We obtain a large class of significant examples of n-random reals (i.e., Martin-Löf random in oracle ∅(n-1)) à la Chaitin. Any such real is defined as the probability that a universal monotone Turing machine performing possibly infinite computations on infinite (resp. finite large enough, resp. finite self-delimited) inputs produces an output in a given set 𝒪⊆ 𝔓(ℕ). In particular, we develop methods to transfer Σ0n or Π0n many-one completeness results of index sets to n-randomness of associated probabilities.
"From index sets to randomness in ∅n: random reals and possibly infinite computations. Part II." J. Symbolic Logic 74 (1) 124 - 156, March 2009. https://doi.org/10.2178/jsl/1231082305