A Note on the Differences of Computably Enumerable Reals

Abstract

Abstract. We show that given any non-computable left-c.e. real α there exists a left-c.e. real β such that α β + γ for all left-c.e. reals and all right-c.e. reals γ. The proof is non-uniform, the dichotomy being whether the given real α is Martin-Löf random or not. It follows that given any universal machine U, there is another universal machine V such that the halting probability Ω U of U is not a translation of the halting probability Ω V of V by a left-c.e. real. We do not know if there is a uniform proof of this fact.

Some Questions of Uniformity in Algorithmic Randomness

“…reals. As explained in [BLP17], if α is random, then β = α + γ − δ is left-c.e. and random, and α − β is neither left-c.e.…”

“…Theorem 1.5 (Barmpalias and Lewis-Pye [BLP17]). For every universal machine U , there is a universal machine V such that Ω U − Ω V is neither left-c.e.…”

Some Questions of Uniformity in Algorithmic Randomness

“…Theorem 1.5 (Barmpalias and Lewis-Pye) [3]. For every universal machine U, there is a universal machine V such that Ω U -Ω V is neither left-c.e.…”

“…This can happen at most finitely many times until the measure of the domain of M is within r d of the current approximation to , and so the requirement re-enters state waiting. 3 The requirement may then later re-enter state restraining if the approximation to s increases too much faster than the measure of the domain of M, but since the measure of the domain of M will increase by at least 1 2 r d every time R d switches from restraining to waiting, R d can only switch finitely many times.…”

“…reals. As explained in [3], if α is random, then = α + -is left-c.e. and random, and α -is neither left-c.e.…”