Ramdom Reals and Possibly Infinite Computations. Part I: Randomness in ∅′ Verónica Becher and Serge Grigorieff

Using possibly infinite computations on universal monotone Turing machines, we prove Martin-Löf randomness in ∅′ of the probability that the output be in some set O ⊆ 2≤ω under complexity assumptions about O. §1. Randomness in the spirit of Rice’s theorem for computability. Let 2∗ be the set of all finite strings in the binary alphabet 2 = {0, 1}. Let 2ω be the set of all infinite binary sequences. For X ⊆ 2ω the Lebesgue set-theoretic measure of X is denoted by μ(X ). For a particular string s ∈ 2∗, μ(s2ω) = 2−|s|. If X ⊆ 2∗ is a prefix-free set then μ(X2ω) = ∑ s∈X 2 −|s| ≤ 1. As usual (cf.[25] p.451), ∅(n) denotes the n-th jump of ∅, which is a Σn complete set of integers. 1.1. A problem about randomness and finite computations. Randomness will mean Martin-Löf randomness (relative to possible oracles), which is equivalent to the definition of randomness given by the theory prefix-free program-size complexity. In this theory one considers Turing machines with prefix-free domains and a particular notion of universality: U is universal by “prefix adjunction” if for every Turing machine M with prefix-free domain, there is a word e such that, ∀p ∈ 2∗ (M(p) halts ⇔ (U(ep) halts and M(p) = U(ep))) All along the paper, U denotes a machine universal by prefix adjunction. As pointed by Chaitin ([11], p.109), his randomness results do rely on the fact that U is universal by prefix-adjunction. Chaitin [9, 11] introduces, for every subset O of 2∗, the real ΩU [O] = μ(U−1(O)2ω) = ∑