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- W1615665212 abstract "Constructive dimension and constructive strong dimension are effectivizations of the Hausdorff and packing dimensions, respectively. Each infinite binary sequence A is assigned a dimension $dim(A) in [0,1]$ and a strong dimension Dim(A) ∈ [0,1]. Let DIM α and ${rm DIM}_{str}^alpha$ be the classes of all sequences of dimension α and of strong dimension α, respectively. We show that DIM0 is properly $Pi^{rm 0}_{rm 2}$ , and that for all $Delta^{rm 0}_{rm 2}$ -computable α ∈ (0,1], DIM α is properly $Pi^{rm 0}_{rm 3}$ . To classify the strong dimension classes, we use a more powerful effective Borel hierarchy where a co-enumerable predicate is used rather than a enumerable predicate in the definition of the $Sigma^{rm 0}_{rm 1}$ level. For all $Delta^{rm 0}_{rm 2}$ -computable α ∈ [0,1), we show that ${rm DIM}_{str}^alpha$ is properly in the $Pi^{rm 0}_{rm 3}$ level of this hierarchy. We show that ${rm DIM}_{str}^1$ is properly in the $Pi^{rm 0}_{rm 2}$ level of this hierarchy. We also prove that the class of Schnorr random sequences and the class of computably random sequences are properly $Pi^{rm 0}_{rm 3}$ ." @default.
- W1615665212 created "2016-06-24" @default.
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- W1615665212 date "2003-01-01" @default.
- W1615665212 modified "2023-09-23" @default.
- W1615665212 title "The Arithmetical Complexity of Dimension and Randomness" @default.
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- W1615665212 doi "https://doi.org/10.1007/978-3-540-45220-1_21" @default.
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