Randomness and halting probabilities

Abstract

We consider the question of randomness of the probability Ω_U[X] that an optimal Turing machine U halts and outputs a string in a fixed set X. The main results are as follows:

• Ω_U[X] is random whenever X is Σ⁰_n-complete or Π⁰_n-complete for some n≥2.

• However, for n≥2, Ω_U[X] is not n-random when X is Σ⁰_n or Π⁰_n. Nevertheless, there exists Δ⁰_n+1 sets such that Ω_U[X] is n-random.

• There are Δ⁰₂ sets X such that Ω_U[X] is rational. Also, for every n≥1, there exists a set X which is Δ⁰_n+1 and Σ⁰_n-hard such that Ω_U[X] is not random.

We also look at the range of Ω_U as an operator. We prove that the set {Ω_U[X] : X⊆2^{< ω}} is a finite union of closed intervals. It follows that for any optimal machine U and any sufficiently small real r, there is a set X⊆ 2^{< ω} recursive in ∅’⊕ r, such that Ω_U[X]=r.

The same questions are also considered in the context of infinite computations, and lead to similar results.