Universal Recursively Enumerable Sets of Strings

The present work clarifies the relation between domains of universal machines and r.e. prefix-free supersets of such sets. One such characterisation can be obtained in terms of the spectrum function sW (n) mapping n to the number of all strings of length n in the set W. An r.e. prefix-free set W is the superset of the domain of a universal machine iff there are two constants c, d such that sW (n) + sW (n + 1) + . . . + sW (n + c) is between 2n-H(n)-d and 2n-H(n)+d for all n; W is the domain of a universal machine iff there is a constant c such that for each n the pair of n and sW (n) + sW (n + 1) + . . . + sW (n + c) has at least the prefix-free description complexity n. There exists a prefix-free r.e. superset W of a domain of a universal machine which is the not a domain of a universal machine; still, the halting probability W of such a set W is Martin-L

• Type: Conference
• Year: 2008
• Where: DLT
• Authors: Cristian S. Calude, André Nies, Ludwig Staiger, Frank Stephan
• Keywords: DLT 2008, Present Work, Spectrum Function SW, Theoretical Computer Science, Universal Machines