Recursively enumerable reals and (2001) [8 citations — 3 self]
Abstract:
Emails � fcristian�hertling�bmk�wangg�cs.auckland.ac.nz A real � is called recursively enumerable if it can be approximated by an in� creasing � recursive sequence of rationals. The halting probability of a universal self� delimiting Turing machine �Chaitin�s � number � �10� � is a random r.e. real. Solovay�s �25 � ��like reals are also random r.e. reals. Solovay showed that any Chaitin � num� ber is ��like. In this paper we show that the converse implication is true as well � any ��like real in the unit interval is the halting probability of a universal self�delimiting Turing machine. Following Solovay �25 � and Chaitin �11 � we say that an r.e. real � dominates an r.e. real � if from a good approximation of � from below one can compute a good approximation of � from below. We shall study this relation and characterize it in terms of relations between r.e. sets. ��like numbers are the maximal r.e. real numbers with respect to this order � that is � from a good approximation to an ��like real one can compute a good approximation for every r.e. real. This property shows the strength of � for approximation purposes. However � the situation is radically di�erent if one wishes to compute digits of the binary expansion of an r.e. real � one cannot compute with a total recursive function the �rst n digits of the r.e. real 0��K �the characteristic sequence of the halting problem � from the �rst g�n � digits of �� for any total recursive function g. � The �rst and third authors were partially supported by AURC A18�XXXXX�62090�F3414056 � 1996.
