December 2013 A perfect set of reals with finite self-information
Ian Herbert
J. Symbolic Logic 78(4): 1229-1246 (December 2013). DOI: 10.2178/jsl.7804130

Abstract

We examine a definition of the mutual information of two reals proposed by Levin in [5]. The mutual information is I(A:B)=logσ,τ2<ω2K(σ)KA(σ)+K(τ)KB(τ)K(σ,τ), 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 Δ20 f 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.

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Ian Herbert. "A perfect set of reals with finite self-information." J. Symbolic Logic 78 (4) 1229 - 1246, December 2013. https://doi.org/10.2178/jsl.7804130

Information

Published: December 2013
First available in Project Euclid: 5 January 2014

zbMATH: 1350.03033
MathSciNet: MR3156522
Digital Object Identifier: 10.2178/jsl.7804130

Rights: Copyright © 2013 Association for Symbolic Logic

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Vol.78 • No. 4 • December 2013
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