Chaitin's halting probability and the compression of strings using oracles
George Barmpalias, Andrew E.M. Lewis
Abstract:
If a computer is given access to an oracle—the characteristic
function of a set whose membership relation may or may not be
algorithmically calculable—this may dramatically aﬀect its ability to
compress information and to determine structure in strings which might
otherwise appear random. This leads to the basic question, "given an
oracle A, how many oracles can compress information at most as well as
A?"
This question can be formalized using Kolmogorov complexity. We say
that B ≤_{LK} A if there exists a constant c such that K^A(σ) <
K^B(σ) + c for all strings σ, where K^X denotes the preﬁx-free
Kolmogorov complexity relative to oracle X. The formal counterpart to
the previous question now is, "what is the cardinality of the set of
≤_{LK}-predecessors of A?"
We completely determine the number of oracles that compress at most as
well as any given oracle A, by answering a question of Miller [Mil10,
Section 5] which also appears in Nies [Nie09, Problem 8.1.13]; the
class of ≤_{LK}-predecessors of a set A is countable if and only if
Chaitin’s halting probability Ω is Martin-Löf random relative to A.