The present work investigates several questions from a recent survey of Miller and Nies related to Chaitin’s Ω numbers and their dependence on the underlying universal machine. It is shown that there are universal machines for which ΩU is just x 21−H(x). For such a universal machine there exists a co-r.e. set X such that ΩU[X] = � p:U(p)↓∈X 2−|p | is neither left-r.e. nor Martin-Löf random. Furthermore, one of the open problems of Miller and Nies is answered completely by showing that there is a sequence Un of universal machines such that the truth-table degrees of the ΩUn form an antichain. Finally it is shown that the members of hyperimmunefree Turing degree of a given Π0 1-class are not low for Ω unless this class contains a recursive set. 1

Abstract. We say that A ≤LR B if every B-random number is Arandom. Intuitively this means that if oracle A can identify some patterns on some real γ, oracle B can also find patterns on γ. In other words, B is at least as good as A for this purpose. We study the structure of the LR degrees globally and locally (i.e. restricted to the computably enumerable degrees) and their relationship with the Turing degrees. Among other results we show that whenever α is not GL2 the LR degree of α bounds 2 ℵ0 degrees (so that, in particular, there exist LR degrees with uncountably many predecessors) and we give sample results which demonstrate how various techniques from the theory of the c.e. degrees can be used to prove results about the c.e. LR degrees. 1.

Abstract. We study algebraic properties of cost functions. We give an application: building sets close to being Turing complete. 1.

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is c ∈ N such that ∀n K(A ↾ n) ≤ K(0 n) + c. This class was introduced by Chaitin [5] and further studied by Solovay (unpublished).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

Abstract. Strong jump traceability has been studied by various authors. In this paper we study a variant of strong jump traceability by looking at a partial relativization of traceability. We discover a new subclass H of the c.e. K-trivials with some interesting properties. These sets are computationally very weak, but yet contains a cuppable member. Surprisingly they cannot be constructed using cost functions, and is the first known example of a subclass of the K-trivials which does not contain any promptly simple member. Furthermore there is a single c.e. set which caps every member of H, demonstrating that they are in fact very far away from being promptly simple. 1.

Abstract. We prove that a set is low for weakly 1-generic iff it has neither dnr nor hyperimmune Turing degree. As this notion is more general than being recursively traceable, we refute a recent conjecture on the characterization of these sets. Furthermore, we show that every set which is low for weakly 1-generic is also low for Kurtz-random. 1

Abstract. For every Scott set F and every nonrecursive set X in F, there is a Y ∈ F such that X and Y are Turing incomparable. 1.

Abstract. For every order h such that P n 1/h(n) is finite, every K-trivial degree is h-jump-traceable. This motivated Cholak, Downey and Greenberg [2] to ask whether this traceability property is actually equivalent to K-triviality, thereby giving the hoped for combinatorial characterisation of lowness for Martin-Löf randomness. We show however that the K-trivial degrees are properly contained in those that are h-jump-traceable for every convergent order h. 1.

Research supported by NSF grants DMS-0600823 and DMS-0652637.

This is a dissertation in the field of Mathematics: Logic: Computability Theory: Algorithmic Randomness (Mathematics Subject Classification 03D80, 68Q30). Its focus is relative randomness as measured by rK-reducibility, a refinement of Turing reducibility defined as follows. An infinite binary sequence A is rK-reducible to an infinite binary sequence B, written A ≤rK B, if ∃d ∀n. K(A ↾ n|B ↾ n) < d, where K(σ|τ) is the conditional prefix-free descriptional complexity of σ given τ. Herein i study the relationship between relative randomness and (standard) absolute randomness and that between relative randomness and computable analysis. i Acknowledgements Foremost, i would like to thank my advisor, Steffen Lempp, for all his words of wisdom and encouragement throughout the long years of the Ph.D. Also, thanks to Frank Stephan who worked with me on some of the questions herein at the Computational Prospects of Infinity Workshop in Singapore, July 2005, and to Joseph Miller who read early drafts of my results, suggested questions, and always offered insightful comments. Lastly, thanks to Rod Downey, Denis Hirschfeldt, Robert Owen, Sasha Rubin, and Reed Solomon for the helpful talks we had. ii There are, it seems, two Muses: the Muse of Inspiration, who gives us inarticulate visions and desires, and the Muse of Realization, who returns again and again to say, “It is yet more difficult than you thought.” iii