Recursively enumerable sets and degrees
Recursively enumerable sets and degrees
Visualization 2001 Conference (Acm
Visualization 2001 Conference (Acm
Computability and Randomness
Kolmogorov complexity of initial segments of sequences and arithmetical definability
Theoretical Computer Science
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The strongly null sets of reals have been widely studied in the context of set theory of the real line. We introduce an effectivization of strong nullness. A set of reals is said to be effectively strongly null if, for any computable sequence {εn}n∈ω of positive rationals, a sequence of intervals In of diameter εn covers the set. We show that for $\Pi^0_1$ subsets of 2ω effective strong nullness is equivalent to another well studied notion called diminutiveness: the property of not having a computably perfect subset. In addition, we also investigate the Muchnik degrees of effectively strongly null $\Pi^0_1$ subsets of 2ω. Let MLR and DNC be the sets of all Martin-Löf random reals and diagonally noncomputable functions, respectively. We prove that neither the Muchnik degree of MLR nor that of DNC is comparable with the Muchnik degree of a nonempty effectively strongly null $\Pi^0_1$ subsets of 2ω with no computable element.