Equivalent characterizations of partial randomness for a recursively enumerable real

Kohtaro Tadaki

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arXiv · 0805.2691

Equivalent characterizations of partial randomness for a recursively enumerable real

Kohtaro Tadaki

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Abstract

A real number αis called recursively enumerable if there exists a computable, increasing sequence of rational numbers which converges to α. The randomness of a recursively enumerable real αcan be characterized in various ways using each of the notions; program-size complexity, Martin-Löf test, Chaitin's Ωnumber, the domination and Ω-likeness of α, the universality of a computable, increasing sequence of rational numbers which converges to α, and universal probability. In this paper, we generalize these characterizations of randomness over the notion of partial randomness by parameterizing each of the notions above by a real number T\in(0,1]. We thus present several equivalent characterizations of partial randomness for a recursively enumerable real number.

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Kohtaro Tadaki. 2008-05-17. Equivalent characterizations of partial randomness for a recursively enumerable real. https://arxiv.org/abs/0805.2691

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