The Arithmetical Complexity of Dimension and Randomness

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) ∈ [0, 1] and a strong dimension Dim(A) ∈ [0, 1]. Let DIMα and DIMα str be the classes of all sequences of dimension α and of strong dimension α, respectively. We show that DIM0 is properly Π0 2, and that for all ∆0 2-computable α ∈ (0, 1], DIMα is properly Π0 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 Σ0 1 level. For all ∆0 2-computable α ∈ [0, 1), we show that DIMα str is properly in the Π0 3 level of this hierarchy. We show that DIM1 str is properly in the Π0 2 level of this hierarchy. We also prove that the class of Schnorr random sequences and the class of computably random sequences are properly Π0 3.

• Type: Conference
• Year: 2003
• Where: CSL
• Authors: John M. Hitchcock, Jack H. Lutz, Sebastiaan Terwijn
• Keywords: Constructive Strong Dimension, CSL 2003, DIMα Str, Strong Dimension