Elementary differences between the degrees of unsolvability and degrees of compressibility

Given two infinite binary sequences A, B we say that B can compress at least as well as A if the prefix-free Kolmogorov complexity relative to B of any binary string is at most as much as the prefix-free Kolmogorov complexity relative to A, modulo a constant. This relation, introduced in [Nie05] and denoted by A LK B, is a measure of relative compressing power of oracles, in the same way that Turing reducibility is a measure of relative information. The equivalence classes induced by LK are called LK degrees (or degrees of compressibility) and there is a least degree containing the oracles which can only compress as much as a computable oracle, also called the `low for K' sets. A classic result from [Nie05] states that these coincide with the K-trivial sets, which are the ones having minimal prefix-free Kolmogorov complexity. We show that with respect to LK , given any non-trivial 0 2 sets X, Y there is a computably enumerable set A which is not K-trivial and it is below X, Y . Th...

• Type: Journal
• Year: 2010
• Where: APAL
• Authors: George Barmpalias
• Keywords: APAL 2010, Lk Degrees, Prefix-free Kolmogorov Complexity, Relative Compressing Power