Recursively enumerable sets and degrees
Recursively enumerable sets and degrees
An introduction to Kolmogorov complexity and its applications
An introduction to Kolmogorov complexity and its applications
Logical depth and physical complexity
The universal Turing machine (2nd ed.)
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Journal of the ACM (JACM)
Recursively enumerable reals and Chaitin &OHgr; numbers
Theoretical Computer Science
Information, Randomness and Incompleteness
Information, Randomness and Incompleteness
Randomness and Recursive Enumerability
SIAM Journal on Computing
Visualization 2001 Conference (Acm
Visualization 2001 Conference (Acm
Solovay reducibility on d-c.e real numbers
COCOON'05 Proceedings of the 11th annual international conference on Computing and Combinatorics
Random reals à la Chaitin with or without prefix-freeness
Theoretical Computer Science
Universal recursively enumerable sets of strings
Theoretical Computer Science
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The present work investigates several questions from a recent survey of Miller and Nies related to Chaitin's @W numbers and their dependence on the underlying universal machine. Furthermore, the notion @W"U[X]=@?"p":"U"("p")"@7"@?"X2^-^|^p^| is studied for various sets X and universal machines U. A universal machine U is constructed such that for all x, @W"U[{x}]=2^1^-^H^(^x^). For such a universal machine there exists a co-r.e. set X such that @W"U[X] is neither left-r.e. nor Martin-Lof random. Furthermore, one of the open problems of Miller and Nies is answered completely by showing that there is a sequence U"n of universal machines such that the truth-table degrees of the @W"U"""n form an antichain. Finally, it is shown that the members of hyperimmune-free Turing degree of a given @P"1^0-class are not low for @W unless this class contains a recursive set.