Differences of halting probabilities

We study the differences of Martin-Löf random left-c.e. reals and show that for each pair of such reals α,β there exists a unique number r>0 such that qα−β is a Martin-Löf random left-c.e. real for each positive rational q>r and a Martin-Löf random right-c.e. real for each positive rational q<r. Based on this result we develop a theory of differences of halting probabilities, which answers a number of questions about Martin-Löf random left-c.e. reals, including one of the few remaining open problems from the list of open questions in algorithmic randomness [21]. The halting probability of a prefix-free machine M restricted to a set X is the probability that the machine halts and outputs an element of X . Becher, Figueira, Grigorieff, and Miller asked whether ΩU(X) is Martin-Löf random when U is universal and X is a View the MathML source set. We apply our theory of differences of halting probabilities to give a positive answer.