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December 2006 Randomness and halting probabilities
Verónica Becher, Santiago Figueira, Serge Grigorieff, Joseph S. Miller
J. Symbolic Logic 71(4): 1411-1430 (December 2006). DOI: 10.2178/jsl/1164060463

Abstract

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

  • ΩU[X] is random whenever X is Σ⁰n-complete or Π⁰n-complete for some n≥2.

  • However, for n≥2, ΩU[X] is not n-random when X is Σ⁰n or Π⁰n. Nevertheless, there exists Δ⁰n+1 sets such that ΩU[X] is n-random.

  • There are Δ⁰₂ sets X such that ΩU[X] is rational. Also, for every n≥1, there exists a set X which is Δ⁰n+1 and Σ⁰n-hard such that ΩU[X] is not random.

We also look at the range of ΩU as an operator. We prove that the set {ΩU[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 ΩU[X]=r.

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

Citation

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Verónica Becher. Santiago Figueira. Serge Grigorieff. Joseph S. Miller. "Randomness and halting probabilities." J. Symbolic Logic 71 (4) 1411 - 1430, December 2006. https://doi.org/10.2178/jsl/1164060463

Information

Published: December 2006
First available in Project Euclid: 20 November 2006

zbMATH: 1152.03038
MathSciNet: MR2275867
Digital Object Identifier: 10.2178/jsl/1164060463

Rights: Copyright © 2006 Association for Symbolic Logic

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Vol.71 • No. 4 • December 2006
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