We define $\Psi $-autoreducible sets given an autoreduction procedure $\Psi $. Then, we show that for any $\Psi $, a measurable class of $\Psi $-autoreducible sets has measure zero. Using this, we show that classes of cototal, uniformly introenumerable, introenumerable, and hyper-cototal enumeration degrees all have measure zero.

By analyzing the arithmetical complexity of the classes of cototal sets and cototal enumeration degrees, we show that weakly 2-random sets cannot be cototal and weakly 3-random sets cannot be of cototal enumeration degree. Then, we see that this result is optimal by showing that there exists a 1-random cototal set and a 2-random set of cototal enumeration degree. For uniformly introenumerable degrees and introenumerable degrees, we utilize $\Psi $-autoreducibility again to show the optimal result that no weakly 3-random sets can have introenumerable enumeration degree. We also show that no 1-random set can be introenumerable.

Suppose that we have a method which estimates the conditional probabilities of some unknown stochastic source and we use it to guess which of the outcomes will happen. We want to make a correct guess as often as it is possible. What estimators are good for this? In this work, we consider estimators given by a familiar notion of universal coding for stationary ergodic measures, while working in the framework of algorithmic randomness, i.e., we are particularly interested in prediction of Martin-Löf random points. We outline the general theory and exhibit some counterexamples. Completing a result of Ryabko from 2009 we also show that universal probability measure in the sense of universal coding induces a universal predictor in the prequential sense. Surprisingly, this implication holds true provided the universal measure does not ascribe too low conditional probabilities to individual symbols. As an example, we show that the Prediction by Partial Matching (PPM) measure satisfies this requirement with a large reserve.

The $\Omega $ numbers—the halting probabilities of universal prefix-free machines—are known to be exactly the Martin-Löf random left-c.e. reals. We show that one cannot uniformly produce, from a Martin-Löf random left-c.e. real $\alpha $ , a universal prefix-free machine U whose halting probability is $\alpha $ . We also answer a question of Barmpalias and Lewis-Pye by showing that given a left-c.e. real $\alpha $ , one cannot uniformly produce a left-c.e. real $\beta $ such that $\alpha - \beta $ is neither left-c.e. nor right-c.e.

Recently, a connection has been established between two branches of computability theory, namely between algorithmic randomness and algorithmic learning theory. Learning-theoretical characterizations of several notions of randomness were discovered. We study such characterizations based on the asymptotic density of positive answers. In particular, this note provides a new learning-theoretic definition of weak 2-randomness, solving the problem posed by (Zaffora Blando, Rev. Symb. Log. 2019). The note also highlights the close connection between these characterizations and the problem of convergence on random sequences.

The aim of this paper is to shed light on our understanding of large scale properties of infinite strings. We say that one string $\alpha $ has weaker large scale geometry than that of $\beta $ if there is color preserving bi-Lipschitz map from $\alpha $ into $\beta $ with small distortion. This definition allows us to define a partially ordered set of large scale geometries on the classes of all infinite strings. This partial order compares large scale geometries of infinite strings. As such, it presents an algebraic tool for classification of global patterns. We study properties of this partial order. We prove, for instance, that this partial order has a greatest element and also possess infinite chains and antichains. We also investigate the sets of large scale geometries of strings accepted by finite state machines such as Büchi automata. We provide an algorithm that describes large scale geometries of strings accepted by Büchi automata. This connects the work with the complexity theory. We also prove that the quasi-isometry problem is a $\Sigma _2^0$ -complete set, thus providing a bridge with computability theory. Finally, we build algebraic structures that are invariants of large scale geometries. We invoke asymptotic cones, a key concept in geometric group theory, defined via model-theoretic notion of ultra-product. Partly, we study asymptotic cones of algorithmically random strings, thus connecting the topic with algorithmic randomness.

