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A Perfect Set of Reals with Finite Self-Information

Published online by Cambridge University Press:  12 March 2014

Ian Herbert
Ian Herbert*
Affiliation:
Department of Mathematics, University of California, Berkeley 970 Evans Hall, Berkeley, CA 94720-3840, USA, E-mail: iherbert@math.berkeley.edu
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Abstract

We examine a definition of the mutual information of two reals proposed by Levin in [5]. The mutual information is

where K(·) is the prefix-free Kolmogorov complexity. A real A is said to have finite self-information if I (A : A) is finite. We give a construction for a perfect Π10 class of reals with this property, which settles some open questions posed by Hirschfeldt and Weber. The construction produces a perfect set of reals with K(σ) ≤+KA (σ) + f (σ) for any given Δ20f with a particularly nice approximation and for a specific choice of f it can also be used to produce a perfect Π10 set of reals that are low for effective Hausdorff dimension and effective packing dimension. The construction can be further adapted to produce a single perfect set of reals that satisfy K(σ) ≤+KA (σ) + f (σ) for all f in a ‘nice’ class of Δ20 functions which includes all Δ20 orders.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 78 , Issue 4 , December 2013 , pp. 1229 - 1246
Copyright
Copyright © Association for Symbolic Logic 2013

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References

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