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Lowness Properties and Randomness
 ADVANCES IN MATHEMATICS
"... The set A is low for MartinLof random if each random set is already random relative to A. A is Ktrivial 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 AmbosSpies and Kucera [2 ..."
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The set A is low for MartinLof random if each random set is already random relative to A. A is Ktrivial 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 AmbosSpies and Kucera [2], showing that each low for MartinLof 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
Lowness Properties of Reals and Randomness
 Advances in Mathematics
, 2002
"... We investigate three properties of the set of natural numbers which have been discovered independently by different... ..."
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Cited by 8 (3 self)
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We investigate three properties of the set of natural numbers which have been discovered independently by different...
Relative Randomness via RKReducibility
, 2006
"... Its focus is relative randomness as measured by rKreducibility, a refinement of Turing reducibility defined as follows. An infinite binary sequence A is rKreducible to an infinite binary sequence B, written A ≤rK B, if ∃d ∀n. K(A ↾ nB ↾ n) < d, where K(στ) is the conditional prefixfree descr ..."
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Cited by 2 (0 self)
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Its focus is relative randomness as measured by rKreducibility, a refinement of Turing reducibility defined as follows. An infinite binary sequence A is rKreducible to an infinite binary sequence B, written A ≤rK B, if ∃d ∀n. K(A ↾ nB ↾ n) < d, where K(στ) is the conditional prefixfree 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
Computability and randomness: Five questions
"... 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 ..."
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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 firstorder 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 truthtable above the halting problem is not MartinLöf 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 Ktriviality, a notion formalizing the intuitive idea of a set of natural numbers that is far from random. To understand Ktriviality, 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 prefixfree description of a string x (sometimes called the prefixfree Kolmogorov complexity of x even though Kolmogorov didn’t introduce it). We say that K(x) is the prefixfree complexity of x. Chaitin [6] defined a set A ⊆ N to be Ktrivial if each initial segment of A has prefixfree complexity no greater than the prefixfree complexity of its length. That is, there is b ∈ N such that, for each n,