From index sets to randomness in ∅n: random reals and possibly infinite computations. Part II

Abstract

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 Σ⁰_n or Π⁰_n many-one completeness results of index sets to n-randomness of associated probabilities.