Randomness and universal machines

• Autores: Figueira, S.; Stephan, F.; Wu, G.
• Año de publicación: 2006
• Idioma: inglés
• Tipo de recurso: artículo
• Fuente: J. Complexity 2006;22(6):738-751
• Materia: Algorithmic randomness; Halting probability; Kolmogorov complexity; Recursion theory; Truth-table degrees; Universal machines; Ω-numbers; Algorithms; Turing machines; Algorithmic randomness; Halting probability; Kolmogorov complexity; Recursion theory; Truth-table degrees; Universal machines; Ω-numbers; Random number generation
• Institución: Universidad Nacional de Buenos Aires. Facultad de Ciencias Exactas y Naturales

The present work investigates several questions from a recent survey of Miller and Nies related to Chaitin's Ω numbers and their dependence on the underlying universal machine. Furthermore, the notion ΩU [X] = ∑p : U (p) ↓ ∈ X 2- | p | is studied for various sets X and universal machines U. A universal machine U is constructed such that for all x, ΩU [{ x }] = 21 - H (x). For such a universal machine there exists a co-r.e. set X such that ΩU [X] is neither left-r.e. nor Martin-Löf random. Furthermore, one of the open problems of Miller and Nies is answered completely by showing that there is a sequence Un of universal machines such that the truth-table degrees of the ΩUn form an antichain. Finally, it is shown that the members of hyperimmune-free Turing degree of a given Π1 0-class are not low for Ω unless this class contains a recursive set. © 2006 Elsevier Inc. All rights reserved.
Fil:Figueira, S. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina.