löf random sequences in so far as every KL-random sequence has arbitrarily dense subsequences that are Martin-Löf random. A key lemma in the proof of this result is that for every effective split of a KL-random sequence at least one of the halves is Martin-Löf random. However, this splitting property does not characterize KL-randomness; we construct a sequence that is not even computably random such that every effective split yields two subsequences that are 2-random. Furthermore, we show for any KL-random sequence A that is computable in the halting problem that, first, for any effective split of A both halves are Martin-Löf random and, second, for any computable, nondecreasing, and unbounded function g and almost all n, the prefix of A of length n has prefix-free Kolmogorov complexity at least n−g. Again, the latter property does not characterize KL-randomness, even when restricted to left-r.e. sequences; we construct a left-r.e. sequence that has this property but is not KL-stochastic and, in fact, is not even Mises–Wald–Church stochastic.Turning our attention to KL-stochasticity, we construct a non-empty class of KL-stochastic sequences that are not weakly 1-random; by the usual basis theorems we obtain such sequences that in addition are left-r.e., are low, or are of hyperimmune-free degree.Our second main result asserts that every KL-stochastic sequence has effective dimension 1, or equivalently, a sequence cannot be KL-stochastic if it has infinitely many prefixes that can be compressed by a factor of α<1. This improves on a result by Muchnik, who has shown that were they to exist, such compressible prefixes could not be found effectively. (shrink)

We prove that there exists a noncomputable c.e. real which is low for weak 2-randomness, a definition of randomness due to Kurtz, and that all reals which are low for weak 2-randomness are low for Martin-Löf randomness.

We answer a question of Ambos-Spies and Kuˇcera in the affirmative. They asked whether, when a real is low for Schnorr randomness, it is already low for Schnorr tests.

We study reals with infinitely many incompressible prefixes. Call $A \in 2^{\omega}$ Kolmogorot random if $(\exists^{\infty}n) C(A \upharpoonright n) \textgreater n - \mathcal{O}(1)$ , where C denotes plain Kolmogorov complexity. This property was suggested by Loveland and studied by $Martin-L\ddot{0}f$ , Schnorr and Solovay. We prove that 2-random reals are Kolmogorov random. Together with the converse-proved by Nies. Stephan and Terwijn [11]-this provides a natural characterization of 2-randomness in terms of plain complexity. We finish with a related characterization of 2-randomness.

We show there is a non-recursive r.e. set A such that if W is any low r.e. set, then the join W $\oplus$ A is also low. That is, A is "almost deep". This answers a question of Jockusch. The almost deep degrees form an definable ideal in the r.e. degrees (with jump.).

We compare various notions of algorithmic randomness. First we consider relativized randomness. A set is n-random if it is Martin-Löf random relative to ∅. We show that a set is 2-random if and only if there is a constant c such that infinitely many initial segments x of the set are c-incompressible: C ≥ |x|-c. The ‘only if' direction was obtained independently by Joseph Miller. This characterization can be extended to the case of time-bounded C-complexity. Next we prove some results (...) on lowness. Among other things, we characterize the 2-random sets as those 1-random sets that are low for Chaitin's Ω. Also, 2-random sets form minimal pairs with 2-generic sets. The r.e. low for Ω sets coincide with the r.e. K-trivial ones. Finally we show that the notions of Martin-Löf randomness, recursive randomness, and Schnorr randomness can be separated in every high degree while the same notions coincide in every non-high degree. We make some remarks about hyperimmune-free and PA-complete degrees. (shrink)

We consider the question of randomness of the probability ΩU[X] that an optimal Turing machine U halts and outputs a string in a fixed set X. The main results are as follows: ΩU[X] is random whenever X is $\Sigma _{n}^{0}$-complete or $\Pi _{n}^{0}$-complete for some n ≥ 2. However, for n ≥ 2, ΩU[X] is not n-random when X is $\Sigma _{n}^{0}$ or $\Pi _{n}^{0}$ Nevertheless, there exists $\Delta _{n+1}^{0}$ sets such that ΩU[X] is n-random. There are $\Delta _{2}^{0}$ sets (...) X such that ΩU[X] is rational. Also, for every n ≥ 1, there exists a set X which is $\Delta _{n+1}^{0}$ and $\Sigma _{n}^{0}$-hard such that ΩU[X] is not random. We also look at the range of ΩU as an operator. We prove that the set {ΩU[X]: X ⊆ 2<ω} is a finite union of closed intervals. It follows that for any optimal machine U and any sufficiently small real r, there is a set X ⊆ 2<ω recursive in ∅′ ⊕ r, such that ΩU[X] = r. The same questions are also considered in the context of infinite computations, and lead to similar results. (shrink)