The Probability of a Computable Output from a Random Oracle

Abstract

Consider a universal oracle Turing machine that prints a finite or an infinite binary sequence, based on the answers to the binary queries that it makes during the computation. We study the probability that this output is infinite and computable when the machine is given a random (in the probabilistic sense) stream of bits as the answers to its queries during an infinitary computation. Surprisingly, we find that these probabilities are the entire class of real numbers in (0,1) that can be written as the difference of two halting probabilities relative to the halting problem. In particular, there are universal Turing machines that produce a computable infinite output with probability exactly 1/2. Our results contrast a large array of facts (the most well-known being the randomness of Chaitin’s halting probability) that witness maximal initial segment complexity of probabilities associated with universal machines. Our proof uses recent advances in algorithmic randomness.