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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
Measures and their random reals
 IN PREPARATION
"... We study the randomness properties of reals with respect to arbitrary probability measures on Cantor space. We show that every nonrecursive real is nontrivially random with respect to some measure. The probability measures constructed in the proof may have atoms. If one rules out the existence of ..."
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We study the randomness properties of reals with respect to arbitrary probability measures on Cantor space. We show that every nonrecursive real is nontrivially random with respect to some measure. The probability measures constructed in the proof may have atoms. If one rules out the existence of atoms, i.e. considers only continuous measures, it turns out that every nonhyperarithmetical real is random for a continuous measure. On the other hand, examples of reals not random for a continuous measure can be found throughout the hyperarithmetical Turing degrees.
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,
COMPARING NOTIONS OF RANDOMNESS
"... Abstract. It is an open problem in the area of computable randomness whether KolmogorovLoveland randomness coincides with MartinLöf randomness. Joe Miller and André Nies suggested some variations of KolmogorovLoveland randomness to approach this problem and to provide a partial solution. We show ..."
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Abstract. It is an open problem in the area of computable randomness whether KolmogorovLoveland randomness coincides with MartinLöf randomness. Joe Miller and André Nies suggested some variations of KolmogorovLoveland randomness to approach this problem and to provide a partial solution. We show that their proposed notion of partial permutation randomness is still weaker than MartinLöf randomness. 1.