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EXTRACTING INFORMATION IS HARD: A TURING DEGREE OF NONINTEGRAL EFFECTIVE HAUSDORFF DIMENSION
"... Abstract. We construct a ∆0 2 infinite binary sequence with effective Hausdorff dimension 1/2 that does not compute a sequence of higher dimension. Introduced by Lutz, effective Hausdorff dimension can be viewed as a measure of the information density of a sequence. In particular, the dimension of A ..."
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Abstract. We construct a ∆0 2 infinite binary sequence with effective Hausdorff dimension 1/2 that does not compute a sequence of higher dimension. Introduced by Lutz, effective Hausdorff dimension can be viewed as a measure of the information density of a sequence. In particular, the dimension of A ∈ 2ω is the lim inf of the ratio between the information content and length of initial segments of A. Thus the main result demonstrates that it is not always possible to extract information from a partially random source to produce a sequence that has higher information density. 1.
Bounded Randomness ⋆
"... Abstract. We introduce some new variations of the notions of being MartinLöf random where the tests are all clopen sets. We explore how these randomness notions relate to classical randomness notions and to degrees of unsolvability. 1 ..."
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Abstract. We introduce some new variations of the notions of being MartinLöf random where the tests are all clopen sets. We explore how these randomness notions relate to classical randomness notions and to degrees of unsolvability. 1
LOWNESS FOR BOUNDED RANDOMNESS
"... In [3], Brodhead, Downey and Ng introduced some new variations of the notions of being MartinLöf random where the tests are all clopen sets. We explore the lowness notions associated with these randomness notions. While these bounded notions seem far from classical notions with infinite tests like ..."
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In [3], Brodhead, Downey and Ng introduced some new variations of the notions of being MartinLöf random where the tests are all clopen sets. We explore the lowness notions associated with these randomness notions. While these bounded notions seem far from classical notions with infinite tests like MartinLöf and Demuth randomness, the lowness notions associated with bounded randomness turn out to be intertwined with the lowness notions for these two concepts. In fact, in one case, we get a new and likely very useful characterization of Ktriviality