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COMPUTING KTRIVIAL SETS BY INCOMPLETE RANDOM SETS
"... Abstract. Every Ktrivial set is computable from an incomplete MartinLöf random set, i.e., a MartinLöf random set that does not compute the halting problem. A major objective in algorithmic randomness is to understand how random sets and computably enumerable (c.e.) sets interact within the Turi ..."
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Abstract. Every Ktrivial set is computable from an incomplete MartinLöf random set, i.e., a MartinLöf random set that does not compute the halting problem. A major objective in algorithmic randomness is to understand how random sets and computably enumerable (c.e.) sets interact within the Turing degrees. At some level of randomness all interesting interactions cease. The lower and upper cones of noncomputable c.e. sets are definable null sets, and thus if a set is “sufficiently” random, it cannot compute, nor be computed by, a noncomputable c.e. set. However, the most studied notion of algorithmic randomness, MartinLöf randomness, is not strong enough to support this argument, and in fact, significant interactions between MartinLöf random sets and c.e. sets occur. The study of these interactions has lead to a number of surprising results that show a remarkably robust relationship between MartinLöf random sets and the class of Ktrivial sets. Interestingly, the significant interaction occurs “at the boundaries”: the MartinLöf random sets in question are close to being nonrandom (in that they fail fairly simple statistical