Numerous learning tasks can be described as the process of extrapolating patterns from observed data. One of the driving intuitions behind the theory of algorithmic randomness is that randomness amounts to the absence of any effectively detectable patterns: it is thus natural to regard randomness as antithetical to inductive learning. Osherson and Weinstein [11] draw upon the identification of randomness with unlearnability to introduce a learning-theoretic framework (in the spirit of formal learning theory) for modelling algorithmic randomness. They define two success criteria—specifying under what conditions a pattern may be said to have been detected by a computable learning function—and prove that the collections of data sequences on which these criteria cannot be satisfied correspond to the set of weak 1-randoms and the set of weak 2-randoms, respectively. This learning-theoretic approach affords an intuitive perspective on algorithmic randomness, and it invites the question of whether restricting attention to learning-theoretic success criteria comes at an expressivity cost. In other words, is the framework expressive enough to capture most core algorithmic randomness notions and, in particular, Martin-Löf randomness—arguably, the most prominent algorithmic randomness notion in the literature? In this article, we answer the latter question in the affirmative by providing a learning-theoretic characterisation of Martin-Löf randomness. We then show that Schnorr randomness, another central algorithmic randomness notion, also admits a learning-theoretic characterisation in this setting.

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.

We investigate the strength of a randomness notion ${\cal R}$ as a set-existence principle in second-order arithmetic: for each Z there is an X that is ${\cal R}$-random relative to Z. We show that the equivalence between 2-randomness and being infinitely often C-incompressible is provable in $RC{A_0}$. We verify that $RC{A_0}$ proves the basic implications among randomness notions: 2-random $\Rightarrow$ weakly 2-random $\Rightarrow$ Martin-Löf random $\Rightarrow$ computably random $\Rightarrow$ Schnorr random. Also, over $RC{A_0}$ the existence of computable randoms is equivalent to the existence of Schnorr randoms. We show that the existence of balanced randoms is equivalent to the existence of Martin-Löf randoms, and we describe a sense in which this result is nearly optimal.

We study genericity and randomness with respect to ITTMs, continuing the work initiated by Carl and Schlicht. To do so, we develop a framework to study randomness in the constructible hierarchy. We then answer several of Carl and Schlicht’s question. We also ask a new question one the equality of two classes of randoms. Although the natural intuition would dictate that the two classes are distinct, we show that things are not as simple as they seem. In particular we show that the categorical analogues of these two classes coincide, in contradiction with the natural intuition. Even though we are not able to answer the question for randomness in this article, we delineate and sharpen its contour and outline.

In this paper, we study the power and limitations of computing effectively generic sequences using effectively random oracles. Previously, it was known that every 2-random sequence computes a 1-generic sequence (as shown by Kautz) and every 2-random sequence forms a minimal pair in the Turing degrees with every 2-generic sequence (as shown by Nies, Stephan, and Terwijn). We strengthen these results by showing that every Demuth random sequence computes a 1-generic sequence and that every Demuth random sequence forms a minimal pair with every pb-generic sequence (where pb-genericity is an effective notion of genericity that is strictly between 1-genericity and 2-genericity). Moreover, we prove that for every comeager ${\cal G} \subseteq {2^\omega }$, there is some weakly 2-random sequence X that computes some $Y \in {\cal G}$, a result that allows us to provide a fairly complete classification as to how various notions of effective randomness interact in the Turing degrees with various notions of effective genericity.

We study randomness beyond ${\rm{\Pi }}_1^1$-randomness and its Martin-Löf type variant, which was introduced in [16] and further studied in [3]. Here we focus on a class strictly between ${\rm{\Pi }}_1^1$ and ${\rm{\Sigma }}_2^1$ that is given by the infinite time Turing machines (ITTMs) introduced by Hamkins and Kidder. The main results show that the randomness notions associated with this class have several desirable properties, which resemble those of classical random notions such as Martin-Löf randomness and randomness notions defined via effective descriptive set theory such as ${\rm{\Pi }}_1^1$-randomness. For instance, mutual randoms do not share information and a version of van Lambalgen’s theorem holds.

Towards these results, we prove the following analogue to a theorem of Sacks. If a real is infinite time Turing computable relative to all reals in some given set of reals with positive Lebesgue measure, then it is already infinite time Turing computable. As a technical tool towards this result, we prove facts of independent interest about random forcing over increasing unions of admissible sets, which allow efficient proofs of some classical results about hyperarithmetic sets.