Metka Selak (2016) *Chaitin's constant Omega - from definition to present*. EngD thesis.

## Abstract

Problems, that led to the discovery of Chaitin's constant Ω are presented in this work. The constant Ω is the halting probability of the universal self-delimited Turing machine. Its definition is given. Its interesting features are presented, among which randomness and uncomputability are the most important. The constant Ω brings randomness and uncomputability to different areas of mathematics and computer science. This is a great problem for the traditional approach to problem solving. Three transformations of famous problems from different areas of mathematics, which relate bits of Ω to solutions to these problems, are given. Recent researches connected whit the constant Ω, mainly about computing its exact initial bits, are presented. A few examples of the halting probability in practise are listed. The presentation of Chaitin's constant Ω tries to be as intuitive as possible and still exact, while using only the necessary mathematics.

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