SOME QUESTIONS OF UNIFORMITY IN ALGORITHMIC RANDOMNESS

Abstract

AbstractThe $\Omega $ numbers—the halting probabilities of universal prefix-free machines—are known to be exactly the Martin-Löf random left-c.e. reals. We show that one cannot uniformly produce, from a Martin-Löf random left-c.e. real $\alpha $ , a universal prefix-free machine U whose halting probability is $\alpha $ . We also answer a question of Barmpalias and Lewis-Pye by showing that given a left-c.e. real $\alpha $ , one cannot uniformly produce a left-c.e. real $\beta $ such that $\alpha - \beta $ is neither left-c.e. nor right-c.e.