Results 1 
3 of
3
Mass problems and measuretheoretic regularity
, 2009
"... Research supported by NSF grants DMS0600823 and DMS0652637. ..."
Abstract

Cited by 4 (3 self)
 Add to MetaCart
Research supported by NSF grants DMS0600823 and DMS0652637.
Propagation of partial randomness
"... Let f be a computable function from finite sequences of 0’s and 1’s to real numbers. We prove that strong frandomness implies strong frandomness relative to a PAdegree. We also prove: if X is strongly frandom and Turing reducible to Y where Y is MartinLöf random relative to Z, then X is strongl ..."
Abstract
 Add to MetaCart
Let f be a computable function from finite sequences of 0’s and 1’s to real numbers. We prove that strong frandomness implies strong frandomness relative to a PAdegree. We also prove: if X is strongly frandom and Turing reducible to Y where Y is MartinLöf random relative to Z, then X is strongly frandom relative to Z. In addition, we prove analogous propagation results for other notions of partial randomness, including nonKtriviality and autocomplexity. We prove that frandomness relative to a PAdegree implies strong frandomness, but frandomness does not imply frandomness relative to a PAdegree. Keywords: partial randomness, effective Hausdorff dimension, MartinLöf randomness, Kolmogorov complexity, models of arithmetic.
Cone avoidance and randomness preservation
"... Let X be an infinite sequence of 0’s and 1’s. Let f be a computable function. Recall that X is strongly frandom if and only if the a priori Kolmogorov complexity of each finite initial segment τ of X is bounded below by f(τ) minus a constant. We study the problem of finding a PAcomplete Turing orac ..."
Abstract
 Add to MetaCart
Let X be an infinite sequence of 0’s and 1’s. Let f be a computable function. Recall that X is strongly frandom if and only if the a priori Kolmogorov complexity of each finite initial segment τ of X is bounded below by f(τ) minus a constant. We study the problem of finding a PAcomplete Turing oracle which preserves the strong frandomness of X while avoiding a Turing cone. In the context of this problem, we prove that the cones which cannot always be avoided are precisely the Ktrivial ones. We also prove: (1) If f is convex and X is strongly frandom and Y is MartinLöf random relative to X, then X is strongly frandom relative to Y. (2) X is complex relative to some oracle if and only if X is random with respect to some continuous probability measure.