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Randomness and Computability: Open Questions

Published online by Cambridge University Press:  15 January 2014

Joseph S. Miller  and
André Nies
Joseph S. Miller
Affiliation:
Department of Mathematics, University of Connecticut, U-3009, 196 Auditorium Road, Storrs, CT 06269-3009, USAE-mail: joseph.miller@math.uconn.edu
André Nies
Affiliation:
Department of Computer Science, Auckland University, Auckland, New ZealandE-mail: andre@cs.auckland.ac.
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Extract

It is time for a new paper about open questions in the currently very active area of randomness and computability. Ambos-Spies and Kučera presented such a paper in 1999 [1]. All the question in it have been solved, except for one: is KL-randomness different from Martin-Löf randomness? This question is discussed in Section 6.

Not all the questions are necessarily hard—some simply have not been tried seriously. When we think a question is a major one, and therefore likely to be hard, we indicate this by the symbol ▶, the criterion being that it is of considerable interest and has been tried by a number of researchers. Some questions are close contenders here; these are marked by ▷. With few exceptions, the questions are precise. They mostly have a yes/no answer. However, there are often more general questions of an intuitive or even philosophical nature behind. We give an outline, indicating the more general questions.

All sets will be sets of natural numbers, unless otherwise stated. These sets are identified with infinite strings over {0, 1}. Other terms used in the literature are sequence and real.

Type
Articles
Information
Bulletin of Symbolic Logic , Volume 12 , Issue 3 , September 2006 , pp. 390 - 410
Copyright
Copyright © Association for Symbolic Logic 2006

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References

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