Solovay showed that there are noncomputable reals ff such that H(ff _ n) 6 H(1n) + O(1), where H is prefix-free Kolmogorov complexity. Such H-trivial reals are interesting due to the connection between algorithmic complexity and effective randomness. We give a new, easier construction of an H-trivial real. We also analyze various computability-theoretic properties of the H-trivial reals, showing for example that no H-trivial real can compute the halting problem. Therefore, our construction of an H-trivial computably enumerable set is an easy, injury-free construction of an incomplete computably enumerable set. Finally, we relate the H-trivials to other classes of "highly nonrandom " reals that have been previously studied.

A positive answer to a question of M. van Lambalgen and D. Zambella whether there exist nonrecursive sets that are low for the class of random sets is obtained. Here a set A is low for the class RAND of random sets if RAND = RAND A . 1 Introduction The present paper is concerned with the notion of randomness as originally defined by P. Martin-Lof in [8]. A set is Martin-Lof-random, or 1-random for short, if it cannot be approximated in measure by recursive means. These sets have played a central role in the study of algorithmic randomness. One can relativize the definition of randomness to an arbitrary oracle. Relativized randomness has been studied by several authors. The intuitive meaning of "A is 1-random relative to B" is that A is independent of B. A justification for this interpretation is given by M. van Lambalgen [7]. In this introduction we review some of the basic properties of sets which are 1-random and we state the main problem. We work in the Cantor space 1 The fi...

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Four classes of sets have been introduced independently by various researchers: low for K, low for ML-randomness, basis for ML-randomness and K-trivial. They are all equal. This survey serves as an introduction to these coincidence results, obtained in [24] and [10]. The focus is on providing backdoor access to the proofs. 1. Outline of the results All sets will be subsets of N unless otherwise stated. K(x) denotes the prefix free complexity of a string x. A set A is K-trivial if, within a constant, each initial segment of A has minimal prefix free complexity. That is, there is c ∈ N such that ∀n K(A ↾ n) ≤ K(0 n) + c. This class was introduced by Chaitin [5] and further studied by Solovay (unpublished). Note that the particular effective epresentation of a number n by a string (unary here) is irrelevant, since up to a constant K(n) is independent from the representation. A is low for Martin-Löf randomness if each Martin-Löf random set is already Martin-Löf random relative to A. This class was defined in Zambella [28], and studied by Kučera and Terwijn [17]. In this survey we will see that the two classes are equivalent [24]. Further concepts have been introduced: to be a basis for ML-randomness (Kučera [16]), and to be low for K (Muchnik jr, in a seminar at Moscow State, 1999). They will also be eliminated, by showing equivalence with K-triviality. All

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,

Two objects are independent if they do not affect each other. Independence is wellunderstood in classical information theory, but less in algorithmic information theory. Working in the framework of algorithmic information theory, the paper proposes two types of independence for arbitrary infinite binary sequences and studies their properties. Our two proposed notions of independence have some of the intuitive properties that one naturally expects. For example, for every sequence x, the set of sequences that are independent (in the weaker of the two senses) with x has measure one. For both notions of independence we investigate to what extent pairs of independent sequences, can be effectively constructed via Turing reductions (from one or more input sequences). In this respect, we prove several impossibility results. For example, it is shown that there is no effective way of producing from an arbitrary sequence with positive constructive Hausdorff dimension two sequences that are independent (even in the weaker type of independence) and have super-logarithmic complexity. Finally, a few conjectures and open questions are discussed.