KL-randomness and effective dimension under strong reducibility
Bj{\o}rn Kjos-Hanssen, David J. Webb
TL;DR
This paper establishes the equivalence of Medvedev degrees for Kolmogorov--Loveland and Martin-Löf randomness, and explores an analogue of complex packing dimension linking weak truth-table Medvedev degrees to Turing degrees.
Contribution
It proves the equivalence of Medvedev degrees for different randomness notions and introduces an analogue of complex packing dimension connecting Medvedev and Turing degrees.
Findings
Medvedev degree of Kolmogorov--Loveland randomness equals that of Martin-Löf randomness.
An analogue of complex packing dimension is developed.
Weak truth-table Medvedev degrees form a structure isomorphic to Turing degrees.
Abstract
We show that the (truth-table) Medvedev degree KLR of Kolmogorov--Loveland randomness coincides with that of Martin L\"of randomness, MLR, answering a question of Miyabe. Next, an analogue of complex packing dimension is studied which gives rise to a set of weak truth-table Medvedev degrees isomorphic to the Turing degrees.
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