Developing an analogue of Solovay reducibility in the higher recursion setting, we show that results from the classical computably enumerable case can be extended to the new context.

Unlike Martin‐Löf randomness and Schnorr randomness, computable randomness has not been defined, except for a few ad hoc cases, outside of Cantor space. This paper offers such a definition (actually, several equivalent definitions), and further, provides a general method for abstracting “bit‐wise” definitions of randomness from Cantor space to arbitrary computable probability spaces. This same method is also applied to give machine characterizations of computable and Schnorr randomness for computable probability spaces, extending the previously known results. The paper contains a (...) new type of randomness—endomorphism randomness—which the author hopes will shed light on the open question of whether Kolmogorov‐Loveland randomness is equivalent to Martin‐Löf randomness. The last section contains ideas for future research. (shrink)

We show that a computable function $f:\mathbb R\rightarrow \mathbb R$ has Luzin’s property if and only if it reflects $\Pi ^1_1$ -randomness, if and only if it reflects $\Delta ^1_1$ -randomness, and if and only if it reflects ${\mathcal {O}}$ -Kurtz randomness, but reflecting Martin–Löf randomness or weak-2-randomness does not suffice. Here a function f is said to reflect a randomness notion R if whenever $f$ is R-random, then x is R-random as well. If additionally f is known to have (...) bounded variation, then we show f has Luzin’s if and only if it reflects weak-2-randomness, and if and only if it reflects $\emptyset '$ -Kurtz randomness. This links classical real analysis with algorithmic randomness. (shrink)

We show that a computable function $f:\mathbb R\rightarrow \mathbb R$ has Luzin’s property if and only if it reflects $\Pi ^1_1$ -randomness, if and only if it reflects $\Delta ^1_1$ -randomness, and if and only if it reflects ${\mathcal {O}}$ -Kurtz randomness, but reflecting Martin–Löf randomness or weak-2-randomness does not suffice. Here a function f is said to reflect a randomness notion R if whenever $f$ is R-random, then x is R-random as well. If additionally f is known to have (...) bounded variation, then we show f has Luzin’s if and only if it reflects weak-2-randomness, and if and only if it reflects $\emptyset '$ -Kurtz randomness. This links classical real analysis with algorithmic randomness. (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)

We prove that degrees that are low for Kurtz randomness cannot be diagonally non-recursive. Together with the work of Stephan and Yu [16], this proves that they coincide with the hyperimmune-free non-DNR degrees, which are also exactly the degrees that are low for weak 1-genericity. We also consider Low(M, Kurtz), the class of degrees a such that every element of M is a-Kurtz random. These are characterised when M is the class of Martin-Löf random, computably random, or Schnorr random reals. (...) We show that Low(ML, Kurtz) coincides with the non-DNR degrees, while both Low(CR, Kurtz) and Low(Schnorr, Kurtz) are exactly the non-high, non-DNR degrees. (shrink)

We introduce agreement reducibility and highlight its major features. Given subsets A and B of, we write if there is a total computable function satisfying for all,.We shall discuss the central role plays in this reducibility and its connection to strong‐hyper‐hyper‐immunity. We shall also compare agreement reducibility to other well‐known reducibilities, in particular s1‐ and s‐reducibility. We came upon this reducibility while studying the computable reducibility of a class of equivalence relations on based on set‐agreement. We end by describing the (...) origin of agreement reducibility and presenting some results in that context. (shrink)

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 classify the asymptotic densities of the sets according to their level in the Ershov hierarchy. In particular, it is shown that for, a real is the density of an n‐c.e. set if and only if it is a difference of left‐ reals. Further, we show that the densities of the ω‐c.e. sets coincide with the densities of the sets, and there are ω‐c.e. sets whose density is not the density of an n‐c.e. set for any.