Optimal asymptotic bounds on the oracle use in computations from Chaitin’s Omega

Chaitin’s number is the halting probability of a universal prefix-free machine, and although it depends on the underlying enumeration of prefix-free machines, it is always Turing-complete. It can be observed, in fact, that for every computably enumerable (c.e.) real �, there exists a Turing functional via which computes �, and such that the number of bits of that are needed for the computation of the first n bits of � (i.e. the use on argument n) is bounded above by a computable function h(n) = n + o (n). We characterise the asymptotic upper bounds on the use of Chaitin’s 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)P