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

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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 ( Notre Dame J. Formal Logic , 2010, 50 , 381–391), which also appears in Nies ( Computability and randomness . Oxford, UK: Oxford University Press, 2009. 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 .