An explicit recursion-theoretic definition of a random sequence or random set of natural numbers was given by Martin-Löf in 1966. Other approaches leading to the notions of n-randomness and weak n-randomness have been presented by Solovay, Chaitin, and Kurtz. We investigate the properties of n-random and weakly n-random sequences with an emphasis on the structure of their Turing degrees. After an introduction and summary, in Chapter II we present several equivalent definitions of n-randomness and weak n-randomness including a new definition in terms of a forcing relation analogous to the characterization of n-generic sequences in terms of Cohen forcing. We also prove that, as conjectured by Kurtz, weak nrandomness is indeed strictly weaker than n-randomness. Chapter III is concerned with intrinsic properties of n-random sequences. The main results are that an (n + 1)-random sequence A satisfies the condition A (n) ≡T A⊕0 (n) (strengthening a result due originally to Sacks) and that n-random sequences satisfy a number of strong independence properties, e.g., if A ⊕ B is n-random then A is n-random relative to B. It follows that any countable distributive lattice can be embedded

We discuss some aspects of algorithmic randomness and state some open problems in this area. The first part is devoted to the question "What is a computably random sequence?" Here we survey some of the approaches to algorithmic randomness and address some questions on these concepts. In the second part we look at the Turing degrees of Martin-Lof random sets. Finally, in the third part we deal with relativized randomness. Here we look at oracles which do not change randomness. 1980 Mathematics Subject Classification. Primary 03D80; Secondary 03D28. 1 Introduction Formalizations of the intuitive notions of computability and randomness are among the major achievements in the foundations of mathematics in the 20th century. It is commonly accepted that various equivalent formal computability notions -- like Turing computability or -recursiveness -- which were introduced in the 1930s and 1940s adequately capture computability in the intuitive sense. This belief is expressed in the w...

Mathematical Logic Quarterly, 53, 2007, pp. 483–492. We examine the concept of almost everywhere domination from the viewpoint of mass problems. Let AED and MLR be the set of reals which are almost everywhere dominating and Martin-Löf random, respectively. Let b1, b2, b3 be the degrees of unsolvability of the mass problems associated with the sets AED, MLR×AED, MLR∩AED respectively. Let Pw be the lattice of degrees of unsolvability of mass problems associated with nonempty Π 0 1 subsets of 2 ω.Let1 and 0 be the top and bottom elements of Pw. We show that inf(b1, 1) andinf(b2, 1) andinf(b3, 1) belongtoPw and that 0 < inf(b1, 1) < inf(b2, 1) < inf(b3, 1) < 1. Under the natural embedding of the recursively enumerable Turing degrees into Pw, weshow that inf(b1, 1) andinf(b3, 1) but not inf(b2, 1) are comparable with some recursively enumerable Turing degrees other than 0 and 0 ′. In order to make this paper more self-contained, we exposit the proofs of some recent

Abstract. It was shown by Cholak, Jockusch, and Slaman that every computable 2-coloring of pairs admits an infinite low2 homogeneous set H. We answer a question of the same authors by showing that H may be chosen to satisfy in addition C ̸≤T H, where C is a given noncomputable set. This is shown by analyzing a new and simplified proof of Seetapun’s cone avoidance theorem for Ramsey’s theorem. We then extend the result to show that every computable 2-coloring of pairs admits a pair of low2 infinite homogeneous sets whose degrees form a minimal pair. 1.

