The Arithmetical Complexity of Dimension and Randomness

TL;DR

This paper analyzes the arithmetical complexity of classes of sequences defined by their constructive dimensions and randomness properties, revealing their placement in the arithmetical hierarchy.

Contribution

It classifies the arithmetical complexity of dimension and randomness classes using effective hierarchies, providing new insights into their logical complexity.

Findings

DIM^0 is properly Pi^0_2

DIM^alpha for alpha in (0,1] is properly Pi^0_3

Randomness classes are properly Pi^0_3

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) in [0,1]. Let DIM^alpha and DIMstr^alpha be the classes of all sequences of dimension alpha and of strong dimension alpha, respectively. We show that DIM^0 is properly Pi^0_2, and that for all Delta^0_2-computable alpha in (0,1], DIM^alpha is properly Pi^0_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^0_1 level. For all Delta^0_2-computable alpha in [0,1), we show that DIMstr^alpha is properly in the Pi^0_3 level of this hierarchy. We show that DIMstr^1 is properly in the Pi^0_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^0_3.