Exact Approximations of omega Numbers

Cristian S. Calude, Michael J. Dinneen. Exact Approximations of omega Numbers. I. J. Bifurcation and Chaos, 17(6):1937-1954, 2007. [doi]

Abstract

A Chaitin Omega number is the halting probability of a universal Chaitin (self-delimiting Turing) machine. Every Omega number is both computably enumerable (the limit of a computable, increasing, converging sequence of rationals) and random (its binary expansion is an algorithmic random sequence). In particular, every Omega number is strongly non-computable. The aim of this paper is to describe a procedure, which combines Java programming and mathematical proofs, for computing the exact values of the first 64 bits of a Chaitin Omega:

0000001000000100000110001000011010001111110010111011101000010000.

Full description of programs and proofs will be given elsewhere.