Universal recursively enumerable sets of strings

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 s W (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 s W (n)+s W (n+1)+...+s W (n+c) is between 2 n-H(n)-d and 2 n-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)+s W (n+1)+...+s W (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 Ω W of such a set W is Martin-Löf random. Furthermore, it is investigated to which extend this results can be transferred to plain universal machines. © 2008 Springer-Verlag Berlin Heidelberg.