An extension of van Lambalgen’s theorem to infinitely many relative 1-random reals. (English) Zbl 1208.03045

Miyabe, Kenshi

Notre Dame J. Formal Logic 51, No. 3, 337-349 (2010).

Summary: Van Lambalgen’s Theorem plays an important role in algorithmic randomness, especially when studying relative randomness. In this paper we extend van Lambalgen’s Theorem by considering the join of infinitely many reals which are random relative to each other. In addition, we study computability of the reals in the range of Omega operators. It is known that \(\Omega^{\varphi'}\) is high. We extend this result showing that \(\Omega^{\varphi^{(n)}}\) is high\(_n\). We also prove that there exists \(A\) such that, for each \(n\), the real \(\Omega^A_M\) is high\(_n\) for some universal Turing machine \(M\) by using the extended van Lambalgen Theorem.

MSC:

03D32	Algorithmic randomness and dimension
03D25	Recursively (computably) enumerable sets and degrees

Keywords:

van Lambalgen’s theorem; martingale; Omega operator; highness; algorithmic randomness; relative randomness