Randomness and universal machines
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 u...
| Authors: | , , |
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| Format: | article |
| Status: | Published version |
| Publication Date: | 2006 |
| Country: | Argentina |
| Institution: | Universidad Nacional de Buenos Aires. Facultad de Ciencias Exactas y Naturales |
| Repository: | Biblioteca Digital (UBA-FCEN) |
| Language: | English |
| OAI Identifier: | paperaa:paper_0885064X_v22_n6_p738_Figueira |
| Online Access: | http://hdl.handle.net/20.500.12110/paper_0885064X_v22_n6_p738_Figueira |
| Access Level: | Open access |
| Keyword: | Algorithmic randomness Halting probability Kolmogorov complexity Recursion theory Truth-table degrees Universal machines Ω-numbers Algorithms Turing machines Random number generation |
| Summary: | 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. |
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