Enumerable Sets of Strings
نویسندگان
چکیده
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 ΩW 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.
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