Chaitin's halting probability and the compression of strings using oracles

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 KA(σ)