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

Journal of Symbolic Logic 74 (1):124-156 (2009)
Abstract
We obtain a large class of significant examples of n-random reals (i.e., Martin-Löf random in oracle $\varphi ^{(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 ⊆(ℕ). In particular, we develop methods to transfer $\Sigma _n^0 $ or $\Pi _n^0 $ or many-one completeness results of index sets to n-randomness of associated probabilities
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DOI 10.2178/jsl/1231082305
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Hierarchies in Φ‐Spaces and Applications.Victor L. Selivanov - 2005 - Mathematical Logic Quarterly 51 (1):45-61.

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