Random numbers as probabilities of machine behavior

@article{Barmpalias2017RandomNA,
  title={Random numbers as probabilities of machine behavior},
  author={George Barmpalias and Douglas A. Cenzer and Christopher P. Porter},
  journal={ArXiv},
  year={2017},
  volume={abs/1605.05838}
}
The Probability of a Computable Output from a Random Oracle
TLDR
Surprisingly, it is found that these probabilities are the entire class of real numbers in (0,1) that can be written as the difference of two halting probabilities relative to the halting problem.
Randomness and uniform distribution modulo one
Aspects of Chaitin's Omega
TLDR
The purpose of this survey is to expose developments and tell a story about Omega, which outlines its multifaceted mathematical properties and roles in algorithmic randomness.
CHAITIN’S Ω AS A CONTINUOUS FUNCTION
TLDR
It is proved that the continuous function $X:2^\omega \to $ for all $X is differentiable exactly at the Martin-Löf random reals with the derivative having value 0; that it is nowhere monotonic.

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