The set A is low for Martin-Lof random if each random set is already random relative to A. A is K-trivial if the prefix complexity K of each initial segment of A is minimal, namely K(n)+O(1). We show that these classes coincide. This implies answers to questions of Ambos-Spies and Kucera [2], showing that each low for Martin-Lof random set is # 2 . Our class induces a natural intermediate # 3 ideal in the r.e. Turing degrees (which generates the whole class under downward closure). Answering

We investigate three properties of the set of natural numbers which have been discovered independently by different...

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

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,