Optimal asymptotic bounds on the oracle use in computations from Chaitin's Omega

TL;DR

This paper characterizes the optimal asymptotic bounds on the use of Chaitin's Omega in computing c.e. reals and sets, linking these bounds to information content measures and providing a comprehensive theoretical framework.

Contribution

It provides a precise characterization of asymptotic bounds on Omega's use in oracle computations, connecting them to information content measures and Solovay functions.

Findings

h(n)-n is an information content measure iff Omega computes all c.e. reals with use bounded by h

g is an information content measure iff Omega computes all c.e. sets with use bounded by g

Establishes equivalences between bounds and information content measures for Omega's computational use

Abstract

Chaitin's number Omega is the halting probability of a universal prefix-free machine, and although it depends on the underlying enumeration of prefix-free machines, it is always Turing-complete. It can be observed, in fact, that for every computably enumerable (c.e.) real, there exists a Turing functional via which Omega computes it, and such that the number of bits of omega that are needed for the computation of the first n bits of the given number (i.e. the use on argument n) is bounded above by a computable function h(n) = n+o(n). We characterise the asymptotic upper bounds on the use of Chaitin's omega in oracle computations of halting probabilities (i.e. c.e. reals). We show that the following two conditions are equivalent for any computable function h such that h(n)-n is non-decreasing: (1) h(n)-n is an information content measure, (2) for every c.e. real there exists a Turing functional via which omega computes the real with use bounded by h. We also give a similar characterisation with respect to computations of c.e. sets from Omega, by showing that the following are equivalent for any computable non-decreasing function g: (1) g is an information-content measure, (2) for every c.e. set A, Omega computes A with use bounded by g. Further results and some connections with Solovay functions are given.