Relativizing Chaitin ’ S Halting Probability

@inproceedings{Hirschfeldt2004RelativizingC,
  title={Relativizing Chaitin ’ S Halting Probability},
  author={Denis R. Hirschfeldt and Joseph S. Miller and Andr{\'e} Nies},
  year={2004}
}
As a natural example of a 1-random real, Chaitin proposed the halting probability Ω of a universal prefix-free machine. We can relativize this example by considering a universal prefix-free oracle machine U . Let ΩU be the halting probability of U A; this gives a natural uniform way of producing an A-random real for every A ∈ 2ω . It is this operator which is our primary object of study. We can draw an analogy between the jump operator from computability theory and this Omega operator. But… CONTINUE READING

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