We characterise the asymptotic upper bounds on the use of Chaitin's ź in oracle computations of halting probabilities (i.e. c.e. reals). We show that the following two conditions are equivalent for any computable function h such that h ( n ) - n is non-decreasing: (1) h ( n ) - n is an information content measure, i.e. the series ź n 2 n - h ( n ) converges, (2) for every c.e. real α there exists a Turing functional via which ź computes α with use bounded by h. We also give a similar characterisation with respect to computations of c.e. sets from ź, by showing that the following are equivalent for any computable non-decreasing function g: (1) g is an information-content measure, (2) for every c.e. set A, ź computes A with use bounded by g. Further results and some connections with Solovay functions (studied by a number of authors 38,3,26,11) are given. Optimal oracle-use of Chaitin's Omega number.Qualification and optimization of the completeness of Chaitin's Omega.Tight bounds on oracle use in the computably enumerable sets and reals.
['George Barmpalias', 'Nan Fang', 'Andrew Lewis-Pye']
Optimal asymptotic bounds on the oracle use in computations from Chaitin's Omega