@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