Computing halting probabilities from other halting probabilities

Abstract

The halting probability of a Turing machine is the probability that the machine will halt if it starts with a random stream written on its one-way input tape. When the machine is universal, this probability is referred to as Chaitin's omega number, and is the most well known example of a real which is random in the sense of Martin-Löf. Although omega numbers depend on the underlying universal Turing machine, they are robust in the sense that they all have the same Turing degree, namely the degree of the halting problem. This means that, given two universal prefix-free machines U,V, the halting probability ΩU of U computes the halting probability ΩV of V. If this computation uses at most the first n+g(n) bits of ΩU for the computation of the first n bits of ΩV, we say that ΩU computes ΩV with redundancy g.In this paper we give precise bounds on the redundancy growth rate that is generally required for the computation of an omega number from another omega number. We show that for each ϵ>1, any pair of omega numbers compute each other with redundancy ϵlog⁡n. On the other hand, this is not true for ϵ=1. In fact, we show that for each omega number ΩU there exists another omega number which is not computable from ΩU with redundancy log⁡n. This latter result improves an older result of Frank Stephan.