1 How were you initially drawn to the study of computation and randomness? My first contact with the area was in 1996 when I still worked at the University of Chicago. Back then, my main interest was in structures from computability theory, such as the Turing degrees of computably enumerable sets. I analyzed them via coding with first-order formulas. During a visit to New Zealand, Cris Calude in Auckland introduced me to algorithmic information theory, a subject on which he had just finished a book [3]. We wrote a paper [4] showing that a set truth-table above the halting problem is not Martin-Löf random (in fact the proof showed that it is not even weakly random [33, 4.3.9]). I also learned about Solovay reducibility, which is a way to gauge the relative randomness of real numbers with a computably enumerable left cut. These topics, and many more, were studied either in Chaitin’s work [6] or in Solovay’s visionary, but never published, manuscript [35], of which Cris possessed a copy. l In April 2000 I returned to New Zealand. I worked with Rod Downey and Denis Hirschfeldt on the Solovay degrees of real numbers with computably enumerable left cut. We proved that this degree structure is dense, and that the top degree, the degree of Chaitin’s Ω, cannot be split into two lesser degrees [9]. During this visit I learned about K-triviality, a notion formalizing the intuitive idea of a set of natural numbers that is far from random. To understand K-triviality, we first need a bit of background. Sets of natural numbers (simply called sets below) are a main topic of study in computability theory. Sets can be “identified ” with infinite sequences of bits. Given a set A, the bit in position n has value 1 if n is in A, otherwise its value is 0. A string is a finite sequence of bits, such as 11001110110. Let K(x) denote the length of a shortest prefix-free description of a string x (sometimes called the prefix-free Kolmogorov complexity of x even though Kolmogorov didn’t introduce it). We say that K(x) is the prefix-free complexity of x. Chaitin [6] defined a set A ⊆ N to be K-trivial if each initial segment of A has prefix-free complexity no greater than the prefix-free complexity of its length. That is, there is b ∈ N such that, for each n,

We propose a formal definition of Wolfram’s notion of computational process based on iterated transducers together with a weak observer, a model of computation that captures some aspects of physics-like computation. These processes admit a natural classification into decidable, intermediate and complete, where intermediate processes correspond to recursively enumerable sets of intermediate degree in the classical setting. It is shown that a standard finite injury priority argument will not suffice to establish the existence of an intermediate computational process.

Abstract. We show that if a set A is computable from every superlow 1-random set, then A is strongly jump-traceable. Together with a result from [9], this theorem shows that the computably enumerable jump-traceable sets are exactly the computably enumerable sets computable from every superlow 1-random set. We also prove the analogous result for superhighness: a c.e. set is strongly jump-traceable if and only if it is computable from any superhigh random set. Finally, we show that for each cost function c with the limit condition there is a random ∆ 0 2 set Y such that each c.e. set A �T Y obeys c. 1.

Abstract. Demuth tests generalize Martin-Löf tests (Gm)m∈N in that one can exchange the m-th component for a computably bounded number of times. A set Z ⊆ N fails a Demuth test if Z is in infinitely many final versions of the Gm. If we only allow Demuth tests such that Gm ⊇ Gm+1 for each m, we have weak Demuth randomness. We show that a weakly Demuth random set can be high, yet not superhigh. Next, any c.e. set Turing below a Demuth random set is strongly jump-traceable. We also prove a basis theorem for non-empty Π 0 1 classes P. It extends the Jockusch-Soare basis theorem that some member of P is computably dominated. We use the result to show that some weakly 2-random set does not compute a 2-fixed point free function. 1.

We survey results relating the computability and randomness aspects of sets of natural numbers. Each aspect corresponds to several mathematical properties. Properties originally defined in very different ways are shown to coincide. For instance, lowness for ML-randomness is equivalent to K-triviality. We include some interactions of randomness with computable analysis. Mathematics Subject Classification (2010). 03D15, 03D32. Keywords. Algorithmic randomness, lowness property, K-triviality, cost function.

Abstract. Demuth tests generalize Martin-Löf tests (Gm)m∈N in that one can exchange the m-th component a computably bounded number of times. A set Z ⊆ N fails a Demuth test if Z is in infinitely many final versions of the Gm. If we only allow Demuth tests such that Gm ⊇ Gm+1 for each m, we have weak Demuth randomness. We show that a weakly Demuth random set can be high and ∆ 0 2, yet not superhigh. Next, any c.e. set Turing below a Demuth random set is strongly jump-traceable. We also prove a basis theorem for non-empty Π 0 1 classes P. It extends the Jockusch-Soare basis theorem that some member of P is computably dominated. We use the result to show that some weakly 2-random set does not compute a 2-fixed point free function. 1.