Universal Recursively Enumerable Sets of Strings

  • Authors:

  • Cristian S. Calude;André Nies;Ludwig Staiger;Frank Stephan

  • Affiliations:

  • Department of Computer Science, The University of Auckland, Auckland, New Zealand;Department of Computer Science, The University of Auckland, Auckland, New Zealand;Institut für Informatik, Martin-Luther-Universität Halle-Wittenberg, Halle, Germany D - 06099;Department of Mathematics and School of Computing, National University of Singapore, Singapore 117543

  • Venue:
  • DLT '08 Proceedings of the 12th international conference on Developments in Language Theory
  • Year:
  • 2008

Quantified Score

Hi-index 0.00

Visualization

Abstract

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 nto the number of all strings of length nin the set W. An r.e. prefix-free set Wis the superset of the domain of a universal machine iff there are two constants c,dsuch that sW(n) + sW(n+ 1) + ... + sW(n+ c) is between 2n茂戮驴 H(n) 茂戮驴 dand 2n茂戮驴 H(n) + dfor all n; Wis the domain of a universal machine iff there is a constant csuch that for each nthe pair of nand 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 Wof a domain of a universal machine which is the not a domain of a universal machine; still, the halting probability 茂戮驴Wof such a set Wis Martin-Löf random. Furthermore, it is investigated to which extend this results can be transferred to plain universal machines.