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Abstract: We prove that any Chaitin # number (i.e., the halting probability of a universal self-delimiting Turing machine) is wtt-complete, but not tt-complete. In this way we obtain a whole class of natural examples of wtt-complete but not tt-complete r.e. sets. The proof is direct and elementary. 1 Introduction Kucera [8] has used Arslanov's completeness criterion 1 to show that all random sets of r.e. T-degree are in fact T-complete. Hence, every Chaitin # number is T-complete. In this paper we... (Update)
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0.5: Chaitin Omega Numbers, Solovay Machines, and Incompleteness - Calude (1999) (Correct)
0.5: Recursively Enumerable Reals and Chaitin Ω Numbers - Calude, Hertling.. (1998) (Correct)
0.4: Randomness Everywhere: Computably Enumerable Reals and.. - Calude (2000) (Correct)
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1.1: Chaitin Numbers and Strong Reducibilities - Calude, Nies (1997) (Correct)
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BibTeX entry: (Update)
@article{ calude97chaitin,
author = "C. S. Calude and A. Nies",
title = "{Chaitin Omega} Numbers and Strong Reducibilities",
journal = "J.UCS: Journal of Universal Computer Science",
volume = "3",
number = "11",
pages = "1162--??",
year = "1997",
url = "citeseer.ist.psu.edu/article/calude97chaitin.html" }
Citations (may not include all citations):161 Recursively Enumerable Sets and Degrees (context) - Soare - 1987
137 Classical Recursion Theory (context) - Odifreddi - 1989
47 Information and Randomness (context) - Calude - 1994
26 Logical depth and physical complexity (context) - Bennett - 1988
14 The Limits of Mathematics - Chaitin - 1997
6 Computational depth and reducibility - Juedes, Lathrop et al. - 1994
6 bit strings with maximum complexity (context) - Chaitin, number - 1993
3 wtt-complete sets are not necessarily tt-complete (context) - Lachlan - 1975
3 Program-size complexity computes the halting problem (context) - Chaitin, Arslanov et al. - 1995
2 Recursively enumerable reals and Chaitin # numbers (context) - Calude, Hertling et al. - 1997
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