Normal numbers and sources for BPP
Theoretical Computer Science
A Theory of Program Size Formally Identical to Information Theory
Journal of the ACM (JACM)
An example of a computable absolutely normal number
Theoretical Computer Science
Randomness and Recursive Enumerability
SIAM Journal on Computing
Theoretical Computer Science
Entropy rates and finite-state dimension
Theoretical Computer Science
Visualization 2001 Conference (Acm
Visualization 2001 Conference (Acm
Turing's unpublished algorithm for normal numbers
Theoretical Computer Science
Computability and Randomness
Randomness, computation and mathematics
CiE'12 Proceedings of the 8th Turing Centenary conference on Computability in Europe: how the world computes
Randomness, computation and mathematics
CiE'12 Proceedings of the 8th Turing Centenary conference on Computability in Europe: how the world computes
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In a manuscript entitled "A note on normal numbers" and written presumably in 1938 Alan Turing gave an algorithm that produces real numbers normal to every integer base. This proves, for the first time, the existence of computable normal numbers and it is the best solution to date to Borel's problem on giving examples of normal numbers. Furthermore, Turing's work is pioneering in the theory of randomness that emerged 30 years after. These achievements of Turing are largely unknown because his manuscript remained unpublished until its inclusion in his Collected Works in 1992. The present note highlights Turing's ideas for the construction of normal numbers. Turing's theorems are included with a reconstruction of the original proofs.