Computing halting probabilities from other halting probabilities

TL;DR

This paper investigates the computational relationship between different Chaitin's omega numbers, establishing bounds on the redundancy needed for one to compute the other, and showing that minimal redundancy is insufficient in some cases.

Contribution

It provides precise bounds on the redundancy growth rate required to compute one omega number from another, improving previous results and clarifying the limits of such computations.

Findings

Pairs of omega numbers can compute each other with redundancy psilon \, ext{log} \, n for psilon > 1

Redundancy psilon = 1 is insufficient for such computation in general

There exists an omega number that cannot be computed from another with redundancy psilon = 1

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-L\"{o}f. 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. 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 $\epsilon >1$, any pair of omega numbers compute each other with redundancy $\epsilon \log n$. On the other hand, this is not true for $\epsilon=1$. In fact, we show that for each omega number there exists another omega number which is not computable from the first one with redundancy $\log n$. This latter result improves an older result of Frank Stephan.