Abstract
AbstractWe examine a definition of the mutual information of two reals proposed by Levin in [5]. The mutual information iswhereK(·) is the prefix-free Kolmogorov complexity. A realAis said to have finite self-information ifI (A : A)is finite. We give a construction for a perfect Π10class of reals with this property, which settles some open questions posed by Hirschfeldt and Weber. The construction produces a perfect set of reals withK(σ)≤+KA(σ)+f (σ)for any given Δ20fwith a particularly nice approximation and for a specific choice of f it can also be used to produce a perfect Π10set 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 satisfyK(σ)≤+KA(σ)+f (σ)for allfin a ‘nice’ class of Δ20functions which includes all Δ20orders.
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
